Pseudo-reductive Groups (New Mathematical Monographs)

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Author: Brian Conrad | Ofer Gabber | Gopal Prasad

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Pseudo-reductive Groups Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect ﬁelds and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classiﬁcation in a usable form. The authors present numerous new results and also give a complete exposition of Tits’ structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, for general smooth connected linear algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book, as it develops tools likely to be used in tackling other problems. Brian Conrad is Professor of Mathematics at Stanford University. Ofer Gabber is Directeur de Recherche CNRS at the Institut des Hautes Études Scientiﬁques (IHÉS). Gopal Prasad is Raoul Bott Professor of Mathematics at the University of Michigan.

NEW MATHEMATICAL MONOGRAPHS Editorial Board Béla Bollobás William Fulton Anatole Katok Frances Kirwan Peter Sarnak Barry Simon Burt Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://www.cambridge.org/uk/series/sSeries.asp?code=NMM

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

M. Cabanes and M. Enguehard Representation Theory of Finite Reductive Groups J. B. Garnett and D. E. Marshall Harmonic Measure P. M. Cohn Free Ideal Rings and Localization in General Rings E. Bombieri and W. Gubler Heights in Diophantine Geometry Y. J. Ionin and M. S. Shrikhande Combinatorics of Symmetric Designs S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures A. Shlapentokh Hilbert’s Tenth Problem G. Michler Theory of Finite Simple Groups A. Baker and G. Wüstholz Logarithmic Forms and Diophantine Geometry P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds B. Bekka, P. de la Harpe and A. Valette Kazhdan’s Property (T) J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory M. Grandis Directed Algebraic Topology G. Michler Theory of Finite Simple Groups II R. Schertz Complex Multiplication S. Bloch Lectures on Algebraic Cycles (Second Edition)

Pseudo-reductive Groups BRIAN CONR AD Stanford University OFER GABBE R Institut des Hautes Études Scientiﬁques GOPAL PRASAD University of Michigan

ca mb ridg e u ni v ersi ty press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521195607 © B. Conrad, O. Gabber and G. Prasad 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library Library of Congress Cataloging-in-Publication data Conrad, Brian David, 1970– Pseudo-reductive groups / Brian Conrad, Ofer Gabber, Gopal Prasad. p. cm. – (New mathematical monographs; 17) Includes bibliographical references and index. 1. Linear algebraic groups. 2. Group theory. I. Gabber, Ofer, 1958– II. Prasad, Gopal. III. Title. IV. Series. QA179.C667 2010 512 .55–dc22 2010022761 ISBN 978-0-521-19560-7 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Jacques Tits, on the occasion of his 80th birthday.

Contents

Introduction Terminology, conventions, and notation Part I

Constructions, examples, and structure theory

page xi xix 1

1 1.1 1.2 1.3 1.4 1.5 1.6

Overview of pseudo-reductivity Comparison with the reductive case Elementary properties of pseudo-reductive groups Preparations for the standard construction The standard construction and examples Main result Weil restriction and ﬁelds of deﬁnition

3 3 11 16 25 34 36

2 2.1 2.2 2.3 2.4

Root groups and root systems Limits associated to 1-parameter subgroups Pseudo-parabolic subgroups Root groups in pseudo-reductive groups Representability of automorphism functors

43 43 59 67 78

3 3.1 3.2 3.3 3.4 3.5

Basic structure theory Perfect normal subgroups of pseudo-reductive groups Root datum for pseudo-reductive groups Unipotent groups associated to semigroups of roots Bruhat decomposition and Levi subgroups Classiﬁcation of pseudo-parabolic subgroups

vii

86 86 94 99 114 130

viii

Contents

Part II

Standard presentations and their applications

147

4 4.1 4.2

Variation of (G , k /k, T , C) Absolutely simple and simply connected ﬁbers Uniqueness of (G , k /k)

149 149 154

5 5.1 5.2 5.3

Ubiquity of the standard construction Main theorem and central extensions Properties of standardness and standard presentations A standardness criterion

162 162 170 179

6 6.1 6.2 6.3

Classiﬁcation results The A1 -case away from characteristic 2 Types A2 and G2 away from characteristic 3 General cases away from characteristics 2 and 3

191 192 198 203

Part III General classiﬁcation and applications

215

7 7.1 7.2 7.3

217 217 228

8 8.1 8.2 9 9.1 9.2 9.3 9.4

The exotic constructions Calculations in characteristics 2 and 3 Basic exotic pseudo-reductive groups Algebraic and arithmetic aspects of basic exotic pseudo-reductive groups Preparations for classiﬁcation in characteristics 2 and 3 Further properties of basic exotic pseudo-reductive groups Exceptional and exotic pseudo-reductive groups The absolutely pseudo-simple groups in characteristic 2 Type A1 Root groups and birational group laws Construction of absolutely pseudo-simple groups with a non-reduced root system Classiﬁcation of absolutely pseudo-simple groups with a non-reduced root system

10 General case 10.1 Factors with non-reduced root system and the generalized standard construction 10.2 Classiﬁcation via generalized standard groups

240 255 255 260 279 280 290 299 318 342 342 351

Contents

11 11.1 11.2 11.3 11.4

Applications Maximal tori in pseudo-reductive groups Pseudo-semisimplicity Unirationality Structure of root groups and pseudo-parabolic subgroups

ix

358 358 364 368 372

Part IV Appendices

389

A A.1 A.2 A.3 A.4

391 392 398 402

A.5 A.6 A.7 A.8

Background in linear algebraic groups Review of deﬁnitions Some results from the general theory Frobenius morphisms and non-afﬁne groups Split reductive groups: Existence, Isomorphism, and Isogeny Theorems Weil restriction generalities Groups without Levi subgroups Lie algebras and Weil restriction Lie algebras and groups of multiplicative type

B B.1 B.2 B.3 B.4

Tits’ work on unipotent groups in nonzero characteristic Subgroups of vector groups Wound unipotent groups The cckp-kernel Torus actions on unipotent groups

473 473 479 482 485

C C.1 C.2 C.3 C.4

Rational conjugacy in connected groups Pseudo-completeness Conjugacy results in the smooth afﬁne case Split unipotent subgroups of pseudo-reductive groups Beyond the smooth afﬁne case

494 494 504 511 519

References Index

407 422 441 446 457

525 527

Introduction

Why go beyond reductive groups? The theory of connected reductive groups over a general ﬁeld, and its applications over arithmetically interesting ﬁelds, constitutes one of the most beautiful topics within pure mathematics. However, it does sometimes happen that one is confronted with linear algebraic groups that are not reductive, and whose structure may be rather mysterious. Example: forms of a variety. Let X be a projective variety over a ﬁeld k. Grothendieck constructed a scheme Aut X /k classifying its automorphisms, but this is hard to understand in general. For example, ﬁnite generation of its component group is unknown, even if k = C. Likewise, little is known about the identity component Aut0X /k . Since H1 (k, Aut X /k ) classiﬁes k-forms of X , there is arithmetic interest in Aut X /k even though our knowledge of its structure is limited. If char(k) = 0 then by a structure theorem of Chevalley for smooth connected groups over perfect ﬁelds, Aut 0X /k is an extension of an abelian variety by a smooth connected afﬁne k-group. If k is imperfect then Chevalley’s theorem does not apply (even if Aut 0X /k is smooth). Nonetheless, some general problems for connected k-group schemes G of ﬁnite type (e.g., G = Aut 0X /k ) reduce to the cases of smooth connected afﬁne k-groups and abelian varieties over k. We do not know any restrictions on the smooth connected afﬁne groups arising in this way when G = Aut 0X /k . Example: local-to-global principle. Let X be a quasi-projective (or arbitrary) scheme over a global ﬁeld k. Suppose that X is equipped with a right action by a linear algebraic k-group H . Choose a point x ∈ X (k). Does H (k) act with only ﬁnitely many orbits on the set of x ∈ X (k) that are H (kv )-conjugate to x for all v away from a ﬁxed ﬁnite set S of places of k?

xi

xii

Introduction

This question reduces to the ﬁniteness of the Tate–Shafarevich sets X 1S (k, G) (of isomorphism classes of right G-torsors over k admitting a kv -point for all v ∈ S) as G varies through the isotropy group schemes Hx of all x ∈ X (k). Such isotropy groups are generally disconnected and non-reductive (and non-smooth if char(k) > 0), even if H is connected and semisimple. Over number ﬁelds k the ﬁniteness of X 1S (k, G) for any linear algebraic k-group G was proved by Borel and Serre [BoSe, Thm. 7.1]. The analogous result over global function ﬁelds reduces to the case of smooth G (see Example C.4.3) and the results in this monograph provide what is needed to settle this case.

Imperfect base ﬁelds Over ﬁelds of characteristic 0, or more generally over perfect ﬁelds (such as ﬁnite ﬁelds), it is usually an elementary matter to reduce problems for general linear algebraic groups to the connected reductive case. The situation is entirely different over imperfect ﬁelds, such as local and global function ﬁelds. The purpose of the theory of pseudo-reductive groups, initiated by Borel and Tits, is to overcome this problem over imperfect ﬁelds. To explain the reason for the difﬁculties over imperfect ﬁelds, consider a smooth connected afﬁne group G over an arbitrary ﬁeld k. Recall that the unipotent radical Ru (Gk ) over an algebraic closure k of k is the unique maximal smooth connected unipotent normal k-subgroup of Gk , and that G is reductive if Ru (Gk ) = 1. If k is perfect then k/k is Galois, so by Galois descent Ru (Gk ) (uniquely) descends to a smooth connected unipotent normal k-subgroup Ru (G) ⊆ G. In particular, for perfect k there is an exact sequence of k-groups 1 → Ru (G) → G → G/Ru (G) → 1

(∗)

with G/Ru (G) a connected reductive k-group. For imperfect k the unipotent radical Ru (Gk ) can fail to descend to a k-subgroup of G (as we will illustrate with many examples), in which case G is not an extension of a connected reductive k-group by a smooth connected unipotent k-group. The k-unipotent radical Ru,k (G) is the maximal smooth connected unipotent normal k-subgroup of G. For an extension ﬁeld k /k, the natural inclusion k ⊗k Ru,k (G) → Ru,k (Gk ) is an equality if k /k is separable but can fail to be an equality otherwise (e.g., for k = k when k is imperfect). Over a global ﬁeld k there are arithmetic problems

Introduction

xiii

(such as the ones mentioned above) that can be reduced to questions about smooth connected afﬁne k-groups H for which the inclusion k ⊗k Ru,k (H ) → Ru (Hk ) may not be an equality when char(k) > 0. A pseudo-reductive group over a ﬁeld k is a smooth connected afﬁne k-group with trivial k-unipotent radical. For any smooth connected afﬁne k-group G, the exact sequence 1 → Ru,k (G) → G → G/Ru,k (G) → 1 uniquely expresses G as an extension of a pseudo-reductive k-group by a smooth connected unipotent k-group. (For perfect k this is (∗).) To use this exact sequence, we need a structure theory for pseudo-reductive groups that is as informative as the one for connected reductive groups.

Structure theory A useful structure theory for pseudo-reductive groups over imperfect ﬁelds cannot be developed by minor variations of the methods used in the reductive case, essentially because pseudo-reductivity is not a geometric property. More speciﬁcally, pseudo-reductivity can be lost under an inseparable extension of the base ﬁeld, such as extension from an imperfect ﬁeld to an algebraically closed ﬁeld. Also, pseudo-reductivity is not inherited by quotients in general, as will be seen in many examples. It seems unlikely that there can be an interesting theory of pseudo-reductive groups over a base ring other than a ﬁeld, in contrast with the theory of connected reductive groups. Pseudo-reductive groups were the topic of courses by J. Tits at the Collège de France in 1991–92 and 1992–93; see [Ti2] and [Ti3]. In these courses, he studied pseudo-reductive groups over any ﬁeld k in terms of pseudo-parabolic k-subgroups and a root datum, and he constructed “non-standard” examples of pseudo-reductive groups over separably closed ﬁelds of characteristics 2 and 3. Most of Tits’ results in his second course were announced by Borel and Tits in [BoTi2] without proofs. Tits’ résumé [Ti3] sketched some of the proofs, and a more detailed exposition was given by Springer in [Spr, Ch. 13–15]. Tits also attempted to classify pseudo-reductive groups over separably closed ﬁelds of nonzero characteristic, but he ran into many complications and so did not write up all of his results. For example, the structure of root groups was shrouded in mystery; their dimension can be rather large. Inspired by [BoTi2], Springer obtained (unpublished) classiﬁcation results in cases with geometric semisimple rank 1, and he made partial progress in characteristic 2.

xiv

Introduction

We discovered a new approach to the structure of pseudo-reductive groups. If k is a ﬁeld with char(k) = 2 then our main result over k is a canonical description of all pseudo-reductive k-groups in terms of two ingredients: connected semisimple groups over ﬁnite extensions of k, and commutative pseudo-reductive k-groups. (These commutative groups turn out to be the Cartan k-subgroups, which are the maximal k-tori in the connected reductive case.) We also get a similar result when char(k) = 2 if k is “almost perfect” in the sense that [k : k 2 ] 2. This includes the most interesting cases for number theory: local and global function ﬁelds over ﬁnite ﬁelds. The case of imperfect ﬁelds of characteristic 2 involves new phenomena. Most of the hardest aspects of characteristic 2 show up in the study of G whose maximal geometric semisimple quotient G ss is SL2 or PGL2 . k

Main results and applications There is a lot of “practical” interest in connected reductive groups (and especially connected semisimple groups), due to applications in the study of interesting problems not originating from the theory of algebraic groups. The reason for interest in pseudo-reductive groups is largely theoretical: they arise naturally in proofs of general theorems. Whenever one is faced with proving a result for arbitrary smooth connected afﬁne groups over an imperfect ﬁeld (e.g., reducing a problem for a non-smooth afﬁne group to an analogous problem for a related abstract smooth afﬁne group by the method of §C.4), pseudo-reductive groups show up almost immediately whereas reductive groups do not. More importantly, experience shows that the structure theory of pseudo-reductive groups often makes it possible to reduce general problems to the semisimple and solvable cases. This is why we believe that pseudo-reductive groups are basic objects of interest in the theory of linear algebraic groups over imperfect ﬁelds. Our main discovery is that, apart from some exceptional situations in characteristics 2 and 3, all pseudo-reductive groups can be canonically constructed in terms of three basic methods: direct products, Weil restriction of connected semisimple groups over ﬁnite extensions of k, and modifying a Cartan subgroup (via a central pushout construction). To explain how this works, ﬁrst consider a ﬁnite extension of ﬁelds k /k and a smooth connected afﬁne k -group G . The Weil restriction G = R k /k (G ) is a smooth connected afﬁne k-group of dimension [k : k] dim G characterized by the property G(A) = G (k ⊗k A) for k-algebras A. When k /k is separable, the functor R k /k has good properties and is very useful. For example, it preserves

Introduction

xv

reductivity and any connected semisimple k-group G that is k-simple and simply connected has the form R k /k (G ) for a canonically associated pair (G , k /k) where k /k is a separable extension and G is absolutely simple and simply connected over k . The functor R k /k exhibits some unusual properties when k /k is not separable: it does not carry tori to tori, nor does it preserve reductivity or perfectness of groups or properness of morphisms in general, and it often fails to preserve surjectivity or ﬁniteness of homomorphisms. However, it does carry connected reductive groups (and even pseudo-reductive groups) to pseudoreductive groups. The most basic examples of pseudo-reductive groups are products i R ki /k (Gi ) for a ﬁnite (non-empty) collection of ﬁnite extensions ki /k and absolutely simple and simply connected semisimple ki -groups Gi ; the simply connectedness of Gi ensures that the k-groups R ki /k (Gi ) are perfect, and such Weil restrictions are non-reductive whenever ki /k is not separable. Cartan k-subgroups in pseudo-reductive groups are always commutative and pseudo-reductive. For example, if k /k is a ﬁnite extension and G is a nontrivial connected reductive k -group given with a maximal k -torus T then R k /k (T ) is a Cartan k-subgroup of R k /k (G ) but it is never a k-torus if k /k is not separable. It seems to be an impossible task to describe general commutative pseudoreductive groups over imperfect ﬁelds, as we will illustrate with numerous examples. (That is, we do not expect there to be any description as convenient as the use of Galois lattices to describe tori.) The representation theory of connected reductive groups rests on the commutative case (tori), so it seems difﬁcult to say anything interesting about the representation theory of general pseudo-reductive groups. The structure of the commutative objects turns out to be the only mystery, in the following sense. We introduce a class of noncommutative pseudo-reductive groups called standard, built as certain central quotients

C

i

R ki /k (Gi ) R ki /k (Ti ) i

with a commutative pseudo-reductive k-group C, absolutely simple and simply connected ki -groups Gi for ﬁnite extensions ki /k, and a maximal ki -torus Ti in Gi for each i. (This also recovers the commutative case by taking the collection {ki } to be empty.) We give a canonical way to choose such data in this “standard” construction, and prove that if char(k) = 2, 3 then every non-commutative pseudo-reductive k-group is standard. In this sense, non-commutative pseudo-reductive k-groups away from characteristics 2 and 3 come equipped with an algebraic invariant

xvi

Introduction

k = ki and can be obtained from semisimple groups over its factor ﬁelds along with an additional commutative group. One of the main ingredients in our proofs is Tits’ theory of root groups, root systems, and pseudo-parabolic subgroups of pseudo-reductive groups; we give a self-contained development because we need scheme-theoretic aspects of the theory. Building on some non-standard examples of Tits, we introduce a “generalized standard” construction in characteristics 2 and 3 as well as an additional exceptional construction in characteristic 2 (with a non-reduced root system). We prove that this gives a classiﬁcation theorem in characteristic 3, as well as in characteristic 2 when [k : k 2 ] 2. (Many of our proofs are characteristic-free, and in characteristic 2 we get quite far before we need to assume [k : k 2 ] 2.) We hope to revisit the general classiﬁcation problem in characteristic 2 in the near future. Our description of pseudo-reductive groups in terms of connected semisimple groups and commutative pseudo-reductive groups isolates the mysteries of the commutative case in a useful manner. For example, we discuss automorphisms of pseudo-reductive groups and we gain new insight into the large dimensions of root groups and the non-completeness of coset spaces G/P for non-reductive pseudo-reductive k-groups G and pseudo-parabolic k-subgroups P that are “not too big”: the root groups are canonically identiﬁed with Weil restrictions of Ga from ﬁnite extensions of k (assuming [k : k 2 ] 2 when char(k) = 2), and the coset spaces G/P are i R ki /k (Gi /Pi ) for semisimple Gi and parabolic ki subgroups Pi for ﬁnite extensions ki /k (apart from some exceptional situations in characteristics 2 and 3). As an indication that the difﬁculties with characteristic 2 are not just a matter of technique, we mention another application of our structure theorem: if char(k) = 2 then a pseudo-reductive k-group is reductive if and only if its Cartan k-subgroups are tori, and this would be false over every imperfect ﬁeld of characteristic 2. In one of the appendices we provide the ﬁrst published account with proofs of Tits’ important work on the structure theory of unipotent groups over arbitrary ﬁelds of nonzero characteristic. This is an essential tool in our work and we expect it to be of general interest to those who work with algebraic groups over imperfect ﬁelds. In another appendix we use our results on root groups (valid in all characteristics, via characteristic-free proofs) to give proofs of theorems of Borel and Tits for arbitrary connected linear algebraic groups G over any ﬁeld k. This includes G(k)-conjugacy of: maximal k-split k-tori, maximal ksplit smooth connected unipotent k-subgroups, and minimal pseudo-parabolic k-subgroups. Another such result is the Bruhat decomposition for G(k) (relative to a minimal pseudo-parabolic k-subgroup of G). In Chapter 3 we prove the

Introduction

xvii

existence of Levi k-subgroups of pseudo-reductive k-groups in the “pseudosplit” case. The most interesting applications of our structure theorem are to be found in number theory, so let us say a few words about this. In [Con2], this structure theorem is the key to proving ﬁniteness results for general linear algebraic groups over local and global function ﬁelds, previously known only under hypotheses related to reductivity or solvability. Among the main results proved in [Con2] are that class numbers, Tate–Shafarevich sets, and Tamagawa numbers of arbitrary linear algebraic groups over global function ﬁelds are ﬁnite. We hope that the reader ﬁnds such arithmetic applications to be as much a source of motivation for learning about pseudo-reductive groups as we did in discovering and proving the results presented in this monograph.

Plan of the monograph We assume the reader has prior familiarity with basics of the theory of connected reductive groups over a ﬁeld (as in [Bo2]), but we do not assume previous familiarity with earlier work on pseudo-reductive groups. We develop everything we need concerning pseudo-reductivity. In Part I we establish some basic results that only require the deﬁnition of pseudo-reductivity and we carry out many fundamental constructions. We begin in Chapter 1 with an introduction to the general theory, some instructive examples, and the construction of “standard” pseudo-reductive groups. In Chapter 2 we deﬁne (following Tits) pseudo-parabolic subgroups and root groups in pseudo-reductive groups over any ﬁeld k (assuming k = ks when it sufﬁces for our needs). In Chapter 3 we give the properties of pseudo-parabolic subgroups and root groups, and deﬁne the root datum associated to a pseudoreductive group. We also establish the Bruhat decomposition and existence of Levi subgroups for pseudo-reductive k-groups when k = ks (to be used in our work in characteristic 2). Part II takes up the ﬁner structure theory of standard pseudo-reductive groups (in arbitrary characteristic). In Chapter 4, we rigidify the choice of data in the “standard” construction. In Chapter 5 we state our main theorem in general and record some useful results on splitting of certain central extensions. We also prove some invariance properties of standardness, including that standardness is insensitive to replacing the base ﬁeld with a separable closure, and we reduce the task of proving standardness for all pseudo-reductive groups (over a ﬁxed separably closed base ﬁeld k) to the special case of pseudo-reductive

xviii

Introduction

k-groups G such that G = D (G) and the maximal geometric semisimple quotient G ss = Gk /R(Gk ) is a simple k-group; such G are called (absolutely) k pseudo-simple. For such G we establish a standardness criterion. This is used in Chapter 6 to prove the general classiﬁcation away from characteristics 2 and 3, building on the rank-1 case that we handle using calculations with SL2 . Part III introduces new ideas needed for small characteristics. In Chapter 7 we construct a class of pseudo-reductive groups (discovered by Tits) called exotic, which only exist over imperfect ﬁelds of characteristics 2 and 3. In Chapter 8 we study the properties of these groups, including their automorphisms and the splitting of certain central extensions. The structure theory in characteristic 2 involves new difﬁculties and requires additional group-theoretic constructions (known to Tits) resting on birational group laws. The absolutely pseudo-simple case is studied in Chapter 9, and the classiﬁcation in characteristics 2 and 3 is completed in Chapter 10 using a “generalized standard” construction. In characteristic 2 we need to assume k is almost perfect (i.e., [k : k 2 ] 2) in order to prove that the constructions we give are exhaustive. (If [k : k 2 ] > 2 then there are more possibilities, and we have not classiﬁed them.) Chapter 11 gives applications of our main results. Appendix A summarizes useful (but largely technical) facts from the theory of linear algebraic groups, with complete proofs or references for proofs. We frequently refer to this appendix in the main text, so the reader may wish to skim some of its results and examples (especially concerning Weil restriction) early in the process of learning about pseudo-reductive groups. Appendix B proves unpublished results of Tits [Ti1] on the structure of smooth connected unipotent groups over arbitrary (especially imperfect) ﬁelds with nonzero characteristic. This is crucial in many proofs of the deeper aspects of the structure theory of pseudo-reductive groups. Appendix C proves many interesting rational conjugacy theorems, and the Bruhat decomposition, announced by Borel and Tits in [BoTi2] for all smooth connected afﬁne groups over arbitrary ﬁelds.

Acknowledgments We are grateful for the encouragement and comments from many people who read earlier versions of this work or saw presentations on it in seminars and workshops, especially Stephen DeBacker, Skip Garibaldi, Johan de Jong, and an anonymous referee. And most of all, we thank Indu Prasad for her patience and support in so many ways throughout our work on this project. The work of the ﬁrst author was supported by a grant from the Alfred P. Sloan Foundation and by NSF grants DMS-0093542 and DMS-0917686. The work of the third author was supported by the Humboldt Foundation, the École Normale Supérieure (Paris), and NSF grant DMS-0653512.

Terminology, conventions, and notation

Algebraic geometry. All rings are understood to be commutative with identity unless we indicate otherwise. For example, if B is such a ring then a B-algebra is understood to be commutative with the “same” multiplicative identity element. For a ﬁeld k, we write ks to denote a separable closure and k to denote an algebraic closure. If X is a scheme over a ring k and R is a k-algebra then XR = X ⊗k R denotes the R-scheme obtained by scalar extension. We use similar notation MR = M ⊗k R for k-modules M . We often work with schemes and maps between them by thinking functorially. In particular, if k is a ring and X is a k-scheme then a point of X often means an element in X (R) for a k-algebra R that we allow to vary. The context should always make clear if we instead mean points of the underlying topological space of a scheme (such as generic points). For a map of schemes f : Y → Y and a locally closed subscheme X → Y we write f −1 (X ) to denote the scheme-theoretic pullback X ×Y Y viewed as a locally closed subscheme of Y . Preimages and intersections of subschemes are always taken in the schemetheoretic sense (and so may be non-reduced) unless we indicate otherwise. The underlying reduced scheme of a scheme X is denoted Xred . Beware that if k is an imperfect ﬁeld and G is an afﬁne k-group scheme of ﬁnite type then Gred can fail to be a k-subgroup scheme of G! Moreover, even when it is a k-subgroup scheme, it may not be k-smooth. See Example A.8.3 for natural examples. (A case with good behavior over any ﬁeld k is that if T is a k-torus and M is a closed k-subgroup scheme of T then Mred is a smooth k-subgroup; see Corollary A.8.2.) An extension of ﬁelds K/k is separable if K ⊗k k is reduced. Note that K/k is not required to be algebraic (such as K = k(x), or K = kv for a global ﬁeld k and place v of k).

xix

xx

Terminology, conventions, and notation

In algebraic geometry over a ﬁeld k, Galois descent is a useful procedure for descending constructions and results from ks down to k. We sometimes need to descend through an inseparable ﬁeld extension. For such purposes, we use faithfully ﬂat descent. The reader is referred to [BLR, 6.1] for an introduction to that theory, covering (more than) everything we need. The reader is referred to [BLR, Ch. 2] for an excellent exposition of the theory of smooth and étale morphisms of schemes. (For example, a map of schemes X → Y is smooth if, Zariski-locally on X , it factors as an étale morphism to an afﬁne space over Y .) We sometimes need to use the functorial criterion for a morphism to be smooth or étale, so we now recall the statement of this criterion in a special case that is sufﬁcient for our needs. If X is a scheme locally of ﬁnite type over a noetherian ring R then X → Spec R is smooth (resp. étale) if and only if X (R ) → X (R /J ) is surjective (resp. bijective) for every R-algebra R and ideal J in R such that J 2 = 0. Algebraic groups. For an integer n 1, a commutative group scheme is ntorsion if it is killed by n. We write C[n] to denote the n-torsion in a commutative group scheme C (with n 1), and μn = GL1 [n]. We also write [n] to denote multiplication by n on a commutative group scheme. For a smooth group G of ﬁnite type over a ﬁeld k, its derived group is D (G) (Deﬁnition A.1.14). We always use the notions of kernel, center, centralizer, central, normalizer, subgroup, normal, and quotient in the scheme-theoretic sense, though sometimes we append the adjective “scheme-theoretic” for emphasis; see Deﬁnitions A.1.8–A.1.11. If S is a scheme (generally afﬁne) and G is an S-group scheme then Aut(G) denotes the functor on S-schemes assigning to any S the group Aut S (GS ) of S -group automorphisms of GS . We likewise deﬁne the functors End(G) when G is commutative and Hom(G, H ) for commutative G and H . If we wish to consider functors respecting additional structure (such as a linear structure on a vector group; see Deﬁnition A.1.1) then we will say so explicitly.

PA RT I Constructions, examples, and structure theory

1 Overview of pseudo-reductivity

1.1 Comparison with the reductive case The notion of pseudo-reductivity is due to Borel and Tits. We begin by deﬁning this concept, as well as some related notions. Deﬁnition 1.1.1 Let k be a ﬁeld, G a smooth afﬁne k-group. The k-unipotent radical Ru,k (G) (resp. k-radical Rk (G)) is the maximal smooth connected unipotent (resp. solvable) normal k-subgroup of G. A pseudo-reductive k-group is a smooth connected afﬁne k-group G such that Ru,k (G) = 1. In the deﬁnition of pseudo-reductivity, it is equivalent to impose the stronger condition that G contains no nontrivial smooth connected k-subgroups U such that Uk ⊆ Ru (Gk ). To prove the equivalence it sufﬁces to show that U ⊆ Ru,k (G), or equivalently that U is contained in a smooth connected unipotent normal k-subgroup. The smooth connected ks -subgroup U in Gks generated by the G(ks )-conjugates of Uks is normal in Gks and satisﬁes U ⊆ Ru (Gk ), k so U is unipotent. By construction, U is Gal(ks /k)-stable, so it descends to a smooth connected unipotent normal k-subgroup of G that contains U , as desired. A consequence of this argument is that if N is a smooth connected normal k-subgroup of a smooth connected afﬁne k-group G, so Ru (Nk ) ⊆ Ru (Gk ), then Ru,k (N ) ⊆ Ru,k (G). In particular, if G is pseudo-reductive then so is N . In particular, every smooth connected k-subgroup of a commutative pseudo-reductive k-group is pseudo-reductive. Pseudo-reductive k-groups are called k-reductive in Springer’s book [Spr]. To deﬁne a related notion of pseudo-semisimplicity, triviality of Rk (G) (assuming connectedness of G) is a necessary but not sufﬁcient condition for the right deﬁnition. The reason is that for any imperfect ﬁeld k, there are smooth connected afﬁne k-groups G such that Rk (G) = 1 and D (G) = G (see Example 11.2.1). We will study pseudo-semisimplicity in §11.2. 3

4

Overview of pseudo-reductivity

Over a perfect ﬁeld k, pseudo-reductivity coincides with reductivity for smooth connected afﬁne k-groups G because Ru,k (G)k = Ru (Gk ) if k is perfect. In contrast, for any imperfect ﬁeld k there are many pseudo-reductive k-groups G that are not reductive. We will see interesting examples in §1.3, but we wish to begin with an elementary commutative example of a non-reductive pseudo-reductive group to convey a feeling for pseudo-reductivity. Our example will rest on the Weil restriction functor R k /k relative to a ﬁnite extension of ﬁelds k /k that is not separable. Weil restriction is a familiar operation in the separable case (where it is analogous to the operation of viewing a complex manifold as a real manifold of twice the dimension), but it is not widely known for general ﬁnite extensions k /k. We will use it extensively with inseparable extensions, so we ﬁrst review a few basic facts about this functor, referring the reader to §A.5 and §A.7 for a more thorough discussion. If B → B is a ﬁnite ﬂat map between noetherian rings (e.g., a ﬁnite extension of ﬁelds) and if X is a quasi-projective B -scheme, then the Weil restriction X = R B /B (X ) is a separated B-scheme of ﬁnite type (even quasi-projective) characterized by the functorial property X (A) = X (B ⊗B A) for all B-algebras A. The discussion of Weil restriction in §A.5 treats the general algebro-geometric setting as well as the special case of group schemes. For any smooth connected afﬁne k -group G , the Weil restriction G = R k /k (G ) is an afﬁne k-group scheme of ﬁnite type characterized by the property G(A) = G (k ⊗k A) functorially in k-algebras A. By Proposition A.5.11, the k-group G = R k /k (G ) is smooth and connected. Its dimension is [k : k] dim G , since Lie(G) is the Lie algebra over k underlying Lie(G ) (Corollary A.7.6). Remark 1.1.2 Beware that if k /k is inseparable then the “pushforward” functor R k /k from afﬁne k -schemes to afﬁne k-schemes has some surprising properties in comparison with the more familiar separable case. This will be illustrated in Examples 1.3.2 and 1.3.5. Example 1.1.3 Now we give our ﬁrst example of a non-reductive pseudoreductive group. Let k be an imperfect ﬁeld of characteristic p > 0, and let k /k be a purely inseparable ﬁnite extension of degree pn > 1. Consider the smooth k-group G = R k /k (GL1 ) of dimension pn . (Loosely speaking, G is “k × viewed as a k-group”.) This canonically contains GL1 as a k-subgroup. The smooth connected quotient G/GL1 of dimension pn − 1 > 0 is killed by n the pn -power map since k p ⊆ k, so it is unipotent and hence G is not reductive. (More explicitly, Gk is the algebraic group of units of k ⊗k k, in which the

1.1 Comparison with the reductive case

5

subgroup of 1-units is a codimension-1 unipotent radical.) However, G(ks ) = (k ⊗k ks )× has no nontrivial p-torsion since k ⊗k ks is a ﬁeld of characteristic p, so the smooth connected commutative unipotent k-subgroup Ru,k (G) must be trivial. That is, G is pseudo-reductive over k. The k-group G naturally occurs (up to conjugacy) inside of GLpn . A non-commutative generalization of this example is given in Example 1.1.11. Remark 1.1.4 Let G be a smooth connected afﬁne group over a ﬁeld k. It is elementary to prove that G/Ru,k (G) is pseudo-reductive over k. In particular, G is canonically an extension of the pseudo-reductive k-group G/Ru,k (G) by the smooth connected unipotent k-group Ru,k (G). To use this extension structure to reduce problems for G to problems for pseudo-reductive k-groups, we need to know something about the structure of smooth connected unipotent k-groups. A general smooth connected unipotent k-group U is hard to describe (especially when char(k) > 0), but any such U admits a characteristic central composition series whose successive quotients are k-forms Vi of vector groups Gani (by [SGA3, XVII, 4.1.1(iii)], or by Corollary B.2.7, Corollary B.3.3, and Theorem B.3.4 when char(k) > 0). We have Vi Gani when k is perfect (by [SGA3, XVII, 4.1.5], or Corollary B.2.7 when char(k) > 0) but Vi is mysterious in general if k is imperfect and U = 1. Nonetheless, such Vi are commutative and p-torsion when char(k) = p > 0, so there are concrete hypersurface models for Vi over inﬁnite k (Proposition B.1.13). This often makes Ru,k (G) tractable enough so that problems for general G can be reduced to the pseudo-reductive case. There are interesting analogies between pseudo-reductive k-groups and connected reductive k-groups. For example, we will prove by elementary methods that the Cartan k-subgroups (i.e., centralizers of maximal k-tori) in pseudo-reductive k-groups are always commutative and pseudo-reductive; see Proposition 1.2.4. We do not know an elementary proof of the related fact that, when char(k) = 2, a pseudo-reductive k-group is reductive if and only if its Cartan k-subgroups are tori. Our proof of this result (Theorem 11.1.1) rests on our main structure theorem for pseudo-reductive groups. It seems unlikely that an alternative proof can be found (bypassing the structure theorem), since the avoidance of characteristic 2 is essential. Indeed, we will show in Example 11.1.2 that over every imperfect ﬁeld k with characteristic 2 there are non-reductive pseudo-reductive k-groups G whose Cartan k-subgroups are tori. In view of the commutativity and pseudo-reductivity of Cartan subgroups in pseudo-reductive groups, a basic reason that the structure of pseudo-reductive groups is more difﬁcult to understand than that of connected reductive groups is that we do not understand the structure of the commutative objects very

6

Overview of pseudo-reductivity

well. For example, whereas k-tori are unirational over k for any ﬁeld k, in Example 11.3.1 we will exhibit commutative pseudo-reductive k-groups that are not unirational over k, where k is any imperfect ﬁeld. Example 1.1.5 The property of pseudo-reductivity over arbitrary imperfect ﬁelds exhibits some behavior that is not at all like the more familiar connected reductive case. We mention two such examples now, the second of which is more interesting than the ﬁrst. (i) Pseudo-reductivity can be destroyed by an inseparable ground ﬁeld extension, such as scalar extension to the perfect closure. This is not a surprise, since by deﬁnition it happens for every non-reductive pseudo-reductive group (of which we shall give many examples). In contrast, by Proposition 1.1.9(1) below, pseudo-reductivity is insensitive to separable extension of the ground ﬁeld. This will often be used when passing to a separably closed ground ﬁeld in a proof. An arithmetically interesting example of a non-algebraic separable extension is kv /k for a global function ﬁeld k and a place v of k. Thus, a group scheme G over a global function ﬁeld k is pseudo-reductive over k if and only if Gkv is pseudo-reductive over kv . (ii) The second, and more surprising, deviation of pseudo-reductivity from reductivity is that it is generally not inherited by quotients. For example, consider a non-reductive pseudo-reductive group G over a ﬁeld k with char(k) = p > 0. By Galois descent, the subgroup Ru (Gk ) in Gk is deﬁned over the perfect closure kp of k. Since kp is perfect, the kp -descent of Ru (Gk ) has a composition series over kp whose successive quotients are kp -isomorphic to −n Ga (Proposition A.1.4). But kp is the direct limit of subﬁelds k p , so if n is sufﬁciently large then Ru (Gk ) descends to a nontrivial unipotent subgroup U −n −n of Gk p−n such that U is k p -split. The pn -power map identiﬁes k p with k −n carrying the inclusion k → k p inside of k over to the pn -power map of k, so Gk p−n is thereby identiﬁed with the target of the n-fold Frobenius isogeny n FG/k,n : G → G (p ) as in Deﬁnition A.3.3. Hence, every non-reductive pseudo-reductive k-group G is a (purely inseparable) isogenous cover of a smooth connected afﬁne k-group G such that the smooth connected unipotent k-group Ru,k (G ) is nontrivial and is a successive extension of copies of Ga over k. Such examples are of limited interest since the structure of ker(G G ) is not easily understood. But there are more interesting examples of pseudo-reductive k-groups G admitting quotients G/H that are not pseudo-reductive. This was illustrated in Example 1.1.3 in the commutative case with H a torus, and we will see examples using (inﬁnitesimal) ﬁnite central multiplicative H (Example 1.3.2 for commutative G and Example 1.3.5 for perfect G) as well as very surprising examples using perfect

1.1 Comparison with the reductive case

7

smooth connected H and perfect G (Example 1.6.4). These examples exist over any imperfect ﬁeld. An important ingredient in our work is the minimal intermediate ﬁeld in k/k over which the geometric unipotent radical Ru (Gk ) inside of Gk is deﬁned. This is a special case of a useful general notion: Deﬁnition 1.1.6 Let X be a scheme over a ﬁeld k, K/k an extension ﬁeld, and Z ⊆ XK a closed subscheme. For an intermediate ﬁeld k /k, Z is deﬁned over k if Z descends (necessarily uniquely) to a closed subscheme of Xk . The (minimal) ﬁeld of deﬁnition (over k) for Z ⊆ XK is the unique such k /k that is contained in all others. (For a thorough discussion of the general existence of such a ﬁeld, we refer the reader to [EGA, IV2 , §4.8], especially [EGA, IV2 , 4.8.11].) Sometimes for emphasis we append the word “minimal” when speaking of ﬁelds of deﬁnition, but in fact minimality is always implicit. Remark 1.1.7 The mechanism underlying the existence of (minimal) ﬁelds of deﬁnition of closed subschemes is a fact from linear algebra, as follows. Consider an extension of ﬁelds K/k, a k-vector space V , and a K-subspace W ⊆ VK . Among all subﬁelds F ⊆ K over k such that W arises by scalar extension from an F-subspace of VF , we claim that there is one such F that is contained in all others. To see that such a minimal ﬁeld exists, choose a k-basis {ei }i∈I of V and a subset B = {ei }i∈J projecting to a K-basis of (K ⊗k V )/W . Then F is generated over k by the coefﬁcients of the vectors {ei mod W }i∈J relative to the K-basis B. We call F the ﬁeld of deﬁnition over k for the K-subspace W . The following lemma records the simple behavior of ﬁelds of deﬁnition under a ground ﬁeld extension. Lemma 1.1.8 Let k be a ﬁeld, L /k /k a tower of extensions, and L an intermediate ﬁeld in L /k. Let X be a k-scheme and Y a closed subscheme of Xk , and F ⊆ k the minimal ﬁeld of deﬁnition over k for Y . The minimal ﬁeld of deﬁnition over L for the closed subscheme YL := Y ⊗k L in Xk ⊗k L = XL ⊗L L is the compositum FL inside of L . In particular, if L = Lk and Y ⊆ Xk has minimal ﬁeld of deﬁnition over k equal to k then the closed subscheme YL ⊆ XL has minimal ﬁeld of deﬁnition over L equal to L . The most useful instance of this lemma for our purposes will be the case when k /k is purely inseparable and L/k is separable (with F = k ) but no

8

Overview of pseudo-reductivity

ambient L is provided. In such cases L ⊗k k is a ﬁeld (as we see by passage to the limit from the case when k /k is ﬁnite) and so we can take L := L ⊗k k . Proof. We ﬁrst reduce to the afﬁne case, using a characterization of the minimal ﬁeld of deﬁnition F/k that is better suited to Zariski-localization: if {Ui } is an open cover of X (e.g., afﬁne opens) then a subﬁeld K of k over k contains F if and only if the closed subscheme Y ∩ (Ui )k ⊆ (Ui )k is deﬁned over K for all i. (This follows from the uniqueness of descent of a closed subscheme to a minimal ﬁeld of deﬁnition.) In other words, F is the compositum of the ﬁelds of deﬁnition over k for each Y ∩ (Ui )k ⊆ (Ui )k . Since the ﬁeld of deﬁnition for YL ⊆ XL over L is the corresponding compositum for the closed subschemes YL ∩ (Ui )L ⊆ (Ui )L , if we can handle each Ui separately then passage to composite ﬁelds gives the global result. Thus, we now may and do assume X is afﬁne, say X = Spec A for a k-algebra A. The ideal of Y in Ak has the form k ⊗F J for an ideal J in AF , and F is the smallest intermediate extension in k /k to which k ⊗F J descends as an ideal. But descent as an ideal is equivalent to descent as a vector subspace (as the property of a subspace of an algebra being an ideal can be checked after an extension of the ground ﬁeld), so F/k is the minimal ﬁeld of deﬁnition over k for the k -subspace V = k ⊗F J inside of Ak . The proof of existence of minimal ﬁelds of deﬁnition for subspaces of a vector space (as in Remark 1.1.7) implies that F is also the minimal ﬁeld of deﬁnition over k for the L -subspace L ⊗k V ⊆ AL . Let K be the minimal ﬁeld of deﬁnition over L for this L -subspace. Our aim is to prove that K = FL. Since the L -subspace L ⊗k V is deﬁned over FL (as it even descends to the F-subspace J in AF ), we have L ⊆ K ⊆ FL. The problem is therefore to prove that F ⊆ K. But L ⊗k V does descend to a K-subspace of AK by deﬁnition of K/L, so by the minimality property for F relative to L /k we get F ⊆ K. 2 Proposition 1.1.9 Let K/k be a separable extension of ﬁelds, and G a smooth connected afﬁne k-group. (1) Inside of GK we have Ru,k (G)K = Ru,K (GK ). In particular, G is pseudoreductive over k if and only if GK is pseudo-reductive over K. (2) Choose a k-embedding k → K of algebraic closures. The ﬁeld of deﬁnition Ek over k for Ru (Gk ) ⊆ Gk is a ﬁnite purely inseparable extension and K ⊗k Ek = EK inside of K. The analogue of Proposition 1.1.9(1) for the maximal k-split smooth connected unipotent normal k-subgroup Rus,k (G) lies deeper, and is given in Corollary B.3.5.

1.1 Comparison with the reductive case

9

Proof. First we prove (1). By Galois descent, if k /k is a Galois extension then Ru,k (Gk ) descends to a smooth connected unipotent normal k-subgroup of G. Such a descent is contained in Ru,k (G), so this settles the case where K/k is Galois. Applying this to separable closures ks /k and Ks /K (with Ks chosen to contain ks over k), we may assume that k is separably closed. Since Ru,k (G)K ⊆ Ru,K (GK ) in general, to prove the equality as in (1) it sufﬁces to prove an inequality of dimensions in the opposite direction when k = ks . More generally, if U is a smooth connected unipotent normal K-subgroup of GK , say with d = dim U 0, we shall apply a specialization argument to U to construct a smooth connected unipotent normal k-subgroup of G with dimension d when k = ks . By expressing K as a direct limit of its ﬁnitely generated subﬁelds over k, there is such a subﬁeld F for which U descends to an F-subgroup of GF that is necessarily smooth, connected, unipotent, and normal in GF . Thus, upon renaming F as K we may assume that K/k is ﬁnitely generated. Separability of K/k then allows us to write K = Frac(A) for a k-smooth domain A. The normality of U in G can be expressed as the fact that the map GK ×U → GK deﬁned by (g, u) → gug −1 factors through U , and the unipotence can be expressed as the fact that for some ﬁnite extension K /K the K -group UK admits a composition series whose successive quotients are Ga . Since K is the direct limit of its k-smooth subalgebras A[1/a] for a ∈ A − {0}, by replacing A with a suitable such A[1/a] we may arrange that U = UK for a closed subscheme U ⊆ GA . Let A be an A-ﬁnite domain such that AK = K . By “spreading out” of properties of U = UK and Spec K = (Spec A )K from the ﬁber over the generic point Spec K of Spec A, upon replacing A with a further localization A[1/a] we can arrange that U is an A-smooth normal A-subgroup of GA with (geometrically) connected ﬁbers of dimension d , and that there is a ﬁnite faithfully ﬂat extension A → A (with generic ﬁber K /K) such that UA admits a composition series by A -smooth normal closed A -subgroups with successive quotients isomorphic to Ga as an A -group. Hence, all ﬁbers of U over Spec A are unipotent. Since A is k-smooth, if k = ks then there exist k-points of Spec A. The ﬁber of U over such a point is a smooth connected unipotent normal k-subgroup of G with dimension d . This proves (1). For (2), we apply Lemma 1.1.8 to the tower K/k/k, the intermediate ﬁeld K, and the k-scheme X = G and k-scheme Y = Ru (Gk ) (for which YK = Ru (GK )). The assertion of the Lemma in this case is that the ﬁeld of deﬁnition EK over K for Ru (GK ) is the composite ﬁeld Ek K inside of K. Provided that Ek /k is purely inseparable, the ring Ek ⊗k K is a ﬁeld (since K/k is separable) and hence the surjective map Ek ⊗k K → Ek K is an isomorphism. It

10

Overview of pseudo-reductivity

therefore remains to prove that Ek /k is a purely inseparable extension of ﬁnite degree. By Galois descent, Ru (Gk ) is deﬁned over the perfect closure kp of k, which is to say that it descends to a kp -subgroup U ⊆ Gkp . Since the ideal of U in the coordinate ring of Gkp is ﬁnitely generated, and every ﬁnite subset of kp is contained in a subextension of ﬁnite degree over k, we can descend U to a k -subgroup of Gk for some ﬁnite purely inseparable extension k /k. Necessarily Ek ⊆ k , so we are done. 2 To make interesting non-commutative pseudo-reductive groups, we need to study the Weil restriction of scalars functor R k /k applied to connected reductive groups over ﬁnite extension ﬁelds k /k. For later purposes, it is convenient to work more generally with nonzero ﬁnite reduced k-algebras k , which is to say k = ki for a non-empty ﬁnite collection {ki } of ﬁelds of ﬁnite degree over k. This generality is better-suited to Galois descent (which we use very often): the functor (·)⊗k ks carries nonzero ﬁnite reduced k-algebras to nonzero ﬁnite reduced ks -algebras, but generally does not carry ﬁelds to ﬁelds. For such general k = ki , a quasi-projective k -scheme X is precisely Xi for quasi-projective ki -schemes Xi , and R k /k (X ) = R ki /k (Xi ). Proposition 1.1.10 Let k be a ﬁeld, k a nonzero ﬁnite reduced k-algebra, and G a k -group whose ﬁbers over Spec k are connected reductive (or more generally, pseudo-reductive). The smooth connected afﬁne k-group G = R k /k (G ) is pseudo-reductive. Proof. By Proposition A.5.2(4) and Proposition A.5.9, the afﬁne k-group G is smooth and connected. Let ι : U → G be a smooth connected unipotent normal k-subgroup of G. We must show that ι is the trivial map (i.e., U = 1). It is equivalent to prove triviality of the k -map ι : Uk → G that corresponds to ι via the universal property of Weil restriction. The image H ⊆ G of ι is a smooth unipotent k -subgroup of G with connected ﬁbers over Spec k , and we claim that it is normal. Once this is proved, then reductivity (or even just pseudo-reductivity) of the ﬁbers of G over Spec k implies that ι is trivial, so G is indeed pseudo-reductive over k. To verify the normality of H in G , we ﬁrst observe that (by construction) ι is the restriction to Uk of the canonical map q : Gk → G that corresponds to the identity map of G = R k /k (G ) under the universal property of Weil restriction. Smoothness of G implies that q is surjective (Proposition A.5.11(1)), so normality of Uk in Gk implies that the k -group H = q(Uk ) is normal in G . 2 The k-group G = R k /k (G ) for a k -group G with connected reductive ﬁbers is especially interesting when G has a nontrivial ﬁber over some factor ﬁeld ki

1.2 Elementary properties of pseudo-reductive groups

11

of k that is not separable over k, because then the pseudo-reductive k-group G is not reductive. This will be proved in Example 1.6.1. Here is a concrete instance of this fact. Example 1.1.11 Consider a ﬁnite extension of ﬁelds k /k and a central simple k -algebra A with dimk A = n2 . The k -group G of units of A is a connected reductive k -group (a k -form of GLn ). The k-group G := R k /k (G ) of units of the ﬁnite-dimensional simple (but non-central when k = k) k-algebra underlying A is pseudo-reductive, by Proposition 1.1.10. When k /k is not separable, the unit group G is not reductive over k; this is a special case of Example 1.6.1. Example 1.1.12 Let k /k be a ﬁnite extension of ﬁelds and G a connected reductive k -group. Let k1 /k be the maximal separable subextension in k /k (so k /k1 is purely inseparable), and ks /k a separable closure. Since R k /k = R k1 /k ◦ R k /k1 , the decomposition of k1 ⊗k ks into a ﬁnite product of copies of ks gives rise to the decomposition R k /k (G )ks

(R k /k1 (G ) ⊗k1 ,σ ks ),

(1.1.1)

σ

with σ ranging through the k-embeddings of k1 into ks . This often reduces general problems for R k /k (G ) to the special case of purely inseparable extension ﬁelds.

1.2 Elementary properties of pseudo-reductive groups We now prove some general properties of pseudo-reductive groups that follow from the deﬁnition of pseudo-reductivity (and from general facts in the theory of reductive groups). To begin, we record a lemma that is extremely useful in many arguments with pseudo-reductivity. Lemma 1.2.1 Let G be a pseudo-reductive k-group, and X a closed subscheme of G such that Xk ⊆ Ru (Gk ) and Xk is reduced and irreducible. Then X is a k-rational torsion point in the center of G, trivial if char(k) = 0 and with p-power order if char(k) = p > 0. In particular, if X contains the identity then it is the identity point. Note that the center of G contains only ﬁnitely many k-rational p-power torsion points when char(k) = p > 0. Indeed, the Zariski closure of the group of such points is a smooth unipotent central k-subgroup of G, so its identity component is normal in G and thus is trivial.

12

Overview of pseudo-reductivity

Proof. We may and do assume that k is separably closed, so X (k) is non-empty. To prove the weaker claim that X is a single k-rational point we may translate by x−1 for some x ∈ X (k) to arrange that X (k) contains the identity. Let H ⊆ G be the smooth connected k-subgroup such that H (k) is generated by X (k). (The existence of such an H is [Bo2, §2.1].) Clearly Hk ⊆ Ru (Gk ). Likewise, the collection of G(k)-conjugates of H generates a smooth connected k-subgroup U ⊆ G, and U is normal in G because G(k) is Zariski-dense in G (as k is separably closed). But Xk ⊆ Ru (Gk ), so U ⊆ Ru (Gk ). In particular, U is unipotent. Since G is pseudo-reductive we conclude that U = 1, so X is the identity point as well. In the original situation (prior to translation), we conclude that the geometrically reduced and geometrically irreducible k-scheme X is a single k-rational point. Obviously X must be the identity point if G is reductive, which includes the case of characteristic 0, so it remains to check that if char(k) = p > 0 then := G(k) ∩ Ru (Gk ) is ﬁnite and central with p-power order. The Zariski closure of in G is a smooth k-subgroup of G each of whose connected components is geometrically irreducible (as k = ks ). Thus, each of these components is a single k-rational point, by what we have already shown. It follows that is ﬁnite. But is stable by G(k)-conjugation, so it is normal in G since G is smooth and k is separably closed. By connectivity of G it follows that the ﬁnite constant group is central in G. Since is unipotent, it has p-power order. 2 A useful application of Lemma 1.2.1 is: Proposition 1.2.2 Let f , f : G ⇒ H be a pair of k-homomorphisms between smooth connected afﬁne k-groups, with H pseudo-reductive. Assume fk and f carry Ru (Gk ) into Ru (Hk ), so they induce maps f red , f red : G red ⇒ H red k

k

k

k

k

between maximal reductive quotients over k. If f red = f red then f = f . k k Any k-homomorphism f from a smooth connected afﬁne k-group G to a commutative pseudo-reductive k-group is determined by its restriction to a maximal k-torus of G, and any action by a smooth connected k-group on a commutative pseudo-reductive k-group is trivial. Proof. Consider the k-scheme map φ : G → H deﬁned by g → f (g)f (g)−1 . Over k this lands in Ru (Hk ). For the schematic closure X of the image of φ over k, Xk is the schematic closure of the image of φk . Since 1 ∈ X (k), Lemma 1.2.1 implies that X = 1, so f = f . For a commutative pseudo-reductive group C with maximal k-torus T , the natural map Tk → C red is an isomorphism. Thus, when H is commutative k

1.2 Elementary properties of pseudo-reductive groups

13

(so f factors through G/D (G), and a maximal k-torus in G maps onto one in G/D (G)) then f is indeed determined by its restriction between maximal k-tori. If a smooth connected k-group G acts on a commutative pseudo-reductive k-group C then the action preserves the maximal k-torus T (as we may check at the level of rational points over ks ). Since the automorphism functor Aut(T ) of T is represented by an étale k-group and G is connected, the homomorphism G → Aut(T ) classifying the G -action on T is trivial, so G acts trivially on T . Thus, the action by G (ks ) on Cks is trivial (due to triviality on Tks ), so G acts trivially on C. 2 The following result is an analogue of two well-known properties of connected reductive groups: Proposition 1.2.3 Let G be a pseudo-reductive k-group such that it either has a unique maximal torus over ks or is solvable. The k-group G is commutative. In particular, a non-commutative pseudo-reductive k-group is non-solvable. Proof. First assume that there is a unique maximal torus over ks , so it is deﬁned over k by Galois descent. Call this torus T . The uniqueness of the maximal torus implies that G is nilpotent [Bo2, 11.5(3)], so G is solvable. It remains to prove that a solvable pseudo-reductive k-group G is commutative. The derived group D (G) ⊆ G is a smooth connected k-subgroup of G, and by the structure of smooth connected solvable groups [Bo2, 10.6(1)] this derived group is unipotent. Thus, D (G) = 1 by the pseudo-reductivity of G over k, so G is commutative. 2 In a connected reductive group, the centralizer of a torus is connected reductive and the Cartan subgroups are tori. We have the following useful extension to the pseudo-reductive case. Proposition 1.2.4 If G is pseudo-reductive over k then for any k-torus T in G the centralizer ZG (T ) is a pseudo-reductive k-group. The k-group ZG (T ) is commutative if T is a maximal k-torus in G (i.e., Cartan k-subgroups of G are commutative). Proof. Note that Tk maps isomorphically onto its image in G red (Example k A.1.7). Since G is a smooth connected afﬁne group over a ﬁeld k, the schemetheoretic centralizer ZG (T ) for the conjugation action by the k-torus T is a smooth connected k-group (Proposition A.2.5). Also, the image of ZG (T )k in the maximal reductive quotient G red = Gk /Ru (Gk ) is ZGred (Tk ), due to the k

k

compatibility of torus centralizers with the formation of quotients of smooth connected afﬁne groups (Proposition A.2.8).

14

Overview of pseudo-reductivity

The centralizer ZGred (Tk ) is a connected reductive group [Bo2, 8.18, 13.19]. k

Hence, any smooth connected normal unipotent k-subgroup U ⊆ ZG (T ) satisﬁes Uk ⊆ Ru (Gk ). This forces U = 1, by Lemma 1.2.1. In other words, ZG (T ) is pseudo-reductive over k. Now assume that T is a maximal k-torus. The centrality of T in the pseudoreductive k-group ZG (T ) forces Tks to be the only maximal ks -torus in ZG (T )ks . Thus, by Proposition 1.2.3, ZG (T ) is commutative. 2 As a consequence of the commutativity of Cartan k-subgroups in pseudoreductive k-groups, we get an analogue for pseudo-reductive groups of the fact that the derived group of a connected reductive group is its own derived group. First we need a general lemma concerning tori in pseudo-reductive groups. Lemma 1.2.5 Let M and N be smooth connected k-subgroups of a pseudoreductive k-group G, T a maximal k-torus in G, and Z the maximal k-subtorus of T which commutes with N . (i) If the images of Mk and Nk in G red commute then M and N commute. k (ii) Assume N is normal in G. For the maximal k-torus S := T ∩ D (N ) in D (N ) there is an almost direct product decomposition T = Z · S, and ZG (T ) ∩ N = ZN (S) (so ZG (T ) ∩ N is a Cartan k-subgroup of N ). (iii) The correspondence T → T ∩ D (G) is a bijection between the sets of maximal k-tori of G and D (G). In general, any maximal k-torus T is the almost direct product of T ∩ D (G) and the maximal central k-torus in G. Note that S in (ii) is a maximal k-torus of D (N ) (in particular, smooth and connected) by Corollary A.2.7 since D (N ) is normal in G when N is. Proof. By Galois descent, the formation of Z commutes with extension of the ground ﬁeld to ks . Thus, we may and do assume k = ks . Let G = G red , so the k

quotient map π : Gk G carries Tk isomorphically onto a maximal k-torus T in G . The commutator subgroup (M , N ) is a smooth connected k-subgroup of G, and since M := π(Mk ) commutes with N := π(Nk ) it follows that (M , N )k ⊆ Ru (Gk ). Thus, by Lemma 1.2.1, we have (M , N ) = 1. This proves (i). To prove (ii), now assume N is normal in G, so N is a smooth connected normal subgroup of G . In particular, N is reductive and so T N is a connected reductive subgroup of G . Let Z ⊆ T be the maximal central torus in T N and S := T ∩ D (T N ) = T ∩ D (N ), so S is a maximal torus in D (N ) and T = Z · S as an almost direct product. Since π : Tk → T is an isomorphism and k = ks , there is an evident inclusion-preserving bijective correspondence between subtori of T and k-subtori of T . Let S and Z be the k-subtori of T

1.2 Elementary properties of pseudo-reductive groups

15

corresponding to S and Z respectively. Thus, T = Z · S as an almost direct product. Applying (i) with M = Z then gives that Z commutes with N , and clearly T ∩ D (N ) ⊆ S. But T ∩ D (N ) is a maximal k-torus in D (N ) (Corollary A.2.7), so T ∩ D (N ) = S. Since Z commutes with N and we have T = Z · S, clearly ZG (T ) ∩ N = ZN (S). This proves (ii). Finally, taking N = G (so Z is the maximal central k-torus in G), (iii) is an immediate consequence of (ii). 2 Proposition 1.2.6 Let G be a smooth connected afﬁne group over a ﬁeld k. For any Cartan k-subgroup C, we have G = C · D (G). If G is pseudo-reductive over k then D (G) is perfect (i.e., D (G) = D (D (G))). Proof. We ﬁrst carry out some general considerations with an arbitrary smooth connected afﬁne k-group H and k-torus S in H . If H H is a quotient map of k-groups and S denotes the k-torus image of S in H then ZH (S) maps onto ZH (S ) (Proposition A.2.8). In particular, if H is commutative then ZH (S) maps onto H . Taking H = H /D (H ), we conclude that H = ZH (S) · D (H ). Taking S to be a maximal k-torus gives that a smooth connected afﬁne k-group is generated by its derived group and any single Cartan k-subgroup. Applying this general result to the smooth connected k-group S · D (H ) gives that S · D (H ) = ZS·D (H ) (S) · D (S · D (H )). Now let H = G be pseudo-reductive over k and choose S = T to be a maximal k-torus, so G = ZG (T ) · D (G) = ZG (T ) · (T · D (G)) = ZG (T ) · D (T · D (G)). Let D 2 (G) = D (D (G)). By Lemma 1.2.5(iii), D (T · D (G)) = D 2 (G). Thus, G/D 2 (G) is a quotient of the commutative ZG (T ), so D (G) ⊆ D 2 (G). The reverse containment is clear, so we are done. 2 Proposition 1.2.7 Let G be a pseudo-reductive group over a ﬁeld k, and G a smooth connected normal k-subgroup of G and N a smooth connected normal k-subgroup of G . If N is perfect then N it is normal in G. In Remark 3.1.10 we will show that the perfectness hypothesis on N in Proposition 1.2.7 can be dropped (by applying Proposition 1.2.7 to the perfect D (N ) and using Lemma 1.2.5 and the description of perfect smooth connected normal k-subgroups of G provided in § 3.1).

16

Overview of pseudo-reductivity

Proof. We may assume that k is separably closed, so G(k) is Zariski-dense in G. Thus, the G(k)-conjugates of N generate a smooth connected normal k-subgroup of G that is contained in G . We may replace G with this subgroup so that the G(k)-conjugates of N generate G . Since N is perfect, G is also perfect. In this case we will prove that N = G . Consider the smooth connected images N of Nk and G of G in G red . Clearly k

k

N is normal in G , which in turn is normal in G red . The classiﬁcation of smooth k

connected normal k-subgroups in connected reductive k-groups implies that N is normal in G red . But G is generated by conjugates of N , so G = N . This k implies that G /Nk is unipotent, so G /N is solvable. But G /N is perfect (since k G is), so G /N = 1. 2

1.3 Preparations for the standard construction Many of the difﬁculties in the classiﬁcation of pseudo-reductive groups are seen in the “simplest” interesting case: for an arbitrary ﬁeld k, consider non-reductive pseudo-reductive k-groups G such that the maximal reductive quotient Gkred = Gk /Ru (Gk ) of Gk is a simple semisimple group (so G is non-solvable and G red coincides k with the maximal semisimple quotient G ss ). k The quotient map π : Gk G ss has a smooth unipotent (scheme-theoretic) k kernel, but this kernel fails to descend to a k-subgroup of G. Let k /k be the ﬁeld of deﬁnition (in the sense of Deﬁnition 1.1.6) for the k-subgroup ker π ⊆ Gk . In Examples 1.3.3 and 1.3.5 we will compute the ﬁeld of deﬁnition over k in some interesting cases. Since the k-subgroup Ru (Gk ) = R(Gk ) of Gk descends to the perfect closure of k in k (by Galois descent from k), the ﬁeld of deﬁnition k for ker π over k is purely inseparable of ﬁnite degree over k. The quotient G of Gk modulo the k -descent of ker π is a connected absolutely simple and semisimple k -group. The quotient map π : Gk G over k corresponds to a k-group map iG : G → R k /k (G ).

(1.3.1)

Note that the deﬁnition of iG (using a descent of Ru (Gk )) makes sense for any smooth connected afﬁne k-group G. In Example 1.6.3 we will exhibit commutative pseudo-reductive G for which ker iG = 1, and Example 5.3.7 provides non-commutative pseudo-reductive G for which dim ker iG is arbitrarily

1.3 Preparations for the standard construction

17

large. There are also interesting examples of pseudo-reductive G for which iG is not surjective; see Example 1.3.3 for commutative G and Example 1.3.5 for non-commutative G. Remark 1.3.1 The connected semisimple quotient G ss above is simple by k hypothesis. It therefore follows from Theorem 1.6.2(2) that the pseudoreductive k-group G arises as the Weil restriction of a connected reductive group over a ﬁnite extension of k if and only if iG is an isomorphism. Our main classiﬁcation theorem leads to a simple characterization of when iG is an isomorphism for such G, provided that we also require G = D (G) and that [k : k 2 ] = 2 when char(k) = 2: it is necessary and sufﬁcient that the order of the fundamental group of G ss is not divisible by char(k), except possibly k when char(k) = p ∈ {2, 3} and the Dynkin diagram of G ss has an edge of k multiplicity p or is a single vertex when p = 2 (i.e., type G2 for p = 3 and types B, C, or F4 for p = 2). This is proved in Proposition 11.1.4 and Remark 11.1.5 (which also explicitly describe all exceptional cases) when Gks has a reduced root system. The case of a non-reduced root system rests on Theorem 2.3.10, Theorem 9.3.10, and several results in §9.4. The construction of commutative examples of non-surjective iG rests on Weil restriction through arbitrary ﬁnite extensions of ﬁelds. The following example shows that inseparable Weil restriction can convert an inﬁnitesimal group into a (non-smooth) group with dimension > 0, and so can ruin the property of a map being an isogeny: Example 1.3.2 Let k /k be a nontrivial purely inseparable ﬁnite extension of ﬁelds with char(k) = p > 0. By Example 1.1.3, G := R k /k (GL1 ) is pseudoreductive over k, and G/GL1 is nontrivial and unipotent. Thus, not only is G not reductive, but we see that pseudo-reductivity can be lost after passage to the quotient by a smooth connected k-subgroup. Let f be the p-power map on GL1 over k (an isogeny), so ker( f ) = μp . Thus, R k /k ( f ) is the p-power map on the pseudo-reductive k-group R k /k (GL1 ), and this map fails to be surjective for dimension reasons since its kernel ker R k /k ( f ) = R k /k (μp ) = R k /k (GL1 )[p] has dimension at least p − 1 > 0. To see this, choose an intermediate ﬁeld k0 ⊆ k of degree p over k. Clearly R k /k (GL1 ) contains R k0 /k (GL1 ), and p−1

(R k0 /k (GL1 )[p]/μp )k0 (R k0 /k (GL1 )/GL1 )k0 Ga

by using the truncation of log(1 + x) in degree < p in characteristic p.

(1.3.2)

18

Overview of pseudo-reductivity

Thus, the isogeny f has Weil restriction R k /k ( f ) that is not an isogeny. Also, the quotient R k /k (GL1 )/μp by the ﬁnite multiplicative k-subgroup μp is not pseudo-reductive since its k-subgroup R k0 /k (GL1 )/μp is nontrivial, smooth, connected, and unipotent by (1.3.2). Here is a commutative example with non-surjective iG . Example 1.3.3 Let k be an imperfect ﬁeld of characteristic p > 0, and let k /k /k be a tower of purely inseparable extensions of degree p such that 2 k p ⊂ k (i.e., k = k(a1/p ) and k = k(a1/p ) for some a ∈ k that is not a p-power). Thus, the only intermediate ﬁelds in k /k are k, k , and k . Consider a smooth connected k-subgroup G of C := R k /k (GL1 ) that strictly contains C := R k /k (GL1 ). (We will construct examples of such G = C at the end. The reason we focus on these G instead of smooth connected proper k-subgroups of C that strictly contain GL1 is that no G of the latter type exist when p = 2, as the codimension of GL1 in C is [k : k] − 1 = p − 1.) Such a k-group G inherits pseudo-reductivity from the commutative pseudo-reductive C . Since Ck is the unit group of the algebra k ⊗k k k [x]/(x p ), it is GL1 × U where U is the group of 1-units. Hence, Gk = GL1 × (U ∩ Gk ), so U ∩Gk must be nontrivial, smooth, and connected. It follows that G is nonreductive with k /k containing the ﬁeld of deﬁnition of its geometric unipotent radical, so this ﬁeld of deﬁnition is either k or k . We now check that the ﬁeld of deﬁnition over k for the k-subgroup Ru (Gk ) ⊂ Gk is k by ruling out the possibility that it is k . The k -group Ck is the unit group of k ⊗k k = k ⊗k (k ⊗k k ) = k [x]/(x p ), so Ck = R k /k (GL1 ) × U for a smooth connected p-torsion k -group U . Observe that the projection π : Ck → R k /k (GL1 ) restricts to the identity on GL1 . Transitivity of Weil restriction provides another quotient map q : Ck = R k /k (R k /k (GL1 ))k R k /k (GL1 ) via the construction in Proposition A.5.7 (with X = R k /k (GL1 ) there), and we claim that π = q. To prove this, observe that both are homomorphisms between the same pair of smooth commutative afﬁne k -groups and their restrictions to the maximal k -torus GL1 coincide (equal to the identity map). This forces the maps to be equal, by Proposition 1.2.2. The equality π = q ensures that π cannot carry Gk into GL1 . To explain this, observe rather generally that if X is a quasi-projective k -scheme and

1.3 Preparations for the standard construction

19

f : Y → R k /k (X ) is a k-map from a k-scheme Y then (by the very deﬁnition of q) the composite map qX ◦fk : Yk → X corresponds under the universal property of Weil restriction to the given map f . Since π = q, the restriction of π to Gk therefore corresponds to the inclusion of G into R k /k (R k /k (GL1 )) = C . Thus, if π|Gk were to land in GL1 then G ⊆ R k /k (GL1 ) = C inside of C . But we assumed that G strictly contains C , so this would be a contradiction. Now we can show that k /k is not the ﬁeld of deﬁnition of the geometric unipotent radical of G. If it were, then Ru,k (Gk ) would be a complement to GL1 inside of Gk , yet π would have to kill it (as the target of π is commutative and pseudo-reductive over k ). Hence, π |Gk would land in GL1 , which we have shown does not happen. Since the ﬁeld of deﬁnition over k for the geometric unipotent radical of G has now been determined to be k /k, the universal property of Weil restriction makes iG correspond to the evident projection from Gk onto its maximal reductive quotient GL1 . But this projection factors as the composite of Gk → Ck and the maximal reductive quotient map for Ck . In other words, iG factors as iC

G → C → R k /k (GL1 ). The map iC is the identity map (since the evident quotient map Ck → GL1 coincides with the map qGL1 ,k /k from Proposition A.5.7), so if G = C then iG is not surjective. It remains to exhibit nontrivial smooth connected proper k-subgroups of U = C /C . This k-group is smooth, connected, commutative, and p-torsion, so by Lemma B.1.10 it admits an étale k-isogeny onto a vector group V . Since p2 −p

V Ga

and p2 −p > 1, we can choose a nonzero smooth connected proper p2 −p

k subgroup of Ga , and the identity component of its preimage in U gives rise to the desired G. Examples of the failure of surjectivity of iG with perfect pseudo-reductive G rest on: Proposition 1.3.4 Let k be a ﬁeld, k a nonzero ﬁnite reduced k-algebra, and G a reductive k -group with connected ﬁbers over Spec k . For any k -subgroup scheme μ ⊆ ZG , the natural k-group map G := R k /k (G )/R k /k (μ) → R k /k (G /μ)

20

Overview of pseudo-reductivity

is a normal subgroup inclusion with commutative cokernel. In particular, G is pseudo-reductive over k. If G → Spec k has simply connected semisimple ﬁbers then R k /k (G ) is perfect, so in such cases G is the derived group of R k /k (G /μ). Proof. The k-group R k /k (G /μ) is pseudo-reductive (Proposition 1.1.10), so pseudo-reductivity of G will follow from the normality claim. To prove the normality, view the central extension 1 → μ → G → G /μ → 1 as a short exact sequence of fppf group sheaves over Spec k . Pushforward along f : Spec k → Spec k yields an exact sequence of low-degree higher direct image sheaves δ

1 → R k /k (μ) → R k /k (G ) → R k /k (G /μ) → R 1 f∗ (μ).

(1.3.3)

The connecting map δ is a homomorphism due to the centrality of μ in G . It follows that R k /k (G )/R k /k (μ) = ker δ, so it is a normal k-subgroup of R k /k (G /μ) in the sense of fppf group sheaves over Spec k. Thus, it is also a normal closed k-subgroup in the sense of k-group schemes (see Proposition A.2.1). When G has simply connected ﬁbers, the perfectness of R k /k (G ) is Corollary A.7.11. In such cases G = D (G), so G is the derived group of R k /k (G /μ) since δ has a commutative target. 2 Proposition 1.3.4 provides examples of non-commutative pseudo-reductive groups G for which iG is not surjective, as we show below. The justiﬁcation rests on results to be proved in §1.6, and also yields examples of inseparable isogenies f between connected semisimple groups such that the Weil restriction of f through a ﬁeld extension is not an isogeny: Example 1.3.5 Let G = R k /k (SLp )/R k /k (μp ) for k /k a nontrivial ﬁnite purely inseparable extension with char(k) = p > 0. By Proposition 1.3.4, G is pseudo-reductive and perfect, and by Proposition A.5.11(1),(2) the maximal geometric semisimple quotient G ss maps isomorphically to the simple k semisimple group SLp /μp = PGLp . We will show that iG is not surjective. Granting this, it then follows from the surjectivity of iRK/k (H ) (Proposition A.5.11(1),(2)) that G is not a k-isogenous quotient of R K/k (H ) for any ﬁnite extension K/k and connected reductive K-group H . Thus, G is “far” from Weil restrictions of connected reductive groups.

1.3 Preparations for the standard construction

21

The ﬁrst issue to settle in the proof of surjectivity of iG is the determination of the (minimal) ﬁeld of deﬁnition over k for the geometric (unipotent) radical of G. We claim that this ﬁeld is k . That is, relative to the evident k-structure on Gk , we claim that the scheme-theoretic kernel of the quotient map π : Gk Gkss = PGLp has (minimal) ﬁeld of deﬁnition over k equal to k ⊆ k. This ﬁeld of deﬁnition over k is contained in k since a k -descent of π is given by the canonical map π : Gk PGLp induced by the composite map q

R k /k (SLp )k −→ SLp → PGLp that kills R k /k (μ)k (due to functoriality of (A.5.3) with respect to the inclusion = μp → SLp ). An application of Theorem 1.6.2(1) with G = PGLp and G SLp implies that the ﬁeld of deﬁnition over k for ker π cannot be a proper subﬁeld of k , so it is equal to k . The identiﬁcation of k /k as the ﬁeld of deﬁnition of ker π implies that the k-map iG : G → R k /k (PGLp ) corresponds to π via the universal property of Weil restriction. For the inseparable k -isogeny f : SLp → PGLp we have R k /k ( f ) = iG ◦ h, where h : R k /k (SLp ) → R k /k (SLp )/R k /k (μp ) is the canonical projection. Left-exactness of R k /k implies that ker(R k /k ( f )) = R k /k (μp ) = ker h, so we conclude that ker iG = {1}. However, iG fails to be surjective for dimension reasons. (Proposition 1.3.4 even identiﬁes its image with D (R k /k (PGLp )).) In particular, although f is an isogeny, the map R k /k ( f ) = iG ◦ h is not surjective. This example exhibits an additional feature, as follows. By Proposition 1.1.10, the k-group R k /k (SLp ) is pseudo-reductive over k. Also, the central quotient G of R k /k (SLp ) by a connected k-subgroup of dimension > 0 is pseudo-reductive over k (Proposition 1.3.4). However, the central quotient R k /k (SLp )/μp of R k /k (SLp ) by a ﬁnite multiplicative k-subgroup is not

22

Overview of pseudo-reductivity

pseudo-reductive over k. Indeed, this central quotient contains the p-torsion central k-subgroup R k0 /k (μp )/μp for any subﬁeld k0 ⊆ k of degree p over k, and this subgroup maps isomorphically onto the (unipotent) smooth connected k-group R k0 /k (GL1 )/GL1 of dimension p − 1. Remark 1.3.6 In Example 1.6.4 we will give a subtle variant on Example 1.3.5: a perfect pseudo-reductive k-group G containing a smooth connected perfect normal k-subgroup N ⊆ G such that G/N is not pseudo-reductive. In particular, the preimage N of Ru,k (G/N ) under the quotient map G G/N is a smooth connected normal k-subgroup of G such that N = D (N ) (as N has a nontrivial unipotent quotient). Note the contrast with the theory of connected semisimple groups. Indeed, if G is a connected semisimple group over a ﬁeld then for all smooth connected normal subgroups N of G both N and G/N are semisimple (and in particular, N is perfect). This illustrates some of the difﬁculties in any attempt to develop a good notion of pseudo-semisimplicity. Let k /k be a nonzero ﬁnite reduced k-algebra. Consider a k -group G whose ﬁbers over Spec k are connected and reductive. Let T be a maximal k -torus in G , so T is a Cartan k -subgroup. (Explicitly, T = ZG (T ).) By Proposition A.5.15, the maximal k-torus T in R k /k (T ) is a maximal k-torus in the pseudo-reductive k-group G = R k /k (G ) and the Cartan k-subgroup C := ZG (T ) is equal to R k /k (T ). Likewise, the center ZG is R k /k (ZG ). We will be interested in smooth connected k-groups equipped with a left action on G that restricts to the identity on C. One example is the conjugation action on G by the commutative C, but not even the action of C/ZG on G explains all such actions (especially when k is not étale over k): Example 1.3.7 Let k /k be a nontrivial purely inseparable ﬁnite extension of ﬁelds of characteristic p > 0, and T a maximal k -torus in SLp with image T in the central quotient PGLp . Let T be the maximal k-torus of R k /k (T ). The action of PGLp on SLp over k makes T act on SLp with trivial action on T . Applying R k /k then deﬁnes an action of R k /k (T ) on G = R k /k (SLp ), and this action ﬁxes the Cartan k-subgroup R k /k (T ) = ZG (T ) =: C pointwise. However, the natural inclusion C/ZG → R k /k (T ) is not surjective, as dim C = dim R k /k (T ) but dim ZG > 0 (Example 1.3.2). Theorem 1.3.9 below implies that it is the action of R k /k (T ) (and not the action of C/ZG !) on G that is universal for the property of ﬁxing C pointwise. In general, for a smooth afﬁne k-group G, a maximal k-torus T in G, and the Cartan k-subgroup C = ZG (T ) in G, we are interested in the following functors

1.3 Preparations for the standard construction

23

on k-schemes: Aut G,T : S {g ∈ Aut S (GS ) | g restricts to the identity on TS }, Aut G,C : S {g ∈ Aut S (GS ) | g restricts to the identity on CS }. Observe that if H is a k-group then to give a k-homomorphism of group functors H → Aut G,C is the same as to deﬁne a left action of H on G that restricts to the identity on C. Example 1.3.8 In an evident manner, Aut G,C is a subgroup functor of Aut G,T . If G is pseudo-reductive over k and F/k is a separable extension ﬁeld then the inclusion Aut G,C (F) ⊆ Aut G,T (F) is an equality. Indeed, separability ensures that GF is pseudo-reductive over F, so we may rename F as k to reduce to showing that a k-automorphism φ of G that restricts to the identity on T also restricts to the identity on C. By Proposition 1.2.4, C is a commutative pseudoreductive group. The automorphism φ of G restricts to an automorphism of C = ZG (T ) that is the identity on the maximal k-torus T in C. Thus, by the ﬁnal part of Proposition 1.2.2, φ is the identity on C as well. In Theorem 2.4.1 we will prove that Aut G,C is represented by an afﬁne k-group scheme of ﬁnite type (denoted Aut G,C ). This k-group contains a maximal smooth k-subgroup ZG,C (via the construction in Lemma C.4.1), and we will prove that if G is pseudo-reductive then ZG,C is commutative with pseudo-reductive identity component. The following result makes ZG,C explicit in a special case. We record it here solely for the purpose of motivating the “standard” construction of pseudo-reductive groups in §1.4; it is not logically necessary at this stage of the theory. Theorem 1.3.9 Let G = R k /k (G ) for a nonzero ﬁnite reduced k-algebra k and a k -group G with connected reductive ﬁbers over Spec k . Choose a Cartan k-subgroup C in G , and let T be the unique maximal k -torus in G for which C = R k /k (T ). Using the left action of R k /k (T /ZG ) on G induced by the natural action of G /ZG on G , the natural map R k /k (T /ZG ) → Aut G ,C is an isomorphism onto the maximal smooth k-subgroup ZG ,C . In particular, ZG ,C is connected and the action of R k /k (T /ZG ) on G is ﬁnal among all smooth k-groups equipped with a left action on G that restricts to the identity on C . The correspondence between the choice of C in G and the choice of Cartan k -subgroup T in G as used above is part of Proposition A.5.15(3).

24

Overview of pseudo-reductivity

Proof. We may and do assume k = ks . The natural map R k /k (T /ZG ) → Aut G ,C uniquely factors through a k-homomorphism f : R k /k (T /ZG ) → ZG ,C between smooth afﬁne k-groups, and we need to prove that this is an isomorphism. It sufﬁces to prove that f is birational. In general, if k is a ﬁeld and h : X → X is a k-morphism between integral k-schemes with respective generic points η and η, and if h is injective on k(η )-valued points and is surjective on k(η)-valued points then h is birational (in the sense that h(η ) = η and the induced map k(η) → k(η ) is an isomorphism). Indeed, surjectivity on k(η )-valued points implies that the canonical map Spec k(η) → X factors j

h

over k as Spec k(η) → X → X , so h is dominant and hence h(η ) = η. The h

j

resulting k-map Spec k(η ) → Spec k(η) → X coincides with the canonical k-map Spec k(η ) → X since both have the same composition with h (which is assumed to be injective on k(η )-points). Hence, j(η) = η with j inverse to h between function ﬁelds. We conclude that it sufﬁces to prove f induces a bijection between F-valued points for every separable extension ﬁeld F/k (for F = k = ks this will ensure that ZG ,C is connected, hence integral, so the above reasoning may be applied). For any separable extension ﬁeld F/k, the nonzero ﬁnite F-algebra k ⊗k F is reduced. Thus, we may rename F as k (and again pass to ks ) to reduce to proving the bijectivity property on k-points. In view of how the maximal smooth k-subgroup ZG ,C is constructed when k = ks , this is the assertion that the map (T /ZG )(k ) → Aut k (G ) is an isomorphism onto the subgroup of k-automorphisms of G that restrict to the identity on the Cartan k-subgroup C = R k /k (T ). We want to pass to factor ﬁelds of k to reduce to the case where k is a ﬁeld, so we ﬁrst need to check that a general k-automorphism f of G = R k /k (G ) that restricts to the identity on C is built from corresponding automorphisms arising separately from ﬁbers of (G , T ) over the factor ﬁelds of k . Here is the result: Lemma 1.3.10 Choose f ∈ Aut G ,C (k). There is a unique k -automorphism f of G that restricts to the identity on T and satisﬁes R k /k ( f ) = f . That is, the map Aut G ,T (k ) → Aut G ,C (k) induced by Weil restriction is bijective. Proof. First we treat the case where k is a ﬁeld. Consider the natural map qG : Gk → G as in (A.5.3), so qG is a smooth surjective map by Proposition A.5.11(1). Since k /k is purely inseparable, it follows from Proposition A.5.11(2) that ker qG is connected and unipotent, so G is the maximal reductive quotient of Gk . Hence, fk induces an automorphism f of G that restricts to the identity on the image of Ck in G . But this image is T (by functoriality of qT and qG with respect to the inclusion of T into G ), so f ∈ Aut G ,T (k ).

1.4 The standard construction and examples

25

Functoriality of qG with respect to f then implies that f and R k /k ( f ) are kautomorphisms of G whose associated k -automorphisms of Gk induce the same automorphism f of the maximal reductive quotient G . This forces f = R k /k ( f ) by Proposition 1.2.2. For general k , let {ki } be the set of factor ﬁelds of k , and Gi the ki -ﬁber of G . Let Gi = R ki /k (Gi ), so G = Gi . Observe that if Ti denotes the ki -ﬁber of T then C = Ci , where Ci is the Cartan k-subgroup R ki /k (Ti ). It sufﬁces to prove f = fi for fi ∈ Aut Gi ,Ci (k), as then the problem is reduced to the settled case where k is a ﬁeld. For each i, consider the centralizer of j=i Cj in G . This is clearly Gi × j=i Cj , and so f must restrict to a k-automorphism of this subgroup (as f restricts to the identity on j=i Cj ). Hence, f restricts to an automorphism of its derived group, which is D (Gi ). By Proposition 1.2.6, Gi = Ci · D (Gi ). Thus, it sufﬁces to check that f restricts to an automorphism of Ci for each i. Even better, f restricts to the identity on Ci = C . 2 Continuing with the proof of Theorem 1.3.9, the description of f as a Weil restriction in Lemma 1.3.10 reduces the problem to a separate argument over each factor ﬁeld of k . That is, we can now assume that k is a ﬁeld. We have to prove that any k -automorphism f of G that restricts to the identity on T is induced by the action of a unique point t ∈ (T /ZG )(k ). The effect of f on the root datum for (G , T ) is trivial since f restricts to the identity on T , so the Isomorphism Theorem for the split pair (G , T ) over the separably closed ﬁeld k (Theorem A.4.6) gives that f is induced by the action of a unique t ∈ (T /ZG )(k ). 2

1.4 The standard construction and examples We now construct pseudo-reductive groups going beyond the examples provided by Proposition 1.1.10 (and motivated by Theorem 1.3.9). These will be the standard pseudo-reductive groups. The ubiquity of this construction will be explained by our main classiﬁcation result for pseudo-reductive groups (stated in a weak form in Theorem 1.5.1). To construct more general pseudo-reductive groups, we need a procedure to modify a given pseudo-reductive group by either passing to a central quotient or expanding a Cartan subgroup without losing pseudo-reductivity. To explain the idea, consider a pseudo-reductive k-group G and a maximal k-torus T in G . Let C be the Cartan k-subgroup ZG (T ). According to Proposition 1.2.4, C is commutative and pseudo-reductive. There is a natural left action of C on G via conjugation, and this restricts to the identity on C . Suppose there is given another commutative pseudo-reductive k-group C with a left action on

26

Overview of pseudo-reductivity

G which ﬁxes C pointwise, and a k-homomorphism φ : C → C compatible with the actions on G . In other words, we are given a factorization diagram of group functors on k-schemes ψ

φ

C −→ C −→ Aut G ,C . Suppose that Aut G ,C is represented by an afﬁne k-group scheme of ﬁnite type, and let ZG ,C denote its maximal smooth k-subgroup, so we get a factorization diagram φ

ψ

C −→ C −→ ZG ,C . In §2.4 we will prove that the representability always holds and that ZG ,C is commutative with pseudo-reductive identity component. (What follows is merely motivation for Deﬁnition 1.4.4, so there is no risk of circularity.) Note that ker φ acts trivially on G and so is central in G . We wish to create another pseudo-reductive group in which C replaces C as a Cartan k-subgroup. A side-effect of this procedure will be to kill ker φ, so the center may be replaced with a quotient even as the Cartan k-subgroup may get bigger. Here is the most important example. Example 1.4.1 Suppose G = R k /k (G ) for a nonzero ﬁnite reduced k-algebra k and a k -group G with connected reductive ﬁbers. Let T be a maximal k -torus in G , and let C be the Cartan k-subgroup R k /k (T ) of G , and T be the maximal k-torus of C . Since G /ZG acts on G , the quotient torus T /ZG acts on G . Hence, R k /k (T /ZG ) naturally acts on G , and the natural map C = R k /k (T ) → R k /k (T /ZG ) is compatible with the actions on G . Observe that this map may not be surjective, as we illustrated in Example 1.3.7. Also, its kernel is R k /k (ZG ) due to left exactness of Weil restriction, and this is the entire center of G (Proposition A.5.15(1)). By Theorem 1.3.9 (which is conditional on §2.4), the natural map R k /k (T /ZG ) → ZG ,C is an isomorphism. Thus, to give a pair (C, φ) consisting of a commutative pseudo-reductive k-group C equipped with a left action on G which ﬁxes C pointwise and a k-homomorphism φ : C = R k /k (T ) → C respecting the actions on G amounts to a factorization φ

ψ

R k /k (T ) −→ C −→ R k /k (T /ZG )

(1.4.1)

of the natural map R k /k (T ) → R k /k (T /ZG ) of k-groups (with C a commutative pseudo-reductive k-group). Since Theorem 1.3.9 is presently conditional

1.4 The standard construction and examples

27

on §2.4, we note that it makes sense to consider factorization diagrams (1.4.1) even without knowing Theorem 1.3.9, since ψ provides a C-action on G that restricts to the identity on C . (The point of Theorem 1.3.9 is that the construction resting on (1.4.1) gives all possibilities for pairs (C, φ).) Using the C-action on G , we get a semidirect product group G C. The pair of homomorphisms φ : C → C, ι : C → G deﬁnes the twisted product map α :C →G C

(1.4.2)

via c → (ι(c)−1 , φ(c)). This is readily checked to be an isomorphism onto a central subgroup. Consider the resulting smooth connected afﬁne quotient group G := (G C)/im(α). Remark 1.4.2 The k-group C C is a Cartan k-subgroup in the semidirect product G C. Hence, in the quotient G its image is a Cartan k-subgroup (Proposition A.2.8). But this image is exactly C with its natural homomorphism into G (that has trivial kernel). This procedure therefore replaces C with C, as desired. Explicitly, the commutative k-group C is the centralizer of the image of T in G, as well as of the maximal k-torus T of C (which in turn must be maximal in G). Note that T is the unique maximal k-torus of G containing the image of T , due to the commutativity of C. We claim that the k-group G = coker(α) is pseudo-reductive. It is convenient to work more generally, without mentioning Cartan subgroups. Consider the following generalization: G is a pseudo-reductive k-group, C is a commutative pseudo-reductive k-subgroup such that C = ZG (C ) (for example, a Cartan k-subgroup), C is a commutative pseudo-reductive group equipped with a left action on G that is trivial on C , and φ : C → C is a k-homomorphism compatible with the actions on G . The k-scheme map α :C →G C via c → (c−1 , φ(c)) is easily checked to be a central k-subgroup inclusion, so we can form the smooth connected afﬁne k-group G = coker α = (G C)/α(C ).

28

Overview of pseudo-reductivity

Proposition 1.4.3 Under the above conditions, the k-group G = (G C)/α(C ) is pseudo-reductive.

Proof. Consider the map of group functors ψ : C → Aut(G ) deﬁned by the action of C on G . We claim that ker ψ is represented by a closed k-subgroup scheme of C. To prove this, by Galois descent we may assume k = ks , so G (k) is Zariski-dense in G . It follows that G (k) is relatively schematically dense in G (in the sense of [EGA, IV3 , 11.10.1]), so by [EGA, IV3 , 11.10.6], for any k-algebra A, an A-automorphism of GA is determined by its effect on G (k). Thus, ker ψ is represented by the schematic intersection g∈G (k) ZC (g) of the scheme-theoretic stabilizers in C of the points g ∈ G (k). (Explicitly, ZC (g) is the preimage of the diagonal under the map C → G × G deﬁned by c → (c.g, g).) Now let Z ⊆ G := G C denote the central k-subgroup α(C ). Another central subgroup scheme in G is ker ψ via inclusion into the second factor of the semidirect product. Observe that Z ∩ ker ψ = 1 since the ﬁrst projection G = G C → G as a map of k-schemes carries Z isomorphically onto C . Hence, Z × ker ψ makes sense as a central k-subgroup scheme of G . The ﬁrst step is to check that Z × ker ψ is the entire scheme-theoretic center of G . We begin by observing that the smooth direct product C × C is a k-subgroup of G C and its centralizer in the latter is ZG (C ) C = C × C. In view of the deﬁnition of Z = α(C ), we can say C × C = Z × C inside of G , so Z × C is its own centralizer in G . Hence, any central element of G has the form z · (1, c) for a unique z ∈ Z and c ∈ C. The centrality of Z in G then reduces us to the case z = 1, and then, by deﬁnition, the centrality of (1, c) implies that c acts trivially on G ; i.e., c ∈ ker ψ. This proves that Z × ker ψ is the center of G = G C. Now we turn to the task of proving that U := Ru,k (G) is trivial. Consider the preimage N of U under the smooth quotient map G = G C G. This is an extension of U by C , so N is smooth, connected, and solvable. We claim that the smooth connected normal k-subgroup (N , G ) is unipotent. In fact, we claim this holds with G any smooth connected afﬁne k-group and N a smooth connected solvable normal subgroup. With such generality (no pseudoreductivity conditions) we can assume the base ﬁeld k is algebraically closed. By replacing G with its maximal reductive quotient G red , we reduce to the case where G is connected reductive. Hence, the smooth connected solvable normal subgroup N is reductive, so N is a torus. A normal torus in a smooth connected afﬁne group is always central (as the automorphism scheme of a torus is étale), so (N , G ) = 1.

1.4 The standard construction and examples

29

Returning to our situation, the semidirect product G = G C is clearly pseudo-reductive since G and C are pseudo-reductive. Hence, the smooth connected unipotent normal k-subgroup (N , G ) of G must be trivial, which is to say that N is central in G . That is, N ⊆ ZG = Z × ker ψ. But N was deﬁned as a preimage under G G /Z = G, so N contains Z inside of G . Thus, N = Z × (N ∩ ker ψ). Since N is smooth and connected, it follows that N ∩ ker ψ is smooth and connected. But N /Z = U by deﬁnition of N , and this is unipotent. It follows that N ∩ ker ψ is a smooth connected unipotent k-subgroup of ker ψ. But ker ψ is a k-subgroup of the commutative pseudo-reductive k-group C, so N ∩ ker ψ = 1. This proves that N = Z, so U = 1. 2 Deﬁnition 1.4.4 A standard pseudo-reductive k-group is a k-group G that is k-isomorphic to the pseudo-reductive k-group coker(α) with α as in (1.4.2) in the setting of Example 1.4.1. That is, G (R k /k (G ) C)/R k /k (T )

(1.4.3)

for a nonzero ﬁnite reduced k-algebra k , a k -group G with connected reductive ﬁbers, a maximal k -torus T in G , and a factorization (1.4.1) with C a commutative pseudo-reductive k-group. We say that G arises from a 4-tuple (G , k /k, T , C) (suppressing explicit mention of the factorization diagram (1.4.1), which is always understood from the context). In the preceding deﬁnition, we only use the description (1.4.3) and not the universal property of R k /k (T /ZG ) from Theorem 1.3.9 (which was conditional on §2.4). At present, this universal property merely motivates the deﬁnition of the standard construction. Later it will be useful in the proof of standardness of all pseudo-reductive groups, apart from some exceptional situations in characteristics 2 and 3. Two special cases of standard pseudo-reductive k-groups are R k /k (G ) (take φ to be the identity map in (1.4.1)) and any commutative pseudo-reductive k-group C (take G = 1). In particular, any solvable pseudo-reductive group is standard because it is commutative (Proposition 1.2.3). Also, if K/k is a ﬁnite extension of ﬁelds and G is a standard pseudo-reductive K-group then R K/k (G) is a standard pseudo-reductive k-group; this follows immediately from the fact that R K/k preserves short exactness for smooth afﬁne groups (Corollary A.5.4) and preserves pseudo-reductivity (Proposition 1.1.10).

30

Overview of pseudo-reductivity

We emphasize (see Remark 1.4.2) that in the standard construction we have C = ZG (T ) for the unique maximal k-torus T of G that contains the image in G of the unique maximal k-torus of R k /k (T ). Remark 1.4.5 The construction of standard pseudo-reductive groups is an instance of a general non-commutative pushout construction, as follows. The general setup to consider, for abstract groups, group sheaves, or group schemes of ﬁnite type over a ﬁeld, is a triple of group objects N , G , and G , equipped with left actions of G on the groups G and N and G -homomorphisms f : N → G and f : N → G , where G acts on itself on the left by conjugation. Assume f (n).g = f (n)g f (n)−1 in G for all n ∈ N and g ∈ G . (An interesting example is N a normal subgroup of G and f a G -equivariant quotient.) Consider the semidirect product G G , deﬁned using the left G -action on G . The map f : N → G G deﬁned by f (n) = ( f (n)−1 , f (n)) is a homomorphism that centralizes the normal subgroup G ⊆ G G and intertwines G -conjugation on G G with the given G -action on N . In particular, the image of N under f is normal, so we can form the quotient Q = (G G )/f (N ) that we view as ﬁtting into a pushout diagram N

/ G

G

/ (G G )/f (N )

If G acts trivially on N (as in the construction of standard pseudo-reductive groups) then f (N ) is central in G G . If all three groups are commutative and G acts trivially on both N and G then the preceding pushout construction is the classical pushout construction as in any abelian category. Finally, if N is a normal subgroup of G and f is a G -equivariant quotient then G is a normal subgroup of the pushout Q, with Q/G = G /N . In the setting of the construction of standard pseudo-reductive groups, consider the central k-subgroup scheme Z := ker φ ⊆ R k /k (ZG ).

1.4 The standard construction and examples

31

The quotient R k /k (T )/Z is a smooth connected k-subgroup of the commutative pseudo-reductive k-group C, so it is pseudo-reductive. If we replace C with this k-subgroup in (1.4.3) then we get a k-subgroup inclusion Q := R k /k (G )/Z → G. By Corollary A.7.11, this k-subgroup coincides with D (G) when the ﬁbers of G → Spec k are semisimple and simply connected (a harmless hypothesis on G , as we will see in §4.1). It seems difﬁcult to describe all k-subgroups Z ⊆ R k /k (ZG ) for which R k /k (T )/Z is pseudo-reductive. Likewise, the quotient R k /k (G )/Z for a general k-subgroup Z of R k /k (ZG ) can fail to be pseudo-reductive over k, even if G is semisimple and simply connected. We saw counterexamples in Example 1.3.5 with G = SLp and Z = μp . Remark 1.4.6 In general, when G is semisimple, Proposition 1.3.4 gives a special class of Z ⊆ R k /k (ZG ) for which R k /k (G )/Z is pseudo-reductive. Let us now check that a central quotient G := R k /k (G )/Z is standard whenever it is pseudo-reductive. Pick a maximal k -torus T in G , so ZG ⊆ T . The smooth connected commutative k-group C = R k /k (T )/Z makes sense and it is pseudo-reductive over k because it is a Cartan k-subgroup in the pseudo-reductive k-group G and Cartan k-subgroups of pseudo-reductive groups are always pseudo-reductive (Proposition 1.2.4). There is an evident isomorphism G := R k /k (G )/Z (R k /k (G ) C)/R k /k (T )

(1.4.4)

in which the right side is coker(α) as in (1.4.2), so G is indeed standard over k. We now give a proof that in some cases the standard pseudo-reductive groups as in Proposition 1.3.4 are not themselves the Weil restriction of a connected reductive group: Example 1.4.7 Let k /k be a nontrivial purely inseparable ﬁnite extension of ﬁelds with char(k) = p > 0. Let G = R k /k (G )/R k /k (μ), where G is a connected semisimple k -group that is absolutely simple and simply connected and μ is a non-étale subgroup of the ﬁnite center ZG . (For example, we can take G = SLp and μ = μp .) Observe that Z := R k /k (μ) has dimension > 0 (because we may take k to be separably closed, so μ contains μp , and in Example 1.3.2 we saw that R k /k (μp ) has dimension > 0). Also, by Proposition 1.3.4, G is pseudo-reductive and perfect (the latter using the simply connected hypothesis on G ).

32

Overview of pseudo-reductivity

We claim that the k-group G is not a k-isogenous quotient of any k-group of the form R K/k (H ) for a nonzero ﬁnite reduced k-algebra K and a K-group H whose ﬁbers over Spec K are connected and reductive. This fact, which will never be used in our development of the theory of pseudo-reductivity, rests on Theorem 1.6.2(2) below, as we now explain. We argue by contradiction, so assume G is a k-isogenous quotient of R K/k (H ) for some K and H , and seek a contradiction. First we claim that K must be a purely inseparable ﬁeld extension of k. If K is a nontrivial product or is a ﬁeld not purely inseparable over k then Gk has maximal reductive quotient that is not a simple semisimple group. However, the maximal reductive quotient of Gk is isogenous to the simple semisimple group G ⊗k k. This also shows that dim H dim G . With K/k a purely inseparable ﬁeld extension, we see that the maximal reductive quotient of Gk is an isogenous quotient of Hk , so dim H = dim G . But dim G < [k : k] dim G since dim Z > 0, so if k admits a k-embedding into K then dim G < [k : k] dim G [K : k] dim H = dim R K/k (H ), yielding a contradiction for dimension reasons. It follows that k is not contained in K over k. Let L ⊆ k be the ﬁeld of deﬁnition over k for the subgroup Ru (Gk ) ⊆ Gk . Since the k-subgroup Ru (R K/k (H )k ) in R K/k (H )k is deﬁned over K, and by hypothesis there is a k-isogeny R K/k (H ) G, we deduce that L ⊆ K over k. We have seen that k is not contained in K over k, so to get a contradiction from the assumed existence of (K/k, H ) it sufﬁces to show L = k as purely inseparable extensions of k. This determination of the ﬁeld of deﬁnition L/k is the goal of the remainder of the argument. Obviously L ⊆ k . The ﬁeld of deﬁnition over k for the unipotent radical of the k-ﬁber of a smooth connected afﬁne k-group is unaffected by an étale k-isogeny, so we can replace Z with Z 0 . That is, we may assume that μ is inﬁnitesimal, so μ ⊆ ZG0 . By Galois descent we may assume that k is separably closed. In this case G descends to a connected semisimple k-group G (Corollary A.4.11), and G is necessarily absolutely simple and simply connected. Also, G is naturally a k-subgroup of R k /k (G ), so Z ∩ ZG0 makes sense. Since ZG0 = (ZG0 )k and ZG0 is a ﬁnite multiplicative group over the separably closed ﬁeld k, Z ∩ ZG0 is the unique (inﬁnitesimal) k-subgroup M ⊆ ZG0 such that Mk = μ in ZG . But the k-subgroup Z/M = Z/(Z ∩ ZG0 ) ⊆ R k /k (ZG0 )/ZG0

1.4 The standard construction and examples

33

is a connected central unipotent subgroup scheme in R k /k (G )/(Z ∩ ZG0 ), so its k-ﬁber is contained in the unipotent radical of (R k /k (G )/M )k even though Z/M is generally not smooth. (Here we use that the center of a connected reductive group is of multiplicative type in the sense of Deﬁnition A.1.5 and so has no nontrivial unipotent subgroups, by Example A.1.7.) Thus, the geometric unipotent radicals of R k /k (G )/M and G = R k /k (G )/Z have the same ﬁeld of deﬁnition over k. It therefore sufﬁces to show that the unipotent radical of (R k /k (G )/M )k has ﬁeld of deﬁnition k over k. The composite quotient map R k /k (G )k → G → G /μ = (G /M )k

(1.4.5)

over k kills Mk = μ, and the resulting quotient map (R k /k (G )/M )k (G /M )k over k descends the maximal reductive quotient over k. Thus, the ﬁeld of deﬁnition over k for the geometric unipotent radical of R k /k (G )/M is the same as the ﬁeld of deﬁnition over k for the kernel of (1.4.5). By Theorem 1.6.2(1) this latter ﬁeld of deﬁnition over k is equal to k , so we are done. Remark 1.4.8 Consider a standard pseudo-reductive k-group G and a 4-tuple (G , k /k, T , C) from which it arises. Let G := G /ZG be the maximal adjoint semisimple quotient of the reductive k -group G with connected ﬁbers, and T = T /ZG . From the construction, there is a natural left action of G on R k /k (G ) lifting its action on the central quotient R k /k (G )/(ker φ) ⊆ G that is normal in G. We claim that there is a natural map f : G → R k /k (G ) with central kernel which respects the left actions of both terms on R k /k (G ).

ψ

Using the compatible maps G → G , T → T , and C → R k /k (T ), we get a map of k-groups f : G = (R k /k (G ) C)/R k /k (T )

→ (R k /k (G ) R k /k (T ))/R k /k (T ) R k /k (G ), and this respects the actions on R k /k (G ). The kernel of f is Q/R k /k (T ), where Q is the group of pairs (g, c) such that g ∈ R k /k (G ) maps to some −1 t ∈ R k /k (T ) ⊆ R k /k (G ) and c ∈ C maps to t . Since the preimage of T in G is T , ker f is identiﬁed with ker(ψ : C → R k /k (T )). This is central in G due to the deﬁnition of how C acts on R k /k (G ). Example 1.4.9 We now give a non-commutative counterpart to Example 1.3.2. Our aim is to construct a non-commutative standard pseudo-reductive k-group

34

Overview of pseudo-reductivity

G and a smooth connected normal k-subgroup N such that G/N is not pseudoreductive. Let G be as in Deﬁnition 1.4.4, with G → Spec k a semisimple and simply connected k -group whose ﬁbers are absolutely simple and connected. Let N ⊆ G be the image of R k /k (G ) in G, so this is a smooth connected normal ksubgroup of G. Observe that N is perfect, by Corollary A.7.11. The quotient G/N is identiﬁed with the commutative coker φ, where φ : R k /k (T ) → C is the map in (1.4.1), so N = D (G). Take C = R k /k (T /ZG ), so φ restricts to an isogeny between maximal k-tori. Hence, coker φ is unipotent, so G/N fails to be pseudo-reductive when coker φ = 1. By dimension reasons, coker φ = 1 precisely when dim(ker φ) > 0. This occurs when k /k is a nontrivial purely inseparable ﬁeld extension and ZG is not k -étale. For example, if k is any imperfect ﬁeld of characteristic p > 0 then we can take G = SLp . Such examples of the failure of G/N to inherit pseudo-reductivity from G are due to G/N being unipotent. It would be more interesting to ﬁnd examples in which G/N is not pseudo-reductive but is its own derived group. In 1.6.4 we will construct such examples, even with perfect G. In any such example, the preimage N of Ru,k (G/N ) in G is a smooth connected normal ksubgroup of G such that N = D (N ). This is surprising, since in the connected reductive case, if G is perfect then so are all of its smooth connected normal subgroups.

1.5 Main result Modulo the intractable case of commutative pseudo-reductive k-groups, the pushout construction of standard pseudo-reductive groups leads to a description of all pseudo-reductive k-groups aside from some exceptional situations in characteristics 2 and 3. Our main result is Theorem 5.1.1, and setting aside some complications in characteristics 2 and 3 it is: Theorem 1.5.1 Let k be a ﬁeld, and let G be a pseudo-reductive k-group. If k is imperfect and p = char(k) ∈ {2, 3} then assume that the Dynkin diagram of the maximal semisimple quotient G ss := Gk /R(Gk ) over k has no edge with k multiplicity p, and no isolated vertices when p = 2. The pseudo-reductive k-group G is standard. In this formulation, the simple factors of G ss ruled out from consideration k over imperfect ﬁelds of characteristic p ∈ {2, 3} are type G2 when p = 3 and

1.5 Main result

35

types B, C, and F4 when p = 2. In Theorem 5.1.1 we make no such restrictions, but we do assume [k : k 2 ] 2 when char(k) = 2. In Chapter 4 we will prove (in arbitrary characteristic) that the data (G , k /k) entering into the construction of standard pseudo-reductive groups in Deﬁnition 1.4.4 can be uniquely chosen to satisfy some additional properties. In this sense, we get a classiﬁcation of pseudo-reductive k-groups up to k-isomorphism in the spirit of the classiﬁcation of connected reductive groups except that (i) we have to treat the commutative case as a black box (corresponding to the unknown structure of the commutative pseudo-reductive centralizers of maximal k-tori), (ii) we have to carry along a new invariant, a nonzero ﬁnite reduced k-algebra k (akin to k in Example 1.3.5) as part of the classiﬁcation, and (iii) over imperfect ﬁelds of characteristics 2 and 3 we need special constructions to handle exceptional phenomena related to type G2 in characteristic 3 and types B, C, and F4 in characteristic 2. (Brieﬂy, in these low characteristics some commutation relations among root groups degenerate or some roots are divisible by p in the character lattice.) Example 1.5.2 Let us explain how connected semisimple groups G over a ﬁeld k are described from the viewpoint of the pushout construction of standard pseudo-reductive groups. This is a variant on the classical description of G as of its simply connected central cover G modulo a ﬁnite central a quotient G/μ (hence multiplicative) k-subgroup scheme μ. The central μ is contained in Pick any maximal k-torus T ⊆ G. T . The by conjugation, and the k-torus image C = T /μ of T in G acts on the left on G natural multiplication map C)/ =G (G T → G/μ C via is an isomorphism, where T embeds as a central k-torus in G α : t → (t −1 , t mod μ). C)/ The quotient presentation (G T of G is what is provided by the general procedure to describe pseudo-reductive k-groups in the proof of Theorem 1.5.1, = R k /k (G ) for a nonzero ﬁnite étale k-algebra k and due to the fact that G k -group G whose ﬁbers are absolutely simple and simply connected (Proposition A.5.14). Note that we can choose T to be the maximal k-torus R k /k (T ) for any maximal k -torus T in G . Also, C := T /μ is a maximal k-torus in G and hence is a Cartan k-subgroup of G. It is the viewpoint of Cartan k-subgroups that indicates the analogue of C in the general pseudo-reductive case.

36

Overview of pseudo-reductivity

Due to the appearance of the inversion in the ﬁrst component of the deﬁnition of C α: T →G (rather than in the second component), the natural map Ext(G, μ) → Ext(G, T) of abelian groups of central extensions (arising from the inclusion of μ into T) carries the extension class →G→1 1→μ→G to the negative of the extension class α C → G → 1. 1→ T →G

C as deﬁned above rather than We prefer to use the embedding α of T into G −1 T . The the embedding t → (t, t mod μ) obtained from α via inversion on reason is that α is the embedding that works in more general situations in which T is replaced with a non-commutative group and we use a semidirect product structure against a left action (as opposed to a right action); cf. Remark 1.4.5.

1.6 Weil restriction and ﬁelds of deﬁnition In order to prove that a pseudo-reductive k-group G arises from the standard construction, at least away from certain exceptional situations in characteristics 2 and 3, we need a method to use G to reconstruct the k-algebra k appearing in such a standard construction. The place to look is in ﬁelds of deﬁnition over k for certain subgroups of Gk . Thus, we now undertake a preliminary study of ﬁelds of deﬁnition associated to Weil restrictions. The most basic example of a ﬁeld of deﬁnition over k that naturally arises in any attempt to classify pseudo-reductive groups over a ﬁeld k is the ﬁeld of deﬁnition over k for the k-subgroup Ru (Gk ) ⊆ Gk . This is interesting when G arises as a purely inseparable Weil restriction: Example 1.6.1 Let k /k be a ﬁnite purely inseparable extension of ﬁelds, and let G be a smooth connected non-unipotent (resp. non-solvable) k -group. An example of interest is when G is nontrivial and reductive (resp. nontrivial and semisimple). Consider the Weil restriction G = R k /k (G ). By Proposition A.7.8(2), the (minimal) ﬁeld of deﬁnition over k for Ru (Gk ) (resp. R(Gk )) as a subgroup of Gk contains k . In particular, if G is nontrivial

1.6 Weil restriction and ﬁelds of deﬁnition

37

and reductive then the k-group G is not reductive when k = k, although it is pseudo-reductive (by Proposition 1.1.10). A natural ﬁrst attempt to describe a general pseudo-reductive k-group G, at least when G red acts irreducibly on its Lie algebra (to avoid G being a Weil k restriction from a nontrivial product of ﬁnite extensions of k or from a separable nontrivial ﬁnite extension of k), is as follows. Let G be any smooth connected afﬁne k-group. Let k ⊆ k be the ﬁeld of deﬁnition over k for the k-subgroup Ru (Gk ) ⊆ Gk . The extension k /k is ﬁnite and purely inseparable. Let Ru (Gk ) denote the descent of Ru (Gk ) to a smooth connected unipotent normal k -subgroup of Gk . For the connected reductive k -group G = Gk /Ru (Gk ) there is a natural homomorphism iG : G → R k /k (G )

(1.6.1)

arising from the πG : Gk → G via the universal property of Weil restriction; we studied this map in some examples in §1.3. Concretely, iG is the composition of R k /k (πG ) with the canonical map G → R k /k (Gk ). It follows from Theorem 1.6.2(2) below that such a G arises as the Weil restriction of a connected reductive group over a purely inseparable extension of k if and only if iG is an isomorphism. Our further study of iG will make extensive use of the natural map qX : R k /k (X )k → X from (A.5.3) for quasi-projective k -schemes X (especially smooth afﬁne k -groups). We refer the reader to Proposition A.5.7 for an explicit description of qX at the level of points valued in k-algebras, and to Proposition A.5.11 for its properties when X is k -smooth (e.g., qX is a smooth surjection). The following theorem is the source of all ﬁne control that we can exert on minimal ﬁelds of deﬁnition over k, a crucial aspect of many of our later proofs concerning the classiﬁcation of pseudo-reductive groups. Theorem 1.6.2 Let k /k be a ﬁnite extension of ﬁelds, G a nontrivial smooth connected k -group, and G = R k /k (G ). G be a surjective k -homomorphism from a smooth connected (1) Let G . Assume that the induced map of Lie algebras is nonzero. Then k -group G the scheme-theoretic kernel of the natural surjective composite k -map qG

)k −→ G −→ G π : R k /k (G has ﬁeld of deﬁnition over k equal to k . (2) If k /k is purely inseparable and G is reductive then the natural quotient map q : Gk = R k /k (G )k → G is surjective with smooth connected

38

Overview of pseudo-reductivity unipotent kernel (so G = Gk /Ru,k (Gk )). The k-subgroup Ru (Gk ) ⊆ Gk has ﬁeld of deﬁnition over k equal to k ⊆ k and the map iG in (1.6.1) is an isomorphism.

(3) Assume G is semisimple and absolutely simple over k . Let q : Gk G be the natural quotient map. The kernel ker q ⊆ Gk has ﬁeld of deﬁnition over k equal to k . As σ varies through all k-embeddings of k into k, the quotients Qσ = G ⊗k ,σ k of Gk are (without repetition and up to unique isomorphism) the maximal simple semisimple quotients of Gk . In particular, the subgroup ker(Gk Qσ ) in Gk has ﬁeld of deﬁnition over k equal to σ (k ) ⊆ k. The case of most interest in the ﬁrst assertion is when G is semisimple and → G is a central isogeny (especially the simply connected central cover of such a G ), in which case the non-vanishing hypothesis always holds. Note that in (2) there is no ambiguity concerning the identiﬁcation of k with the ﬁeld of deﬁnition over k in k for Ru (Gk ) since k /k is purely inseparable (so k has a unique k-embedding into k).

G

Proof. We begin by proving the ﬁrst assertion in which G can be any smooth connected k -group (not necessarily connected reductive, nor even afﬁne). Since q := qG π is is a smooth surjection (Proposition A.5.11), it follows that nonzero on Lie algebras. Thus, V = Lie(ker π ) is a proper k -subspace of Lie(R k /k (G )k ) = k ⊗k Lie(G ) that contains Lie(ker q) = ker(Lie( q)) = ker(k ⊗k Lie(G ) → Lie(G )). The ﬁeld of deﬁnition over k for ker π certainly contains the ﬁeld of deﬁnition over k for the k -subspace V in k ⊗k Lie(G ), so by Lemma A.7.7 (with K = k ) this ﬁeld is k . Turning to (2), assume that k /k is purely inseparable. According to Proposition A.5.11(2) the natural quotient map q : Gk → G is surjective with smooth connected unipotent kernel. Thus, the map q over k descends the maximal reductive quotient of R k /k (G )k , so (ker q)k = Ru (Gk ). By Proposition A.7.8 the ﬁeld of deﬁnition over k for Ru (Gk ) is k /k, so by deﬁnition iG corresponds (under the universal property of Weil restriction) to the k -descent q of the projection Gk G red onto the maximal reductive quotient. But the deﬁnition k of q is that it corresponds (under the universal property of Weil restriction) to R k /k (id G ), so iG is an isomorphism since id G is an isomorphism. This completes the proof of (2). Finally, we consider (3), so now k /k is any ﬁnite extension. Let K/k be the separable closure of k in k , so by applying (2) to k /K we see that the geometric unipotent radical of the K-group R k /K (G ) has ﬁeld of deﬁnition over K equal

1.6 Weil restriction and ﬁelds of deﬁnition

39

to k . (This identiﬁcation is unambiguous because k has a unique K-embedding into an algebraic closure of K, as k /K is purely inseparable.) The geometric ﬁber Gk = G ⊗k k is (R K/k (R k /K (G )))k =

σ

R k /K (G ) ⊗K,σ k,

(1.6.2)

with σ ranging through the k-embeddings of K into k. Each such embedding σ uniquely lifts to a k-embedding σ : k → k, so the maximal semisimple quotient of Gk is σ (G ⊗k ,σ k). This is a product of k-simple semisimple groups, so any choice of maximal k-simple semisimple quotient Gk H is uniquely isomorphic to the projection σ from Gk onto some factor G ⊗k ,σ k. The projection σ is q ⊗k ,σ k, and the determination of the ﬁeld of deﬁnition of ker q over k is a special case of Proposition A.7.8(1) (taking K = k and H = ker q there). 2 The criterion in Theorem 1.6.2(2) yields some commutative pseudo-reductive groups G in characteristic p > 0 with the interesting property that their ﬁnite subgroup of rational p-torsion points is nontrivial and ker iG = 1. Example 1.6.3 Let k /k be a nontrivial purely inseparable ﬁnite extension of ﬁelds with char(k) = p > 0, and assume k ⊆ k 1/p . Let pn = [k : k]. Let C = R k /k (GL1 ). By Example 1.3.2, C is a pseudo-reductive k-group and U := C/GL1 is a smooth connected unipotent k-group of dimension pn −1 > 0. The isogenous quotient C/μp of C fails to be pseudo-reductive over k because it is commutative and contains the k-group scheme R k /k (μp )/μp that maps isomorphically onto U (since U is p-torsion, as k p ⊆ k). Loosely speaking, C is k × viewed as a k-group. Choose a p-basis {a1 , . . . , an } p of k over k, so ai ∈ k is not a p-power in k and k = k(a1 , . . . , an ) with [k : k] = pn . Thus, we have a k -algebra isomorphism k ⊗k k k [t1 , . . . , tn ]/(t1 , . . . , tn ), p

p

where ti corresponds to 1 ⊗ ai − ai ⊗ 1. In other words, C is a k-descent p of the unit group (k [t1 , . . . , tn ]/(ti ))× viewed as a k -group and U is a kpn −1 as k -groups. descent of the subgroup of 1-units. We claim that Uk Ga To prove this, ﬁrst note that the maximal ideal m of k ⊗k k is killed by the p-power map. Hence, for x ∈ m the truncation exp

40

Overview of pseudo-reductivity

conclude that the group Ext k (Uk , Z/pZ) of commutative extensions over k n is isomorphic to Ext k (Ga , Z/pZ) p −1 , so there exist connected commutative k -group extensions E of Uk by Z/pZ (via Artin–Schreier theory). Choose such an extension E , so we can view E as a ﬁnite étale cover of Uk (with a canonical trivialization over the identity) and view E × E as a ﬁnite étale cover of Uk × Uk = (U × U )k . The group law on E corresponds to a map E × E m∗Uk (E ) over (U ×U )k satisfying an evident associativity condition, where mUk : Uk × Uk → Uk is the multiplication map. Since k /k is purely inseparable, for any k-scheme S (e.g., Spec k, U , or U × U ), it follows from [SGA1, IX, Thm. 4.10] that pullback along Sk → S induces an equivalence of étale sites. The extension structure E thereby canonically descends to a commutative extension E of U 1 (U , Z/pZ) = H1 (U , Z/pZ).) by Z/pZ over k. (This makes explicit that Hét k ét In particular, E is a smooth connected unipotent commutative k-group. The ﬁber product G = E ×U C is an extension of E by ker(C U ) = GL1 , so G is a smooth connected commutative afﬁne k-group for which Ru (Gk ) Ek . Also, G → C is an étale isogeny with covering group Z/pZ, so G is pseudoreductive over k (since C is) and the ﬁeld of deﬁnition over k for the subgroup Ru (Gk ) ⊆ Gk is the same as that of Ru (Ck ) ⊆ Ck , which is to say k (by Theorem 1.6.2(2)). Via the isomorphism Gk /Ru (Gk ) Ck /Ru (Ck ) = GL1 , the map iG is the natural map G C that is an isogeny with kernel Z/pZ, so ker iG = 1. It also follows that G cannot be k-isomorphic to R k /k (G ) for a nonzero ﬁnite reduced k-algebra G and a k -group G with connected reductive ﬁbers. Indeed, otherwise the commutativity of G(k) = G (k ⊗k ks ) would force G to be a k -torus, which contradicts that G(k)[p] = 1. Example 1.6.4 Let k be an imperfect ﬁeld with char(k) = p > 0. We now make an example of a perfect pseudo-reductive k-group G and perfect smooth connected normal k-subgroup N such that G/N is not pseudo-reductive. As we pointed out at the end of Example 1.4.9, such a G contains a smooth connected

1.6 Weil restriction and ﬁelds of deﬁnition

41

normal k-subgroup N that is not perfect, which never happens in connected semisimple groups. Let k /k be a nontrivial purely inseparable ﬁnite extension such that k ⊆ 1/p k . Let G be a connected semisimple k -group that is absolutely simple and simply connected such that ZG contains a k -subgroup μ that is a k -form of μp2 . (Some such G with ZG = μp2 are G = SLp2 for any p and the spin double covering of SO2n for odd n > 3 when p = 2.) Let T be a maximal k -torus in G , and deﬁne the k-subgroup Z ⊆ R k /k (T ) × R k /k (T ) to correspond to the k-subgroup inclusion R k /k (μ ) → R k /k (T ) × R k /k (T ) deﬁned by ζ → (ζ , ζ p ). Deﬁne k = k × k , the k -group G = G G , and its maximal k torus T = T T . Observe that Z ⊆ R k /k (ZG ) inside of R k /k (T ). We wish to use the smooth connected commutative k-group C = R k /k (T )/Z to construct a standard pseudo-reductive k-group, so let us now check that C is pseudo-reductive over k. Using the exact sequence of k-groups j2

q1

1 → R k /k (T ) → R k /k (T ) → R k /k (T ) → 1 deﬁned via j2 (t ) = (1, t ) and q1 (t1 , t2 ) = t1 , since the image of j2 meets Z trivially we get an induced exact sequence 1 → R k /k (T )→C→R k /k (T )/R k /k (μ ) → 1.

(1.6.3)

To prove that C is pseudo-reductive it sufﬁces to prove pseudo-reductivity for the outer terms. For the ﬁrst term this is clear, and the third term is a smooth connected k-subgroup of the commutative pseudo-reductive k-group R k /k (T /μ ), so it is pseudo-reductive. Hence, C is indeed pseudo-reductive over k. Using the natural quotient map φ : R k /k (T ) C, we deﬁne the standard pseudo-reductive k-group G = (R k /k (G ) C)/R k /k (T ). By Corollary A.7.11, R k /k (G ) is its own derived group. But since φ is surjective, we see that G = R k /k (G )/Z, so G is perfect. Let N ⊆ G be the smooth connected normal k-subgroup that is the image in G of the ﬁrst factor R k /k (G ) of R k /k (G ), so N is perfect. We shall prove that the quotient G/N is not pseudo-reductive over k.

42

Overview of pseudo-reductivity

Consider the left-exact sequence 1 → μ [p] → μ → T in which the ﬁnal map is the p-power map on T restricted to μ . Applying the left-exact Weil restriction functor to k then gives a left-exact sequence 1 → R k /k (μ [p]) → R k /k (μ ) → R k /k (T ). Hence, we get an induced map M := R k /k (μ )/R k /k (μ [p]) → R k /k (T ) that has trivial kernel, so it is a closed k-subgroup inclusion (by Proposition A.2.1). In view of the construction of G and N , as well as the exact sequence (1.6.3), there is a k-group isomorphism G/N (R k /k (G ) C)/R k /k (T )

(1.6.4)

with C = R k /k (T )/M . This presentation of G/N looks like the standard construction but beware that we have not addressed whether or not C is pseudoreductive over k (and it will not be). We shall now use (1.6.4) to exhibit a nontrivial smooth connected central unipotent k-subgroup of G/N (even contained in C), thereby proving that G/N is not pseudo-reductive over k. The k-subgroup M → R k /k (T ) ⊆ R k /k (G ) is central, as it is the (scheme-theoretic) image of the p-power map on the central k-subgroup scheme R k /k (μ ). The natural map R k /k (G ) → G/N has kernel equal to M and so gives rise to the central k-subgroup Q = R k /k (μ )/M in G/N . It therefore sufﬁces to prove that Q contains a nontrivial smooth connected unipotent k-subgroup. If such a k-subgroup exists then there is a unique maximal one containing all others, so by Galois descent we may and do assume k = ks . In particular, μ = μp2 . It remains to prove that the quotient Q = R k /k (μp2 )/[p](R k /k (μp2 )) contains a nontrivial smooth connected unipotent k-group. Since k p ⊆ k, the geometric unipotent radical of R k /k (GL1 ) is killed by p. That is, the p-power map on R k /k (GL1 ) lands inside of the unique GL1 -subgroup. More speciﬁcally, we have a canonical isomorphism of k-groups R k /k (GL1 )k = GL1 × U with pU = 0. Hence, for any n 1 the pn -torsion R k /k (μpn )k of R k /k (GL1 )k is μpn × U . Taking n = 2, the scheme-theoretic image of R k /k (μp2 )k under the p-power map is the canonical subgroup μp over k, so likewise the k-subgroup μp inside of R k /k (μp2 ) is the scheme-theoretic image of the p-power map. It follows that Q = R k /k (μp2 )/μp . This contains R k /k (μp )/μp , and we saw in Example 1.6.3 that this k-group is isomorphic to the k-group R k /k (GL1 )/GL1 that is nontrivial, smooth, connected, and unipotent.

2 Root groups and root systems

The category of pseudo-reductive groups over a ﬁeld k has weak stability properties under basic operations on group schemes, such as extension of the base ﬁeld and passage to quotients (even by ﬁnite central multiplicative or smooth normal subgroups). Fortunately, pseudo-reductive groups do admit many properties generalizing those of connected reductive groups, and this chapter establishes such properties for convenient use later on. Following Tits, we use an interesting construction of unipotent groups to construct a theory of root systems and root groups for pseudo-reductive groups over any separably closed ﬁeld, or more generally for pseudo-split pseudoreductive groups, i.e., the ones which contain a split maximal torus (over an arbitrary ground ﬁeld). Tits developed a version of this theory without requiring a split maximal torus, but it will sufﬁce for our needs to have this theory for pseudo-split groups. In [Spr, Ch. 13–15] an exposition of Tits’ theory is given with an arbitrary base ﬁeld. We provide an exposition in order to make our work more self-contained and to incorporate some scheme-theoretic improvements into the theory. In contrast with the connected reductive case, root groups in pseudo-split pseudo-reductive groups over a ﬁeld k may have dimension larger than 1, though they turn out to always be vector groups over k (i.e., k-isomorphic to a power of Ga ). Our structure theorem for pseudo-reductive groups explains a posteriori why root groups can have large dimension: they turn out to naturally be Weil restrictions of 1-dimensional vector groups over a ﬁnite extension of k (if we assume [k : k 2 ] 2 when char(k) = 2.)

2.1 Limits associated to 1-parameter subgroups The construction of root groups rests on a generalization to pseudo-reductive groups of the theory of parabolic subgroups and their unipotent radicals in 43

44

Root groups and root systems

connected reductive groups. Recall that a k-subgroup P in a smooth connected afﬁne group G over a ﬁeld k is called parabolic if P is smooth and G/P is k-proper. This deﬁnition turns out not to be so useful in the pseudo-reductive case, essentially because Weil restriction through a nontrivial purely inseparable ﬁnite extension of ﬁelds never preserves parabolicity (for proper subgroups), as it never preserves properness for smooth quasi-projective schemes of positive dimension; see Example A.5.6. The viewpoint that works well in the pseudo-reductive case is a variant on the classiﬁcation of parabolic k-subgroups of connected reductive groups in terms of relative root systems and centralizers of k-split tori. This variant involves choosing a 1-parameter k-subgroup λ : GL1 → G (i.e., a k-homomorphism) and studying the limiting behavior of λ(t)gλ(t)−1 as t → 0 in an appropriate sense. Actually, we can consider more generally the case of an abstract left action of GL1 on the k-group G and study the limiting behavior of t → t.g as t → 0 in an appropriate sense. This general case with abstract actions can be studied in more concrete terms using conjugation by 1-parameter subgroups in the semidirect product GL1 G deﬁned by the given abstract action. For clarity of exposition, we therefore initially carry out the development for limits associated to 1-parameter subgroups, and in Remark 2.1.11 we address the generalization to limits associated to abstract left actions by GL1 . The basic features of the theory of such limits are best illustrated by two examples (each of which will be useful in later general considerations). Example 2.1.1 Let k be a ring, and G = GL(V ) for a ﬁnitely generated projective k-module V . Let λ : GL1 → G be a k-homomorphism. By Lemma A.8.8, there is a canonical decomposition

V = Vχi , where {χi } is the ﬁnite set of distinct weights for the GL1 -action on V via λ. In particular, the Vχi are ﬁnitely generated projective k-modules. The pairings χi , λ := χi ◦ λ ∈ End(GL1 ) are locally constant Z-valued functions on Spec k. Working Zariski-locally on Spec k for now, we may arrange that these pairings take constant values and that the Vχi are free k-modules. In particular, we may and do arrange the χi so that the values χi , λ are strictly decreasing in i. Assume also that V has rank > 0. Consider the parabolic k-subgroup PG (λ) consisting of automorphisms of V respecting the increasing ﬁltration by subspaces Fj = ij Vχi . Choose an

2.1 Limits associated to 1-parameter subgroups

45

ordered k-basis {v1 , . . . , vn } of V adapted to this ﬁltration: each vr is in a weight space Vχir and i1 . . . in . Relative to such a basis we have G = GLn with λ(t) = diag(t e1 , . . . , t en ) for er = χir , λ and e1 . . . en . If g = (ars ) in GL(V ) GLn then g is a point of PG (λ) if and only if g(Vχj ) ⊆ Fj for all j 1, which is to say that ars = 0 whenever ir > is . But it is elementary to compute that for any g, λ(t)gλ(t)−1 = (t er −es ars ) = (t χir ,λ−χis ,λ ars ),

(2.1.1)

so the rs-entry has negative t-exponent precisely when ir > is . Now returning to the case of general k, V , and λ (without assuming the existence of bases or constancy of the χi , λ as above), we conclude that PG (λ) represents the functor of points g ∈ G(R) (for any k-algebra R) such that the R-map (GL1 )R → GR deﬁned by t → λ(t)gλ(t)−1 extends (necessarily uniquely) to an R-map AR1 → GR . We summarize this latter condition by saying that lim λ(t)gλ(t)−1

t→0

exists, and the image of 0 in G(R) is considered to be the limit. This limit procedure functorially characterizes the parabolic k-subgroup PG (λ) entirely in terms of the 1-parameter k-subgroup λ : GL1 → G without any mention of the coset space G/PG (λ) (or its properties). This paves the way for a viewpoint that will be useful when G is replaced with a general smooth afﬁne k-group whose ﬁbers over Spec k are connected. We can give a similar functorial description of the “k-split unipotent radical” UG (λ) of the parabolic k-subgroup PG (λ): it consists of automorphisms acting trivially on the graded space gr • (V ) =

(Fj /Fj−1 )

Vχj

for the ﬁltration {Fj } of V . Indeed, by (2.1.1) we see that for any k-algebra R, if g ∈ PG (λ)(R) then limt→0 λ(t)gλ(t)−1 lies in the subgroup GL(Vχj ) and is the automorphism induced by g on gr • (V ). The k-subgroup UG (λ) represents the functor of points of PG (λ) with trivial action on gr • (V ), so UG (λ) is the functor of points g of G such that λ(t)gλ(t)−1 → 1 as t → 0. This functorial description of UG (λ) only uses the 1-parameter subgroup λ and makes no reference to unipotence (or to PG (λ)).

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Example 2.1.1 can be pushed a bit further, by using the reciprocal 1-parameter k-subgroup −λ : GL1 → G deﬁned by t → λ(1/t). The scheme-theoretic intersection PG (λ)∩PG (−λ) is the subgroup of block matrices GL(Vχj ) ⊆ G that is exactly the scheme-theoretic centralizer ZG (λ) of λ in G. By inspection, the natural multiplication map ZG (λ) UG (λ) → PG (λ) is an isomorphism of k-groups; i.e., ZG (λ) serves as a quotient PG (λ)/UG (λ). Before moving on to the next example, we directly prove the following result without reference to the structure theory of reductive groups (such as Bruhat cells): Proposition 2.1.2 Let k be a ring and V a ﬁnitely generated projective k-module. For G = GL(V ) and a k-homomorphism λ : GL1 → G, the multiplication map of k-schemes μ : UG (−λ) × PG (λ) → G is an open immersion. In Proposition 2.1.8(3) this will be proved for any smooth afﬁne group over any ring, and any afﬁne group of ﬁnite type over any ﬁeld. Proof. The intersection UG (−λ) ∩ PG (λ) is trivial by inspection, so μ is a functorial injection (i.e., monomorphism). By [EGA, IV4 , 17.9.1], an open immersion is the same thing as an étale monomorphism, so to prove that μ is an open immersion it sufﬁces to prove that it is étale. Since each of UG (−λ), PG (λ), and G are smooth, μ is étale if it is an isomorphism on tangent spaces at all rational points of UG (−λ) × PG (λ) in geometric ﬁbers over Spec k. Hence, we can assume that k is an algebraically closed ﬁeld, and by translation considerations it sufﬁces to check the tangential isomorphism property at a single k-point of UG (−λ) × PG (λ). We choose the identity point. At the identity, the desired tangential isomorphism property says that the addition map dμ(1,1) : Lie(UG (−λ)) ⊕ Lie(PG (λ)) → Lie(G) is an isomorphism of k-vector spaces. By inspection, Lie(UG (−λ)) is the span of the negative weight spaces in Lie(G) for the GL1 -action induced by g → λ(t)gλ(t)−1 . Likewise, Lie(PG (λ)) is the span of the non-negative weight spaces for this GL1 -action. Hence, dμ(1,1) is an isomorphism. 2 To indicate how Example 2.1.1 will carry over to pseudo-reductive groups, we consider a special case.

2.1 Limits associated to 1-parameter subgroups

47

Example 2.1.3 Let k /k be a ﬁnite extension of ﬁelds, and let G = R k /k (SL2 ). This naturally contains SL2 as a k-subgroup. Let λ : GL1 → G be the k-homomorphism that is the standard 1-parameter diagonal subgroup of this SL2 ; that is, deﬁne λ(t) = diag(t, 1/t). Inside of the k -group SL2 , let D be the diagonal k -subgroup, B the upper triangular Borel k -subgroup, and U the unipotent radical of B . Note that if k /k is purely inseparable then the k-subgroup SL2 in G is a Levi k-subgroup and λ(GL1 ) is a maximal k-torus in G. Consider the subgroups D = R k /k (D ), B = R k /k (B ), and U = R k /k (U ) in G. If k /k is a nontrivial purely inseparable extension then the quotient B/U = D is pseudo-reductive but not reductive (so U = Ru,k (B) is not a kdescent of Ru (Bk )) and G/B = R k /k (G /B ) = R k /k (P1k ) is not proper over k (see Example A.5.6). However, for any k /k and any k-algebra R we have

x y x t2y λ(t) λ(t)−1 = −2 z w t z w in G(R) = SL2 (k ⊗k R) for any t ∈ R× = GL1 (R). Thus, in general D is the centralizer of λ in G, B represents the functor of points g of G such that λ(t)gλ(t)−1 has a limiting value as t → 0, and U represents the functor of points g of G such that this limit exists and equals 1. In this sense, B is analogous to a parabolic subgroup, U is analogous to its unipotent radical, and D is analogous to a Levi subgroup of B. Motivated by Examples 2.1.1 and 2.1.3, we seek to study limits limt→0 t.v for the action of GL1 on an afﬁne k-scheme V for any ring k. Since GL1 is “dual” to Z, by Lemma A.8.8 the coordinate ring k[V ] of V admits an associated weight space decomposition

k[V ] = k[V ]n (2.1.2) n∈Z

with k[V ]n consisting of the f ∈ k[V ] such that f (t.v) = t n f (v) in R for all k-algebras R, v ∈ V (R), and t ∈ GL1 (R) = R× ; such an f is called an n-eigenvector for the GL1 -action on V . Lemma 2.1.4 Let k be a ring and V be an afﬁne k-scheme equipped with a GL1 -action. Deﬁne the ideal J := k[V ] · k[V ]n n<0

in k[V ]. Let V = Spec(k[V ]/J ) denote the zero scheme of J in V .

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Root groups and root systems

The closed subscheme V in V represents the subfunctor of V that associates to any k-algebra R the set of points v ∈ V (R) such that limt→0 t.v exists in the sense that the orbit map αv : (GL1 )R → VR over R deﬁned by t → t.v extends (necessarily uniquely) to an R-morphism AR1 → VR . If K is any k-algebra then VK represents the corresponding subfunctor of VK on the category of Kalgebras, and the formation of V commutes with passage to any GL1 -stable closed subscheme Z ⊆ V (i.e., Z = Z ∩ V ). For v ∈ V (R), the image of 0 in V (R) under the extended map AR1 → VR is denoted limt→0 t.v in what follows. Proof. For f ∈ k[V ], let fn be its component along k[V ]n (so all but ﬁnitely many fn vanish). The R-map αv corresponds to the k-algebra map αv∗ : k[V ] → R[t, 1/t] deﬁned by f → n∈Z t n fn (v). Thus, αv extends to an R-morphism AR1 → VR if and only if αv∗ has image contained in the subring R[t] ⊆ R[t, 1/t]. This containment says exactly that for all n ∈ Z we have n 0 whenever f (v) = 0 for some f ∈ k[V ]n . Equivalently, the necessary and sufﬁcient condition is that whenever n < 0 we have f (v) = 0 for all f ∈ k[V ]n . In other words, J vanishes at v ∈ V (R), which is to say that v ∈ V (R). This establishes the functorial characterization of V . By the functorial characterization (or the construction via J ), it follows that the formation of V commutes with any extension of the base ring. The compatibility with passage to GL1 -stable closed subschemes is handled similarly. (In concrete terms, the coordinate ring of such a subscheme is a quotient of k[V ] compatible with the Z-grading.) 2 Let V = G be an afﬁne group scheme over a ring k and let λ : GL1 → G be a k-group homomorphism. Endow G with a GL1 -action via t.g = λ(t)gλ(t)−1 . By Lemma 2.1.4, there is a closed subscheme PG (λ) ⊆ G representing the functor of points g of G for which limt→0 t.g exists in G. Moreover, the lemma ensures that the formation of PG (λ) is compatible with any extension of the base ring and that if H ⊆ G is a closed k-subgroup scheme through which λ factors then H ∩ PG (λ) = PH (λ) (using λ to also denote the homomorphism GL1 → H ). Following standard conventions with coroots in connected reductive groups, we write −λ to denote the reciprocal homomorphism t → λ(1/t) = λ(t)−1 . Lemma 2.1.5 The closed subscheme PG (λ) ⊆ G is a k-subgroup scheme, and ZG (λ) := PG (λ) ∩ PG (−λ)

2.1 Limits associated to 1-parameter subgroups

49

is the scheme-theoretic centralizer in G for the GL1 -action via conjugation by λ (i.e., its R-points for a k-algebra R are the elements g ∈ G(R) that centralize the R-homomorphism λR ). The subfunctor of G deﬁned by R {g ∈ G(R) | lim t.g = 1} t→0

is represented by a closed normal k-subgroup scheme UG (λ) ⊆ PG (λ), and if G is of ﬁnite type over k then UG (λ) has unipotent ﬁbers over Spec k. The formation of UG (λ) commutes with any extension of the base ring, and if H ⊆ G is a closed k-subgroup scheme through which λ factors then H ∩ UG (λ) = UH (λ). If G is ﬁnitely presented over k then so are PG (λ), UG (λ), and ZG (λ). Proof. To check that PG (λ) is a subgroup scheme we compute on R-points for any k-algebra R, as follows. Choose g, g ∈ G(R) such that t.g → g0 and t.g → g0 as t → 0. That is, there are (necessarily unique) R-morphisms f , f : AR1 ⇒ GR whose restrictions to (GL1 )R are respectively t → t.g and t → t.g , and they satisfy f (0) = g0 and f (0) = g0 in G(R). Use the group law on G to deﬁne a product map f · f : AR1 → GR over R. This carries 0 to g0 g0 , and its restriction to (GL1 )R is t → (t.g)(t.g ) = t.(gg ). Hence, gg ∈ PG (λ)(R). A similar argument shows that PG (λ) is stable under inversion and contains the identity section, so it is a k-subgroup of G. This argument also shows that g → limt→0 t.g is a multiplicative k-map PG (λ) → G. Such a map is necessarily a k-group homomorphism, and its scheme-theoretic kernel is exactly the desired UG (λ). The compatibility of UG (λ) with extension of scalars is clear. Explicitly, the ideal of UG (λ) is generated by the n-eigenvectors f ∈ k[G] (i.e., f ∈ k[G]n ) for n 0 such that f (1) = 0 (i.e., f lies in the augmentation ideal of k[G]). The equality H ∩ UG (λ) = UH (λ) holds because H ∩ PG (λ) = PH (λ). To check unipotence of the ﬁbers of UG (λ) over Spec k when G is of ﬁnite type, we may assume k is a ﬁeld. In this case we can choose a k-subgroup inclusion j : G → GLn , so UG (λ) = G ∩ UGLn ( j ◦ λ). Thus, the unipotence of UG (λ) is reduced to the special case G = GLn , where it was seen in Example 2.1.1.

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Root groups and root systems

Consider ZG (λ) in general. To show that it is the centralizer of the GL1 action, we have to prove that for any k-algebra R and g ∈ G(R), g centralizes the map λR : (GL1 )R → GR if and only if g ∈ ZG (λ)(R). Centralizing λR means precisely that t.g = g for all functorial points t of (GL1 )R , and the membership g ∈ ZG (λ)(R) means precisely that limt→0 t.g and limt→0 t −1 .g both exist. Clearly if g centralizes λR then both limits exist (and equal g). Conversely, if both limits exist then the maps t → t.g and t → t −1 .g glue to deﬁne an R-morphism P1R → GR from the projective line to an afﬁne R-scheme. The only such map is a constant R-map, which forces t.g to be independent of t, so t.g = g for all functorial points t of (GL1 )R . If G is ﬁnitely presented over k then G and λ can be descended to a noetherian subring of k. Hence, the compatibility of the formation of PG (λ), UG (λ), and ZG (λ) with scalar extension on k reduces the ﬁnite presentation claim to the case of noetherian k, where it is clear. 2 Example 2.1.6 Let k be a ﬁeld. It is a well-known fact that for a nonzero ﬁnite-dimensional k-vector space V , the parabolic k-subgroups of GL(V ) correspond to ﬂags of subspaces of V . Thus, by Example 2.1.1, in G = GLn the groups PG (λ) as λ varies are exactly the parabolic k-subgroups, with UG (λ) its unipotent radical and ZG (λ) a Levi k-subgroup of PG (λ). For G = R k /k (SL2 ) and the standard λ as in Example 2.1.3, the k-subgroup PG (λ) is the Weil restriction of the standard upper triangular Borel subgroup of the k -group SL2 and the k-subgroup UG (λ) is the Weil restriction of the unipotent radical of this Borel subgroup. Also, ZG (λ) is the Weil restriction of the diagonal k -subgroup of SL2 . Remark 2.1.7 In the setup of Lemma 2.1.5, for any integer n 1 we have PG (λn ) = PG (λ) and UG (λn ) = UG (λ). Indeed, suppose that λ(t n )gλ(t n )−1 → g ∈ G(R) as t → 0 for some k-algebra R and g ∈ G(R). That is, we assume that the R-map (GL1 )R → GR deﬁned by t → λ(t n )gλ(t n )−1 extends to an n) R-map φ : AR1 → GR , with g = φ(0). The problem is to show that φ = φ(x : A1 → GR such that φ| (GL )R is t → λ(t)gλ(t)−1 and for some R-map φ 1 R φ(0) = g . First we address the factorization of φ through the nth-power map. By descent theory, it is necessary and sufﬁcient that φ(x) = φ(y) whenever xn = yn (for functorial points x and y of the afﬁne line over R). The residue classes t, t ∈ R[T , T ]/(T n − T n ) of T and T respectively are not zero divisors, so it sufﬁces to consider only points x and y of the multiplicative group instead of the afﬁne line, in which case the hypothesis on φ gives the desired equality.

2.1 Limits associated to 1-parameter subgroups

51

n ) = φ(x), faithful ﬂatness of the nth-power map on (GL1 )R implies Since φ(x = λ(t)gλ(t)−1 for points t of (GL1 )R . We also have φ(0) = φ(0) = g . that φ(t) The next task is to relate PG (λ), ZG (λ), and UG (λ) to a GL1 -action on the tangent space to G at the identity. We ﬁrst review some elementary facts related to such tangent spaces for general group schemes over any ring. (We will only need the case of afﬁne group schemes.) Let G be a group scheme over a ring k, with e : Spec k → G the identity section. Since sections to scheme maps are locally closed immersions, the section e is the zero locus of a unique quasicoherent ideal IG near e, so IG /IG2 corresponds to a k-module Cot e (G) (of ﬁnite presentation if G is locally of ﬁnite presentation over k), the cotangent space to G along e. Its formation commutes with any base change on k. For the linear dual g = Tane (G) := Homk (Cot e (G), k), there is a natural linear base change morphism k ⊗k Tane (G) → Tanek (Gk )

(2.1.3)

that is an isomorphism whenever G is locally of ﬁnite presentation over k and either k → k is ﬂat or Cot e (G) is k-ﬂat (e.g., k a ﬁeld or G smooth over k). When (2.1.3) is an isomorphism for every k-algebra k , we say that the formation of Tane (G) commutes with any base change on k. As in the classical case over a ﬁeld, there is a natural bijection of sets g = Tane (G) ker(G(k[]) → G(k)) that is functorial in G, and by working locally along e it is easy to check that this is a homomorphism and carries the k-action on the left over to the functorial action through k-scaling on on the right. It identiﬁes the k-linear map Tane (G) → Tanek (Gk ) underlying (2.1.3) with the map induced by G(k[]) → G(k []) for any k-algebra k . We generally write Lie(G) to denote Tane (G) because in §A.7 we endow it with a natural Lie algebra structure over k. However, in this section we will never use the Lie algebra structure. If the formation of the k-module Lie(G) commutes with any base change on k then the conjugation action of G on itself induces an action of G(k ) on Lie(Gk ) = Lie(G)k functorially in k-algebras k . This is the adjoint representation of G on Lie(G). Proposition 2.1.8 Let k be a ring, and G an afﬁne k-group scheme of ﬁnite presentation. Let λ : GL1 → G be a k-homomorphism, and let GL1 act on g = Lie(G) through the adjoint representation of the action t.g = λ(t)gλ(t)−1

52

Root groups and root systems

on G. Deﬁne g+ = ⊕n>0 gn where gn := {v ∈ g | t.v = t n v} for n ∈ Z, and deﬁne g− and g0 similarly. (1) Inside of g we have Lie(ZG (λ)) = g0 , Lie(UG (±λ)) = g± , and Lie(PG (λ)) = g0 ⊕ g+ . (2) The multiplication map ZG (λ) UG (λ) → PG (λ) is an isomorphism of k-groups. (3) Assume k is a ﬁeld. The multiplication map UG (−λ) × PG (λ) → G is an open immersion of k-schemes and if j : G → G is a k-subgroup inclusion into an afﬁne k-group G of ﬁnite type then G ∩ (UG (−λ ) × PG (λ )) = UG (−λ) × PG (λ), where λ = j ◦ λ. The open immersion property holds in general over any ring k if G is k-smooth, in which case PG (λ), UG (λ), and ZG (λ) are all smooth. (4) The ﬁbers of UG (λ) over Spec k are connected, and the same holds for PG (λ) and ZG (λ) if G has connected ﬁbers. Note that Proposition 2.1.8 only uses the underlying k-modules of the Lie algebras. Also, since GL1 is k-ﬂat, the GL1 -action on g makes sense, and the existence of the weight space decomposition of g is a consequence of Lemma A.8.8. Proof. Since ZG (λ) = PG (λ)∩PG (−λ) as subschemes of G, we have the same relation on Lie algebras inside of g. That is, Lie(ZG (λ)) = Lie(PG (λ)) ∩ Lie(PG (−λ)). Hence, to prove (1) it sufﬁces to compute the Lie algebras of UG (λ) and PG (λ). We shall compute using points valued in the k-algebra of dual numbers k[]. Step 1. By the construction of PG (λ), the set PG (λ)(k[]) consists of the points g ∈ G(k[]) that kill k[G]n for all n < 0. But Lie(PG (λ)) = PG (λ)(k[]) ∩ Lie(G) inside of G(k[]). Hence, Lie(PG (λ)) consists of the maps k[G] → k[] of the form f → f (1) + D( f ) that kill k[G]n for all n < 0, with D : k[G] → k a k-linear point derivation at the identity. Since 1 ∈ PG (λ)(k) (i.e., f (1) = 0 for all f vanishing on PG (λ)),

2.1 Limits associated to 1-parameter subgroups

53

the necessary and sufﬁcient condition is that D kills k[G]n for all n < 0. Under the weight space decomposition of g for the GL1 -action, the weight-m subspace gm (consisting of D such that t.D = t m D for all points t of GL1 ) consists of exactly those D killing k[G]n for all n = m. Hence, Lie(PG (λ)) = g0 ⊕ g+ , as desired. Next, we compute Lie(UG (λ)) by using the functorial interpretation of UG (λ). For any v ∈ Lie(PG (λ)) with corresponding weight decomposition 1 n v = n0 vn we have t.v = n0 t vn . Thus, the map Ak[] → Gk[] deﬁned by x →

xn vn

n 0

is the unique extension of the action map t → t.v over k[]. In particular, limt→0 t.v = v0 inside of g ⊆ G(k[]). The property v ∈ Lie(UG (λ)) therefore says exactly that v0 = 0, or equivalently v ∈ g+ . This completes the proof of (1). Step 2. For the proof of (2), recall the k-group homomorphism f : PG (λ) → G deﬁned by g → limt→0 t.g as in the proof of Lemma 2.1.5, where the kernel was identiﬁed with UG (λ). We claim that this k-homomorphism is a projector onto the k-subgroup ZG (λ), from which the desired semidirect product decomposition is immediate. The map f clearly restricts to the identity on ZG (λ), so we just have to check that it factors through ZG (λ). Consider any k-algebra R and g ∈ PG (λ)(R), and deﬁne g = limt→0 t.g in G(R). We want g ∈ ZG (λ)(R). For any R-algebra R and t ∈ GL1 (R ) we have t .g = limt→0 (t t).g because the GL1 -action on itself by translation extends to an action of GL1 on the afﬁne line preserving the origin. This limit equals g , due to the limiting property for g, so t .g = g for all t . That is, g ∈ ZG (λ)(R) as desired. Step 3. The deduction of (3) lies deeper. We ﬁrst prove the open immersion property in general over a ﬁeld by using the crutch of GLn , and then (in Step 4) we will use a more sophisticated method to handle the case of smooth G over any ring k (which gives another proof when k is a ﬁeld). With k now assumed to be a ﬁeld, by Proposition A.2.3 there is a closed k-subgroup inclusion G → GL(V ) for a nonzero ﬁnite-dimensional k-vector space V . By Example 2.1.1 and Proposition 2.1.2, (3) holds for GL(V ). Thus, it sufﬁces to treat the case where there is a subgroup functor inclusion j : G → G such that G satisﬁes (3) relative to λ := j ◦ λ. We will show that the open

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Root groups and root systems

subscheme G ∩ (UG (−λ ) × PG (λ )) ⊆ G is identiﬁed with UG (−λ) × PG (λ) via multiplication. Consider a k-algebra R and points ∈ UG (−λ )(R), u+ ∈ UG (λ )(R), z ∈ ZG (λ )(R) u− u z = g ∈ G(R). It sufﬁces to prove that u , u , z ∈ G(R) since such that u− + + − we have already proved (2) for G. By Proposition A.2.4, there is a ﬁnite-dimensional k-vector space V , a k-homomorphism ρ : G → GL(V ), and a line L in V such that G is the schemetheoretic stabilizer of L in G . Let v ∈ L be a basis element, so ρ(g)(v) = cv u z , we get in VR = R ⊗k V for a unique c ∈ R× . Since g = u− + −1 z )(v) = cρ((u− ) )(v) ρ(u+

(2.1.4)

in VR . For any point t of GL1 valued in an R-algebra R , the point λ (t) of G (R ) lies in G(R ) and so acts on v (through ρ) by some R × -scaling. Hence, we can replace v with ρ(λ (t)−1 )(v) on both sides of (2.1.4). Now act on both sides of (2.1.4) by ρ(λ (t)), and then commute ρ(λ (t)−1 ) past ρ(z ) (as we may, since z ∈ ZG (λ )(R)) to get the identity −1 )z )(v) = cρ(t.(u− ) )(v) ρ((t.u+

(2.1.5)

as points of the afﬁne space V R over R covariantly associated to VR . Viewing the two sides of (2.1.5) as R-scheme maps (GL1 )R → V R , the left side extends to an R-map P1R − {∞} = AR1 → V R and the right side extends to an R-map P1R − {0} → V R . By combining these, we arrive at an R-map P1R → V R from the projective line to an afﬁne space over R. The only such map is a constant R-map to some v0 ∈ V R (R) = VR , so both sides of (2.1.5) are independent of t (and equal to v0 ). Passing to the limit as t → 0 on the left side and as t → ∞ on the right side yields ρ(z )(v) = v0 = cv. We have proved that z carries v to an R× -multiple of itself. Thus, the point z ∈ G (R) is an R-point of the functorial stabilizer of L inside of V . This stabilizer is exactly G, by the way we chose ρ, so z is an R-point of G ∩ ZG (λ ) = ZG (λ). Since ρ(z )(v) = cv, by cancellation of c on both sides of the identity (2.1.5) we get −1 )(v) = ρ(t.(u− ) )(v) ρ(t.u+

2.1 Limits associated to 1-parameter subgroups

55

with both sides independent of t and equal to c−1 v0 = v. Taking t = 1, this lies in the stabilizer G of v, so u is an R-point of G ∩ U (±λ ) = says that u± G ± UG (±λ), as required. This completes the proof that UG (−λ) × PG (λ) → G is an open immersion when k is a ﬁeld. The connectedness of UG (λ) in such generality follows from the fact that every point u of UG (λ) is joined to the identity by the morphism A1 → UG (λ) that extends t → t.u on GL1 . If G is connected then it is irreducible (as may be checked over k), so the open subscheme UG (−λ) × PG (λ) is connected for such G (using (3)). Thus, if G is connected then PG (λ) is connected, and so by using (2) we deduce that ZG (λ) is also connected. This proves (4) by passing to ﬁbers. Step 4. Now assume that G is smooth over an arbitrary ring k, and we need to establish the open immersion property as in (3) and the smoothness of UG (λ) and ZG (λ) (from which the smoothness of PG (λ) follows via (2)). By construction, UG (−λ) is cut out by the n-eigenvectors of the augmentation ideal of G with n 0, and PG (λ) is cut out by the n-eigenvectors with n < 0. Together these span the augmentation ideal as a k-module, so UG (−λ) ∩ PG (λ) is the identity section. Thus, the multiplication map μ : UG (−λ) × PG (λ) → G in (3) is a monomorphism. By [EGA, IV4 , 17.9.1], an open immersion is the same thing as an étale monomorphism, so to prove that μ is an open immersion it sufﬁces to show that it is étale. Granting for a moment that UG (−λ) and PG (λ) are smooth, let us prove that μ is étale. We may assume k is strictly henselian with an algebraically closed residue ﬁeld, and only need to check étaleness at rational points of the special ﬁber. But these lift to k-points by smoothness, so translations reduce the problem to étaleness at the identity. The k-smoothness hypothesis reduces étaleness at the identity to the isomorphism property on tangent spaces along the identity section, which in turn is just the weight space decomposition for the Lie algebras as in (1). It remains to prove that UG (λ) and ZG (λ) are smooth when G is smooth. Smoothness of the torus centralizer ZG (λ) is a special case of Proposition A.8.10(2) (taking H = GL1 , Y = G, and Y = Spec k), or alternatively is a special case of [SGA3, XI, 5.4]. For UG (λ), we may again assume that k is strictly henselian with algebraically closed residue ﬁeld, and it sufﬁces to prove smoothness at rational points in the special ﬁber. Since UG (λ) has geometrically irreducible ﬁbers (by (4)), if V is any open subscheme of UG (λ) around the identity section then the multiplication map V × V → UG (λ) is surjective. Thus, if there is such a V that is smooth then UG (λ)(k)-translates of V cover the special ﬁber, so smoothness of UG (λ) will follow. In other words,

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to prove smoothness of UG (λ) it sufﬁces to do so along the identity, and for this we may assume that k is local noetherian, and even complete. Thus, the completion of the local ring of G at the identity point of the special ﬁber is identiﬁed with the I -adic completion A of the coordinate ring of G, where I is the augmentation ideal. Consider the cotangent space I /I 2 along the identity. This is a ﬁnite free module (dual to Lie(G)), and as such it has a linear GL1 -action that is induced by the GL1 -action on the entire coordinate ring of G via conjugation through λ. The weight space decomposition of I /I 2 therefore lifts to I , which is to say that we can ﬁnd GL1 -eigenvectors h1 , . . . , hd ∈ I that lift a k-basis of I /I 2 . Thus, there are integers ai ∈ Z such that t.hi = t ai hi for all points t of GL1 (i.e., hi ∈ k[G]ai for all i), so the local k-algebra isomorphism k[[x1 , . . . , xd ]] A deﬁned by xi → hi is GL1 -equivariant modulo each power of the augmentation ideal when using the action t.xi = t ai xi . Since the ideal of UG (λ) in k[G] is generated by the n-eigenvectors in I for n 0, and weight space decompositions are compatible with equivariant quotients (such as the projection k[G] k[G]/I r+1 for each r 0), the completion of UG (λ) along the identity section is the quotient of A by the ideal generated by monomials xiei (all ei 0 and some ei > 0) such that ai ei 0. Any such monomial must have a coefﬁcient ei0 > 0 for some i0 such that ai0 0, so the ideal of UG (λ) in the completion A is generated by the xi for which ai 0. This proves the smoothness of UG (λ). (The same method proves the smoothness of ZG (λ) near the identity, using the ideal generated by xi for which ai = 0, but does not sufﬁce to reprove smoothness of ZG (λ) away from its ﬁbral identity components.) 2 We now record how the formation of the k-subgroups PG (λ), UG (λ), and ZG (λ) behaves with respect to surjective k-homomorphisms π : G → G between smooth connected afﬁne groups over a ﬁeld k. Since a surjective homomorphism G → G between smooth connected k-groups is automatically ﬂat (by [Mat, 23.1]), it is natural to avoid smoothness hypotheses on G or G but to assume that π is ﬂat: Corollary 2.1.9 Let π : G → G be a ﬂat surjective homomorphism between connected afﬁne groups of ﬁnite type over a ﬁeld k, and let λ : GL1 → G be a k-homomorphism. For λ := π ◦ λ, π restricts to ﬂat surjective maps PG (λ) → PG (λ ), UG (λ) → UG (λ ), and ZG (λ) → ZG (λ ). In particular, π(PG (λ)) = PG (λ ) and similarly for UG (λ) and ZG (λ).

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Proof. It is clear from functorial considerations (and calculation of limits as t → 0) that π carries PG (λ) into PG (λ ), UG (λ) into UG (λ ), and ZG (λ) into ZG (λ ). Since π : G → G is ﬂat, the induced map UG (−λ) × ZG (λ) × UG (λ) = UG (−λ) × PG (λ) → UG (−λ ) × PG (λ ) = UG (−λ ) × ZG (λ ) × UG (λ ) between open subschemes is ﬂat. Hence, the maps between the factors are ﬂat. But a ﬂat map between k-schemes of ﬁnite type is open, and any homomorphism between afﬁne k-groups of ﬁnite type has closed image. By Proposition 2.1.8(4) the groups PG (λ), UG (λ), and ZG (λ) are connected (since G is connected), and likewise for G , so the maps PG (λ) → PG (λ ), UG (λ) → UG (λ ), and ZG (λ) → ZG (λ ) are surjective. 2 In the theory of connected reductive groups over a ﬁeld k, it is an important fact that the unipotent radical (over k) of any parabolic k-subgroup is deﬁned over k as a subgroup and is k-split in the sense that it admits a composition series over k with successive quotients k-isomorphic to Ga . We have an analogous result for UG (λ): Proposition 2.1.10 For a smooth connected afﬁne group G over a ﬁeld k and k-homomorphism λ : GL1 → G, the smooth connected unipotent k-group UG (λ) is k-split. Proof. By Theorem B.3.4, any smooth connected unipotent k-group U contains a unique k-split smooth connected normal k-subgroup Usplit such that U /Usplit is k-wound in the sense of Deﬁnition B.2.1 (i.e., admits no non-constant kmorphism from Ak1 ), and moreover the formation of Usplit commutes with separable extension on k. The uniqueness of (Uks )split = (Usplit )ks implies that if U is a k-subgroup of an afﬁne k-group H of ﬁnite type then Usplit is normalized by NH (U )(ks ). Applying this to H = G and U := UG (λ), we conclude that Usplit is normalized by λ(GL1 ). Thus, Lie(U /Usplit ) is a GL1 -equivariant quotient of Lie(U ). A k-wound smooth connected unipotent k-group admits no nontrivial action by a k-torus (Corollary B.4.4) so the GL1 -action on U /Usplit via conjugation through λ must be trivial. Hence, Lie(U /Usplit ) has only the trivial weight. But Proposition 2.1.8(1) implies that Lie(U ) has only nontrivial weights for its GL1 -action, so Lie(U /Usplit ) = 0. This says U = Usplit ; i.e., UG (λ) is k-split. 2 Remark 2.1.11 The theory of PG (λ) and UG (λ) can be generalized to the case of an abstract GL1 -action on G, not just action through conjugation by a

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1-parameter k-subgroup. Indeed, consider an afﬁne group scheme G of ﬁnite presentation over a ring k, and a left action λ : GL1 × G → G over k. We write −λ for the composition of λ with inversion on GL1 . For any k-algebra R we deﬁne PG (λ)(R) = {g ∈ G(R) | lim λ(t, g) exists}, t→0

UG (λ)(R) = {g ∈ G(R) | lim λ(t, g) = 1}, t→0

and also deﬁne ZG (λ)(R) to be the set of g ∈ G(R) that are invariant under the action. Observe that if G is a closed k-subgroup of G stable under the GL1 -action then PG (λ) = G ∩ PG (λ), UG (λ) = G ∩ UG (λ), and ZG (λ) = G ∩ ZG (λ) as subfunctors of G. Let H = GL1 G denote the semidirect product deﬁned by the action λ, so H is k-smooth (resp. has connected ﬁbers over Spec k) whenever the same holds for G. For the canonical inclusion μ : GL1 → H viewed as a 1-parameter k-subgroup, we have (as subfunctors of H ) PH (μ) = GL1 PG (λ), ZH (μ) = GL1 ZG (λ), UH (μ) = UG (λ). In particular, PG (λ) = G ∩ PH (μ) and ZG (λ) = G ∩ ZH (μ), so each of PG (λ), ZG (λ), and UG (λ) is represented by a closed k-subgroup of G whose formation commutes with any scalar extension on k. The conclusions in Proposition 2.1.8, Corollary 2.1.9, and Proposition 2.1.10 continue to hold for PG (λ), UG (λ), and ZG (λ) by applying those results to H = GL1 G and μ. Proposition 2.1.12 Let G be a connected afﬁne group scheme of ﬁnite type over a ﬁeld k, and λ : GL1 × G → G a left action of GL1 on G. Assume that (Gk )red is solvable, and deﬁne PG (λ) and UG (−λ) as in Remark 2.1.11. (1) The open immersion UG (−λ) × PG (λ) → G deﬁned by multiplication is an isomorphism of k-schemes. (2) Let H be a k-subgroup scheme of G that is stable under the GL1 -action. Then the multiplication map UH (−λ) × PH (λ) → H is an isomorphism, and H is connected if and only if ZH (λ) is connected. Proof. We may assume k is algebraically closed, so for (1) the problem is to show that the map of sets UG (−λ)(k) × PG (λ)(k) → G(k) is surjective. The closed subscheme G = Gred is a smooth k-subgroup of G since k = k, and λ factors through G , so PG (λ) = G ∩ PG (λ) and UG (−λ) = G ∩ UG (−λ). But PG (λ) and UG (−λ) are smooth since G is smooth, so PG (λ) = PG (λ)red and UG (−λ) = UG (−λ)red . We may therefore replace G with G so that G

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is smooth and solvable. The same arguments apply for (2) (using Hred ), so for that we can assume H is smooth. If G is commutative then the image of UG (−λ) × PG (λ) is an open subgroup of G (Proposition 2.1.8(3)) and hence by connectedness it equals G, so (1) is clear in such cases. To prove (1) in general, we induct on dim G and may assume that G is not commutative. The derived series of G is stable under the GL1 action, so if D denotes the ﬁnal nontrivial term of this series then the result holds for the commutative normal subgroup D, and by induction on dimension the result also holds for G/D, with their induced GL1 -actions. Applying Corollary 2.1.9 to G G/D gives that G(k) = UG (−λ)(k) · PG (λ)(k) · D(k) = UG (−λ)(k) · D(k) · PG (λ)(k). But D(k) = UD (−λ)(k) · PD (λ)(k) by the settled commutative case, so (1) is proved. For (2), we ﬁrst observe that by part (1) and Proposition 2.1.8(3) (and Remark 2.1.11) applied to H → G we have H = H ∩ (UG (−λ) × PG (λ)) = UH (−λ) × PH (λ) via multiplication. But UH (±λ) are connected and PH (λ) = ZH (λ) UH (λ), so clearly H is connected if and only if ZH (λ) is connected. 2

2.2 Pseudo-parabolic subgroups The following deﬁnition is a useful generalization to arbitrary smooth connected afﬁne groups of the notion of a parabolic subgroup of a connected reductive group. Deﬁnition 2.2.1 Let G be a smooth connected afﬁne group over a ﬁeld k. A k-subgroup of G is pseudo-parabolic if it has the form PG (λ)Ru,k (G) for some k-homomorphism λ : GL1 → G. Since λ(GL1 ) is contained in a maximal k-torus T of G, so T ⊆ ZG (λ) ⊆ PG (λ), the maximal k-tori in a pseudo-parabolic k-subgroup of G are also maximal k-tori of G. In Proposition 2.2.9 we will relate pseudo-parabolic subgroups and parabolic subgroups in the connected reductive case, but beware that in general if G is pseudo-reductive over k then G/P may not be k-proper for pseudo-parabolic k-subgroups P of G (see Example 2.1.3 with P = B there). Also, it can happen that P is a pseudo-parabolic k-subgroup of G but Pk is not

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a pseudo-parabolic k -subgroup of Gk for an inseparable extension of the base ﬁeld k /k (see Example 2.1.3 with k /k a purely inseparable nontrivial extension). Behavior of pseudo-parabolicity with respect to separable extension of the base ﬁeld works out nicely, as we will see in Proposition 3.5.2. Example 2.2.2 If G is pseudo-reductive over k then the pseudo-parabolic k-subgroups are the ones of the form PG (λ) for a k-homomorphism λ : GL1 → G. When G is not pseudo-reductive over k then it is important to remember that Ru,k (G) is put inside of pseudo-parabolic k-subgroups by deﬁnition. For example, consider the standard semidirect product G = GL1 Ga (which is not pseudo-reductive over k). Letting χ : G → GL1 be the standard projection (t, x) → t, we see that PG (λ) = G when χ , λ 0 and that PG (λ) = λ(GL1 ) when χ , λ < 0. Hence, PG (λ)Ru,k (G) = G for all λ, so there is no proper pseudo-parabolic k-subgroup of this G. Lemma 2.2.3 Let G be a smooth connected afﬁne group over a ﬁeld k. (1) There exists a proper pseudo-parabolic k-subgroup of G if and only if G/Ru,k (G) contains a non-central k-split k-torus. (2) If G is pseudo-reductive over k then a pseudo-parabolic k-subgroup PG (λ) is a proper k-subgroup of G if and only if λ(GL1 ) is not central in G. Example 2.2.2 shows the necessity of the pseudo-reductivity hypothesis in the second part of the lemma. Proof. We ﬁrst address part (2). If λ(GL1 ) is central in G then clearly PG (λ) = G. Conversely, if PG (λ) = G then UG (λ) is a smooth connected unipotent normal k-subgroup of the pseudo-reductive k-group G. In such cases UG (λ) must be trivial, so ZG (λ) = PG (λ). This says that λ(GL1 ) is central in G. Now assume that G admits a proper pseudo-parabolic k-subgroup P. By the deﬁnition of pseudo-parabolicity, P contains Ru,k (G). Hence, P/Ru,k (G) is a proper k-subgroup of the pseudo-reductive k-group G := G/Ru,k (G), and by Corollary 2.1.9 this proper k-subgroup is a pseudo-parabolic k-subgroup. In view of the settled part (2), it follows that G contains a non-central k-split torus. Conversely, assume that G contains a non-central k-split torus. Such a torus is generated by copies of GL1 over k, so there must be a non-central GL1 subgroup in G . Let H ⊆ G be the preimage of a ﬁxed non-central GL1 ⊆ G under the quotient map G G . This H is a smooth connected afﬁne k-group that is an extension of GL1 by Ru,k (G). Since the formation of maximal k-tori in smooth connected afﬁne k-groups is compatible with passage to quotients (Proposition A.2.8), a maximal k-torus T in H must map isomorphically onto

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the quotient GL1 . In particular, T = GL1 and this is non-central in G since it maps onto a non-central GL1 in the quotient G . We have proved the existence of a non-central k-homomorphism λ : GL1 → G such that the composite map λ : GL1 → G is also non-central. The proof of part (2) implies that PG (λ ) is a proper k-subgroup of G . Since PG (λ) maps into PG (λ ), clearly PG (λ)Ru,k (G) is a proper k-subgroup of G. This is a pseudo-parabolic k-subgroup of G, so we are done. 2 There is an alternative equivalent deﬁnition of pseudo-parabolicity that is useful, so we now present it and prove the equivalence of the two deﬁnitions. Let Rus,k (G) denote the maximal k-split smooth connected unipotent normal k-subgroup of G. (This exists due to dimension reasons and the normality requirement. In Corollary B.3.5 we give another characterization of it in terms of Theorem B.3.4 applied to Ru,k (G).) For any k-homomorphism λ : GL1 → G, (λ) be the smooth connected k-subgroup of G generated by the smooth let PG connected k-subgroups M ⊆ G that are normalized by λ(GL1 ) and for which the weights a of λ(GL1 ) on Lie(M ) satisfy a, λ 0. By Proposition 2.1.8(1) this weight condition says precisely that Lie(M ) ⊆ Lie(PG (λ)). The following result implies that an equivalent deﬁnition of pseudo-parabolic k-subgroups (λ)R of G is that these are the k-subgroups of the form PG us,k (G) for some k-homomorphism λ : GL1 → G. Proposition 2.2.4 For any smooth connected afﬁne k-group G and k (λ) and P (λ)R (G) = homomorphism λ : GL1 → G, we have PG (λ) = PG G u,k PG (λ)Rus,k (G) inside of G. Proof. The description of Lie(PG (λ)) in Proposition 2.1.8(1) implies that (λ) by deﬁnition of P (λ). To prove that this inclusion of smooth PG (λ) ⊆ PG G connected k-subgroups of G is an equality, it sufﬁces to prove that PG (λ) contains every smooth connected k-subgroup M ⊆ G that is normalized by λ(GL1 ) and for which all weights a of λ(GL1 ) on Lie(M ) satisfy a, λ 0, as such (λ) by deﬁnition. M generate PG Since λ(GL1 ) normalizes M , the smooth connected k-subgroups PM (λ), UM (±λ), and ZM (λ) make sense by Remark 2.1.11. By the hypothesis on Lie(M ) we see that Lie(UM (−λ)) = 0, so UM (−λ) = 1. Proposition 2.1.8(3) applied to M (via Remark 2.1.11) therefore gives that M = PM (λ) ⊆ PG (λ), (λ). as desired. This completes the proof that PG (λ) = PG The desired equality ?

PG (λ)Ru,k (G) = PG (λ)Rus,k (G)

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means that PG (λ)Rus,k (G) contains Ru,k (G). Let G = G/Rus,k (G) and let λ = π ◦ λ, where π : G G is the canonical quotient map. By Corollary 2.1.9, the natural map PG (λ) → PG (λ ) induced by π is surjective. We may therefore replace (G, λ) with (G , λ ) to reduce to the case where Rus,k (G) = 1, which is to say (by Corollary B.3.5) that Ru,k (G) is k-wound in the sense of Deﬁnition B.2.1. It remains to show that Ru,k (G) ⊆ PG (λ) if Ru,k (G) is k-wound. By normality in G, conjugation by λ(GL1 ) deﬁnes an action on Ru,k (G) by the k-torus λ(GL1 ). There is no nontrivial action of a k-torus on a k-wound smooth connected unipotent k-group (Corollary B.4.4), so Ru,k (G) ⊆ ZG (λ) ⊆ PG (λ). 2 Corollary 2.2.5 With notation as in Proposition 2.2.4, for the pseudo-parabolic k-subgroup P := PG (λ)Ru,k (G) we have Ru,k (P) = UG (λ)Ru,k (G), Rus,k (P) = UG (λ)Rus,k (G). In particular, if G is pseudo-reductive then Rus,k (P) = UG (λ) = Ru,k (P). Proof. Since the k-split property of smooth connected unipotent k-groups is inherited by quotients (Proposition A.1.4), and the k-group UG (λ) is a smooth connected unipotent normal k-subgroup of PG (λ) that is k-split (Proposition 2.1.10), it follows that UG (λ)Rus,k (G) is a k-split smooth connected unipotent k-subgroup of G. Also, P = PG (λ)Rus,k (G) by Proposition 2.2.4, so UG (λ)Rus,k (G) is normal in P and hence is contained in Rus,k (P). A simpler variant of this argument shows that UG (λ)Ru,k (G) ⊆ Ru,k (P). We have to prove the reverse inclusions. Let G = G/Ru,k (G) and P = P/Ru,k (G), and let λ : GL1 → G be the composition of λ with the quotient map G G . Corollary 2.1.9 implies that P = PG (λ ) and that UG (λ) has image UG (λ ) in G . Since G is pseudoreductive, so ZG (λ ) is pseudo-reductive, we have UG (λ ) = Ru,k (P ) = Ru,k (P)/Ru,k (G). Hence, Ru,k (P) = UG (λ) · Ru,k (G). To prove that the inclusion UG (λ)Rus,k (G) ⊆ Rus,k (P) is an equality, we may replace G with G/Rus,k (G) so that Rus,k (G) = 1. Thus, Ru,k (G) is k-wound. In particular, as we saw in the proof of Proposition 2.2.4, Ru,k (G) ⊆ ZG (λ), so P = PG (λ) and UG (λ) ∩ Ru,k (G) = 1. It follows that the product group Ru,k (P) = UG (λ) · Ru,k (G) is a semidirect product. Therefore, the k-split UG (λ) is normal in Ru,k (P) with quotient Ru,k (G) that is k-wound. Hence, UG (λ) = Rus,k (P) by Corollary B.3.5. 2 Before we explore pseudo-parabolicity further, it is convenient to recall some useful notions from the general theory of root systems.

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Deﬁnition 2.2.6 Let be a root system. A subset ⊆ is closed if a +b ∈ for any a, b ∈ such that a + b ∈ . If moreover = ∪ − then is called parabolic. An element a ∈ is multipliable if 2a ∈ and divisible if a/2 ∈ . We deﬁne the notions of non-multipliable and non-divisible similarly. The reason for restricting to doubling and halving in the deﬁnitions of multipliability and divisibility is that the only Q-multiples of a root a apart from ±a that can be roots are ±2a or ±a/2 (and not both). Closed sets of roots admit the following useful characterization: Proposition 2.2.7 Let be a root system spanning a Q-vector space V , and a subset of . Then is closed if and only if = ∩ A for a subsemigroup A ⊆ V . In such cases, is uniquely determined by its subset nd = {a ∈ | a/2 ∈ }. The empty set in V is considered to be a subsemigroup. Proof. If is closed then the elements of not in nd are precisely those a ∈ such that a/2 ∈ nd . Hence, in the closed case is uniquely determined by nd . It is clear that if = ∩ A for a subsemigroup A in V then is closed. For the converse, we argue similarly to the proof of [Bou2, VI, §1.7, Prop. 22], as follows. Without loss of generality, we can require A to be the subsemigroup generated by . Thus, the problem is to show that if a1 , . . . , aq ∈ (q 1) and b := a1 + · · · + aq ∈ then b ∈ . The case q = 1 is trivial, and the case q = 2 is the deﬁnition of closedness. In general we proceed by induction on q. We can assume b = ai for all i. Let W = W () be the Weyl group of the root system, and choose a W -invariant Q-valued inner product (·, ·) on V . Since (b, b) > 0, we have (b, ai0 ) > 0 for some i0 . Hence, b − ai0 ∈ by [Bou2, VI, §1.3, Thm. 1, Cor.]. By induction on q, we then have b − ai0 ∈ , so the identity b = (b − ai0 ) + ai0 forces b ∈ by closedness. 2 The structure of parabolic sets of roots is given by: Proposition 2.2.8 Let be a root system spanning a Q-vector space V , and a subset of . The following conditions are equivalent:

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(1) The subset is parabolic. (2) There is a positive system of roots + in and a subset I of simple positive roots such that = + ∪ [I ] where [I ] denotes the set of roots that are Z-linear combinations of elements in I . In particular, ∩ −+ is the set of roots in [I ] whose coefﬁcients with respect to I are 0. (3) There is a linear form λ : V → Q such that = λ0 := {a ∈ | λ(a) 0}. In case (3), the subset λ>0 = {a ∈ | a, λ > 0} is contained in a positive system of roots in . Proof. The equivalence of (1) and (2) is [Bou2, VI, §1.7, Prop. 20]. It is clear from the deﬁnition that sets λ0 as in (3) are parabolic. Conversely, suppose is as in (2). Since the simple positive roots are a basis of V , we can choose λ to vanish on I and to be positive on all other simple positive roots. It is then clear that = λ0 . 2 To justify the terminology “pseudo-parabolic”, we prove the following result that will be quite useful later on. Proposition 2.2.9 Let k be a ﬁeld and G a connected reductive k-group. A k-subgroup P ⊆ G is pseudo-parabolic if and only if it is parabolic. That is, the parabolic k-subgroups of G are precisely the subgroups PG (λ) for k-homomorphisms λ : GL1 → G. Proof. We ﬁrst treat the problem over an algebraically closed ﬁeld, and then pass to the general case. Thus, ﬁrst assume that k = k. The parameterization (via parabolic sets of roots) of the set of parabolic subgroups Q in G containing a ﬁxed maximal torus T is a very basic fact in the theory of connected reductive groups over an algebraically closed ﬁeld, and for the convenience of the reader we now recall how this goes so that we can make a link with pseudo-parabolic subgroups. By parabolicity, Q contains a Borel subgroup B of G. By conjugacy of maximal tori in smooth connected afﬁne groups over an algebraically closed ﬁeld [Bo2, Cor. 11.3], B can be chosen in Q so that it contains T . Let + = (B, T ) be the positive system of roots corresponding to B. By Proposition 2.2.8(2), the parabolic sets of roots in = (G, T ) containing + are parameterized by the sets I of simple roots in + . By [Bo2, 14.18] the set of parabolic subgroups Q containing B is also parameterized by such I , with (Q, T ) equal to the parabolic set of roots I := + ∪ [I ]. Explicitly, Q = ZG (TI ) UI where

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TI ⊆ T is the maximal subtorus killed by all a ∈ I (equivalently, all a ∈ [I ]) and UI = Ru (Q) is directly spanned by the root groups Ua for a ∈ + not in [I ]. We wish to relate this description of Q to a pseudo-parabolic subgroup of G. Choose λ ∈ X∗ (T ) such that I = (G, T )λ0 , so [I ] = (G, T )λ=0 and hence the map λ : GL1 → T is valued in TI . Thus, ZG (TI ) ⊆ ZG (λ). To prove the reverse containment, which is to say that the natural TI -action on ZG (λ) is trivial, it sufﬁces (by Corollary A.8.11) to show that Lie(ZG (λ)) is contained in the subspace of TI -invariants in Lie(G). This Lie subalgebra is spanned by Lie(ZG (T )) = Lie(G)T and the root spaces in Lie(G) for a ∈ (G, T )λ=0 = [I ]. By deﬁnition of TI , its action on such weight spaces is trivial. Passing to the unipotent radical, UI is directly spanned by the Ua for a ∈ such that a, λ > 0, which is to say that Ua ⊆ UG (λ). Thus, UI ⊆ UG (λ). This inclusion of smooth connected subgroups induces an equality on Lie algebras, so it is an equality of subgroups. We conclude that Q = ZG (λ) UG (λ) = PG (λ). Thus, by the equivalent descriptions of parabolic sets of roots in Proposition 2.2.8 and the fact that the group W (G, T )(k) = W ((G, T )) acts simply transitively on the set of Borel subgroups B containing T , the parabolic subgroups of G containing T are precisely the pseudo-parabolic subgroups of G containing T when k = k. Since maximal tori in parabolic subgroups and in pseudo-parabolic subgroups are also maximal in G, this settles the case of an algebraically closed ground ﬁeld. Now consider a general ground ﬁeld. Consider a pseudo-parabolic ksubgroup P in G, so P = PG (λ) for a k-homomorphism λ : GL1 → G. Since Pk = PGk (λk ) is pseudo-parabolic in Gk , it is also parabolic. But parabolicity is a geometric property, so P is parabolic in G. Conversely, let P be a parabolic k-subgroup of G, and let T be a maximal k-torus in P, so T is also maximal in G and Pk = PGk (λ ) for some λ ∈ X∗ (Tk ) (by the argument used over k). We just need to ﬁnd such λ : GL1 → Tk that is deﬁned over k. Let k /k be a ﬁnite Galois extension that splits T and let = Gal(k /k). The homomorphism λ descends to a k -homomorphism GL1 → Tk (again denoted as λ ), so Pk = PGk (λ ). The key point is that the set of roots (Gk , Tk )λ 0 = (Pk , Tk ) is -stable inside of X(Tk ) since (Pk , Tk ) descends to (P, T ). Hence, for any a ∈ (Gk , Tk ), whether a, λ is 0 or < 0 is the same across all members of the -orbit of a. But γ (a), λ = a, γ −1 (λ ) for all γ ∈ , so if we deﬁne λ=

γ ∈

γ (λ ) ∈ Homk (GL1 , T )

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then a, λ 0 if and only if a, λ 0. Thus, the parabolic k -subgroups PG (λ)k and PGk (λ ) = Pk containing Tk have Lie algebras supporting the same sets of Tk -weights, so the parameterization of parabolic subgroups of Gk containing Tk implies that PG (λ)k = Pk , and hence P = PG (λ). 2 The theory of pseudo-parabolic subgroups is primarily of interest in the pseudo-reductive case, due to the fact that Ru,k (G) is always contained in pseudo-parabolic k-subgroups (by deﬁnition). The following result makes the emphasis on the pseudo-reductive case more precise. Proposition 2.2.10 Let G be a smooth connected afﬁne group over a ﬁeld k, and let π : G G = G/Ru,k (G) be its maximal pseudo-reductive quotient over k. Then P → π(P), P → π −1 (P ) are inverse bijections between the sets of pseudo-parabolic k-subgroups of G and G . If P and P correspond under this bijection then the natural map G/P → G /P is an isomorphism. Likewise, if T is a maximal k-torus of G then π −1 (T ) contains a maximal k-torus T of G, and P contains T if and only if P contains T . Proof. By deﬁnition, a pseudo-parabolic k-subgroup P in G has the form P = PG (λ)Ru,k (G) for a 1-parameter k-subgroup λ : GL1 → G. For the induced 1-parameter k-subgroup λ = π ◦ λ : GL1 → G , we have π(P) = π(PG (λ)) = PG (λ ) by Corollary 2.1.9, and this is a pseudo-parabolic k-subgroup of G since G is pseudo-reductive over k. Conversely, let P ⊆ G be a pseudo-parabolic k-subgroup, so P = PG (λ ) for a 1-parameter k-subgroup λ : GL1 → G . We claim that P := π −1 (P ) is a pseudo-parabolic k-subgroup of G. If λ = 1 then P = G , so there is nothing to do. Now suppose λ = 1, so λ (GL1 ) is a 1-dimensional k-split torus in G . In this case H = π −1 (λ (GL1 )) is an extension of λ (GL1 ) by Ru,k (G). Since H λ (GL1 ) has unipotent kernel and must carry maximal k-tori (isomorphically) onto maximal k-tori, H is a smooth connected solvable k-group whose maximal k-tori are all k-split of dimension 1. Pick λ : GL1 → H such that λ = π ◦ λ. Then P = PG (λ ) = π(PG (λ)), so the k-subgroup π −1 (P ) of G

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equals the pseudo-parabolic k-subgroup PG (λ)Ru,k (G). This clearly reverses the ﬁrst construction, and the isomorphism between coset spaces is obvious. Finally, consider the claim concerning maximal k-tori. Let T be a maximal k-torus in G , so π −1 (T ) is an extension of T by Ru,k (G). Hence, π −1 (T ) is a smooth connected solvable k-group and every maximal k-torus T in π −1 (T ) maps isomorphically onto T . If P contains T then certainly P := π −1 (P ) contains T . Conversely, if P contains T then its image P = π(P) contains π(T ) = T . 2 We conclude our initial discussion of pseudo-parabolic subgroups by addressing the issue of how we can change λ without changing PG (λ). Proposition 2.2.11 Let G be a smooth connected afﬁne group over a separably closed ﬁeld k, and let λ : GL1 → G be a k-homomorphism. Let P = PG (λ). For any g ∈ P(k), the k-homomorphism g.λ : t → gλ(t)g −1 satisﬁes PG (λ) = PG (g.λ), and if T is a maximal k-torus in P then there exists g ∈ P(k) such that g.λ has image contained in T . Proof. For any point x of P, the conjugate λ(t)(g −1 xg)λ(t)−1 = g −1 ((g.λ)(t))x((g.λ)(t))−1 g has a limiting value as t → 0 since g −1 xg is a point of P. Hence, PG (λ) = P ⊆ PG (g.λ). In particular, g −1 ∈ PG (g.λ)(k). We can apply the same argument to g −1 and g.λ to get the reverse inclusion. By Proposition A.2.10 applied to P, there exists g ∈ P(k) such that λ(GL1 ) is contained in the maximal k-torus g −1 Tg, so g.λ has image contained in T . 2

2.3 Root groups in pseudo-reductive groups The constructions PG (λ), UG (λ), and ZG (λ) will now enable us to deﬁne and prove the uniqueness of “root groups” in pseudo-reductive groups over a separably closed ﬁeld. We ﬁrst set forth some terminology and notation. Deﬁnition 2.3.1 A smooth connected afﬁne group G over a ﬁeld k is pseudosplit (over k) if it contains a maximal k-torus T that is k-split; we call (G, T ) a pseudo-split pair. If G is connected reductive then this recovers the usual notion of being ksplit. It is true in general that all maximal k-split k-tori in G are G(k)-conjugate, so in the pseudo-split case all k-split maximal k-tori are G(k)-conjugate and

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all maximal k-split k-tori are maximal as k-tori. We give a proof of this general G(k)-conjugacy in Theorem C.2.3, resting on the structure theory of root groups in pseudo-split pseudo-reductive groups that is developed below (so we do not use such G(k)-conjugacy in what follows). Fix a pseudo-split pair (G, T ). Since T is k-split, its representation on Lie(G) is a direct sum of k-rational characters of T (Lemma A.8.8). Deﬁne (G, T ) ⊂ X(T ) to be the ﬁnite set of nontrivial weights for the action by T on Lie(G). Example 2.3.2 Let k be a nonzero ﬁnite reduced algebra over a ﬁeld k, and let G be a smooth afﬁne k -group with connected ﬁbers. Thus, G := R k /k (G ) is a smooth afﬁne connected k-group. Assume that all factor ﬁelds of k are purely inseparable over k (as always happens when k = ks ). Let T be a maximal k-torus in G, so by Proposition A.5.15(2) it is the maximal k-torus in R k /k (T ) for a unique maximal k -torus T in G . Thus, if {ki } is the set of factor ﬁelds of k and (Gi , Ti ) is the ki -ﬁber of (G , T ), then G = Gi and T = Ti where Gi := R ki /k (Gi ) and Ti is the maximal k-torus in R ki /k (Ti ). The pure inseparability hypothesis on k /k implies that T is k-split if and only if T is k -split (i.e., Ti is ki -split for all i), so G is pseudo-split if and only if G is pseudo-split (in the sense that it admits a k -split maximal k -torus, which can be checked on ﬁbers over Spec k ). When this condition holds, the k-split maximal k-tori of G are in natural bijection with the k -split maximal k -tori of G . Assume T is k-split, so (G, T ) = (Gi , Ti ) compatibly with the identi ﬁcation X(T ) = X(Ti ). To relate (Gi , Ti ) and (Gi , Ti ), we may rename ki as k so that k is a ﬁeld. Since k /k is purely inseparable, the natural map Tk → T is an isomorphism. This identiﬁes X(T ) with X(T ). Under the identiﬁcation of Lie(R k /k (G )) with the Lie algebra over k underlying Lie(G ) (CorollaryA.7.6), we then see that the root spaces for T acting on Lie(R k /k (G )) are the underlying k-vector spaces of the root spaces for T acting on Lie(G ). This identiﬁes (G, T ) with (G , T ) inside of X(T ) = X(T ). Returning to the general setup, for each nonzero a ∈ X(T )Q let Ta be the corresponding codimension-1 k-subtorus (ker na)0red for any integer n 1 such that na ∈ X(T ) (the choice of which does not matter). The centralizer ZG (Ta ) is a smooth connected k-subgroup of G (Proposition A.2.5), it is pseudo-reductive when G is pseudo-reductive (Proposition 1.2.4), and its Lie algebra is the subspace of Ta -invariants in Lie(G) (Proposition A.2.5). Thus, for any nonzero rational multiple of a in X(T )Q that is contained in (G, T ), the associated root space in Lie(G) is contained in Lie(ZG (Ta )). Consider the adjoint action by the (k-split) maximal k-torus T of ZG (Ta ) on Lie(ZG (Ta )). The codimension-1 subtorus Ta acts trivially, and the weight

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spaces in Lie(ZG (Ta )) apart from Lie(ZG (T )) are the root spaces in Lie(G) for the rational multiples of a which lie in (G, T ). To separate the positive rational multiples of a occurring in (G, T ) from the negative ones, we will use the following device: choose a k-homomorphism λa : GL1 → T such that a, λa > 0 (so λa (GL1 ) is an isogeny complement to Ta in T ). Lemma 2.3.3 The k-subgroup U(a) := UZG (Ta ) (λa ) ⊆ G is independent of λa , and Lie(U(a) ) is the span of the root spaces for T in Lie(G) for all roots in (G, T ) that are positive rational multiples of a in X(T )Q . In particular, U(a) = 1 precisely when some positive rational multiple of a lies in (G, T ). The k-group G is generated by the k-subgroups ZG (T ) and U(a) for all a ∈ (G, T ). Proof. Since λa (GL1 ) is an isogeny-complement to Ta in T , the composite map λa

λa : GL1 → λa (GL1 ) → T /Ta is an isogeny. Conjugation by T on ZG (Ta ) factors through an action of the quotient T /Ta , so U(a) represents the functor of points R {g ∈ ZG (Ta )(R) | (λa )R (t).g → 1 as t → 0 in (GL1 )R }. Hence, to prove that U(a) is independent of λa , it sufﬁces to show that replacing λa with λna for an integer n > 0 does not affect the deﬁnition of U(a) . This invariance follows from Remark 2.1.7. By Proposition 2.1.8(1), Lie(U(a) ) is the span of the λa (GL1 )-weight spaces in Lie(ZG (Ta )) with weight > 0. But Lie(ZG (Ta )) is the Ta -centralizer in Lie(G) (Proposition A.2.5), so Lie(U(a) ) is the span of the root spaces in Lie(G) for the elements c in (G, T ) that kill Ta and satisfy c, λa > 0. Any a ∈ (G, T ) killing Ta is a nonzero rational multiple of a, and if a = ca in X(T )Q with c ∈ Q× then a , λa = ca, λa has the same sign as c. Thus, Lie(U(a) ) has the desired description. The k-group G is generated by ZG (T ) and the U(a) for all a ∈ (G, T ) since these groups are all smooth and connected and Lie(G) is spanned by Lie(ZG (T )) and the Lie(U(a) ) (due to the weight space decomposition for Lie(G) relative to its T -action). 2

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G , or U Deﬁnition 2.3.4 For a ∈ X(T )Q − {0} and any λa as above, U(a) (a) if G is clear from the context, is the smooth connected unipotent k-subgroup UZG (Ta ) (λa ) in G (with Ta = (ker na)0red for any n ≥ 1 such that na ∈ X(T )).

Remark 2.3.5 The k-subgroup U(a) in G is normalized by ZG (T ) since ZG (T ) centralizes λa . From the deﬁnition, U(a) only depends on a up to a positive rational multiple in X(T )Q . Remark 2.3.6 Let G be a pseudo-split smooth connected afﬁne group over a ﬁeld k, and let T be a k-split maximal k-torus in G. For any smooth connected k-subgroup H of G that contains T (so H is pseudo-split) and any a ∈ (H , T ), since ZH (Ta ) = H ∩ ZG (Ta ) for Ta := (ker a)0red we see (via Lemma 2.1.5) G in Z (T ) and that the associated smooth connected unipotent subgroups U(a) G a H in Z (T ) satisfy U H = H ∩ U G . In particular, H ∩ U G is smooth U(a) H a (a) (a) (a) and connected. Also, if f : G G is a quotient map over k such that (ker fk )0red is unipotent and T := f (T ) (so T is a k-split maximal k-torus in G ), then under the identiﬁcation X∗ (T )Q = X∗ (T )Q we can deﬁne Ta and it is isogenous to Ta via G onto U G for any a ∈ (G, T ), by Corollary 2.1.9 and f . Thus, f carries U(a) (a)

G might be trivial; that is, a may not Proposition A.2.5. (In this generality, U(a) have a positive rational multiple in (G , T ).) If G is pseudo-reductive over k and H is a smooth connected k-subgroup of G containing T then for H = f (H ) H (by Lemma 1.2.1). Thus, for and any a ∈ (H , T ), the map f cannot kill U(a)

H is nontrivial, which is to say that the elepseudo-reductive G the k-group U(a) ment a ∈ (H , T ) cannot completely “disappear” in (H , T ). That is, some positive rational multiple of a lies in (H , T ) when G is pseudo-reductive. Pushing the calculation a bit further, no new weights up to positive rational multiple can be “created” in (H , T ) as well. That is, for any a ∈ (H , T ) there exists a ∈ (H , T ) that is a positive rational multiple of a (so the image H in H is U H , also equal to U H as noted in Remark 2.3.5). To see this, of U(a) (a) (a ) let S ⊆ T be the codimension-1 subtorus corresponding to (ker a )0red in the isogenous quotient T of T . Let a be any nontrivial character of T that is trivial on S, and by replacing it with its negative if necessary we arrange that a is a H onto U H = U H = 1, positive rational multiple of a . The map f carries U(a) (a) (a ) H = 1. Thus, some positive rational multiple of a lies in (H , T ). so U(a)

Example 2.3.7 An interesting class of such f that arise in the classiﬁcation of pseudo-reductive groups is central quotients R K/k (G ) → R K/k (G )/Z for a nonzero ﬁnite reduced k-algebra K, a semisimple K-group G with connected ﬁbers, and a k-subgroup scheme Z ⊆ R K/k (ZG ) = ZRK/k (G ) (see Proposition

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A.5.15(1)). To prove that (Z 0 )red is unipotent it sufﬁces to treat the case Z = k

R K/k (ZG ), so the problem reduces to proving the unipotence of R A/k (μn )0red for an algebraically closed ﬁeld k, a nonzero ﬁnite k-algebra A, and any n 1. Since R A/k (μn ) is the n-torsion in R A/k (GL1 ) = GL1 × U for a smooth connected unipotent k-group U , and (μn )0red = 1, the result is clear. We will prove shortly (in Theorem 2.3.10) that if char(k) = 2 and G is pseudo-reductive then (G, T ) is naturally identiﬁed with the reduced root system (G red , Tk ) in X(Tk ) = X(T ), so Lie(U(a) ) is the a-root space in k Lie(G). When char(k) = 2 and k is imperfect this can fail (i.e., Lie(U(a) ) may support more than one T -weight), as we will see when we study the classiﬁcation of pseudo-reductive groups in characteristic 2. Lemma 2.3.8 Let U be a smooth connected commutative unipotent group over a ﬁeld k, and if char(k) = p > 0 then assume that U is p-torsion. Let T be a split k-torus equipped with an action on U such that the trivial character of T is not a weight for the T -action on Lie(U ). The k-group U is a vector group, and it admits a T -equivariant linear structure (in the sense of Deﬁnition A.1.1). Such a linear structure on U is unique if T acts on Lie(U ) with a single weight. The uniqueness can fail when char(k) = p > 0 if the T -action on Lie(U ) has more than one weight. For example, if T = GL1 acts on V = Ga × Ga via c.(x, y) = (cx, c p y) then (x, y) → (x, y + x p ) is a non-linear k-automorphism of V that is T -equivariant. Proof. If char(k) = 0 then the algebraic exponential isomorphism with the Lie algebra equips U with a functorial linear structure, so it is T -equivariant. If char(k) = p > 0 then since T acts on Lie(U ) with only nontrivial weights, it follows from Theorem B.4.3 that U is a vector group and admits a T -equivariant linear structure. This settles the existence in all characteristics. To prove the uniqueness of the linear structure when T acts on Lie(U ) with a single weight χ = 1, we may assume k is algebraically closed. Any T equivariant linear structure on U must have χ as its unique weight, since the Lie algebra can detect the weight space decomposition relative to the linear structure. For any c ∈ k × we may write c = χ (t) for some t ∈ T (k), so the linear structure on U must satisfy cu = χ (t)u = t.u for all u ∈ U (k). Since t.u does not depend on the linear structure, we are done. 2 Proposition 2.3.9 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and let T be a k-split maximal k-torus in G.

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For each a ∈ (G, T ), the k-group U(a) is a vector group and admits a T equivariant linear structure. If a ∈ (G, T ) is such that qa + q a ∈ (G, T ) inside of X(T )Q for all q, q > 0 in Q then U(a) and U(a ) commute with each other. Moreover, there is a unique positive rational multiple of a in X(T ) that lies in (G red , Tk ). k

Proof. By Lemma 1.2.1, to prove that the commutator (U(a) , U(a) ) vanishes (i.e., U(a) is commutative) it sufﬁces to show that (U(a) , U(a) )k has trivial image in G red . By applying Remark 2.3.6 to the quotient map Gk G red with unipok k tent kernel, some positive rational multiple a of a occurs in the reduced root system (G red , Tk ) and (U(a) )k has image in G red equal to the a-root group. k k This root group is commutative, so U(a) is commutative. We conclude that for any a ∈ (G, T ) we can deﬁne qa ∈ Q to be the unique rational number > 0 such that a := qa · a ∈ (G red , Tk ) and (U(a) )k maps onto the a-root group in G red . k For linearly independent a, a ∈ (G, T ), if

k

qa · a + qa · a ∈ (Gkred , Tk ) then the root groups in G red corresponding to qa · a and qa · a commute. These k

latter root groups are the respective images of (U(a) )k and (U(a ) )k in G red . By k Lemma 1.2.1, we conclude that qa · a + qa · a ∈ (Gkred , Tk ) ⇒ (U(a) , U(a ) ) = 1. This applies when qa + q a ∈ (G, T ) for all q, q ∈ Q× >0 because (G red , Tk ) ⊆ (G, T ) inside of X(Tk ) = X(T ) (as Lie(G red ) is a k k Tk -equivariant quotient of Lie(Gk ) = Lie(G)k ). Finally, we prove that U(a) is a vector group admitting a T -equivariant linear structure. Observe that U(a) is a smooth connected commutative k-group, and the T -action (through T /Ta ) on Lie(U(a) ) has no occurrence of the trivial weight (due to Proposition 2.1.8(1)). Thus, in view of Lemma 2.3.8, we just need to check that if char(k) = p > 0 then U(a) is p-torsion. This goes as the proof of commutativity: the root group image of (U(a) )k in G red is p-torsion, so k p · (U(a) )k ⊆ Ru (Gk ). Hence, p · U(a) = 1 by Lemma 1.2.1. 2 Theorem 2.3.10 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and let T be a k-split maximal k-torus in G. If char(k) = 2 then (G red , Tk ) = k

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(G, T ) inside of X(T ) = X(Tk ). If char(k) = 2 then 1 · (Gkred , Tk ) 2

(Gkred , Tk ) ⊆ (G, T ) ⊆ (Gkred , Tk )

inside of X(T ) = X(Tk ). When char(k) = 2, the case (G, T ) = (G red , Tk ) can only happen when k

k is imperfect and G red has a connected simple semisimple normal subgroup k that is simply connected of type Cn with n 1 (i.e., isomorphic to Sp2n ). Proof. By Proposition 2.3.9, our problem is to prove that the only positive rational multiples of a in X(T ) that can occur in (G, T ) aside from a are a/2 or 2a, and that if this happens then (i) k is imperfect with char(k) = 2, (ii) exactly one of a/2 or 2a occurs, (iii) G red has a simple smooth connected normal of simply connected type Cn k for some n 1. The assertion concerning simply connected type Cn , granting the rest, is due to the fact that in the classiﬁcation of reduced and irreducible root data, this is the only case for which there is a root that is divisible (by 2, in fact) in the character lattice. For the remaining assertions we only need to consider elements of (G, T ) that kill the codimension-1 torus Ta = (ker a)0red . Hence, it is harmless to replace G with ZG (Ta ). This brings us to the case where there is a codimension-1 torus T in T that is central in G. This is the maximal central torus, since T acts on Lie(G) with the nontrivial weight a. The quotient G/D (G) is commutative, so its Lie algebra has no nontrivial T -weight. It follows that Lie(D (G)) contains the root spaces in Lie(G) relative to T and (by Lemma 1.2.5(iii)) that the k-split maximal k-torus S = T ∩ D (G) of D (G) acts nontrivially on all of these root spaces. Hence, there is no loss of generality by replacing (G, T ) with (D (G), S). By Proposition 1.2.6, we also now have that G = D (G). In particular, G red = G ss . k k Since a central torus of Gk has trivial image in G ss , we conclude that G ss k k has a 1-dimensional maximal torus. Thus, this semisimple group is of type A1 (i.e., it is SL2 or PGL2 ). Also, since Tk is a maximal torus in G red = G ss , it k k follows that dim T = 1. But (G ss , Tk ) ⊆ (G, T ), so we may replace a with k a positive rational multiple to arrange that (G ss , Tk ) = {±a}. The maximal k semisimple quotient G ss is isomorphic to either SL2 or PGL2 . k

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Fix a T -equivariant linear structure on the vector group U(a) and consider the weight space decomposition U(a) =

Vχi

according to this linear structure. Choose the unique isomorphism λ : GL1 T such that a, λ > 0, so there is an isomorphism of G ss onto SL2 or PGL2 carryk ing Tk over to the diagonal subgroup D such that λ is carried to the isomorphism t → diag(t, 1/t) when G ss is simply connected and is carried to the isomork phism t → diag(t, 1) when G ss is adjoint. Hence, by Corollary 2.1.9 applied to k the quotient map Gk G ss , we see that (U(a) )k is carried onto the strictly upper k triangular unipotent subgroup U + of G ss via the chosen isomorphism with SL2 k or PGL2 . This is equivariant with respect to the isomorphism Tk D. Let π be the composition of the quotient map Gk G ss with the chosen k isomorphism onto SL2 or PGL2 . Since we arranged for a to lie in (G ss , Tk ), k for any points t of Tk and u of (U(a) )k we have π(tut −1 ) = a(t) · π(u) via the canonical linear structure on the 1-dimensional vector group U + (using any choice of k-group isomorphism U + Ga ). To deduce information about positive rational multiples of a in (G, T ) aside from a, we shall study the restriction of π : (U(a) )k → U + to the Tk -weight spaces (Vχj )k relative to the linear structure we have chosen on U(a) (making T act linearly). The additive map π : (Vχj )k → U + satisﬁes π(χj (t)vj ) = a(t) · π(vj ) for all t in Tk and vj in (Vχj )k . The map π cannot vanish on (Vχj )k . Indeed, if it does then the quotient map Gk G ss would kill (Vχj )k , which is to say k (Vχj )k ⊆ R(Gk ) = Ru (Gk ). By Lemma 1.2.1 it would follow that Vχj = 0, a contradiction. Choose vj ∈ (Vχj )k such that π(vj ) = 0. We therefore obtain a nonzero Tk -equivariant map Ga (χj ) → Ga (a), where Ga (χ ) denotes Ga equipped with a Tk -action through scaling via a k-rational character χ of T . Since a = 1, this ×

self-map of Ga must be x → cj xdj with cj ∈ k and dj = psj for some integer

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sj 0 and p equal to the characteristic exponent of k. Moreover, a = χ dj . This shows that a is the only χj when char(k) = 0, and that {χj } is a set of p-power roots of a when char(k) = p > 0. In particular, we now may and do assume char(k) = p > 0. ps If some χj = a occurs then χj = a in X(T ) = X(Tk ) with some s 1, so the root a ∈ (G ss , Tk ) is divisible by ps in the character group X(Tk ) Z. But k the roots for PGL2 are generators of the character group and the roots for SL2 are twice generators of the character group. Hence, additional positive rational multiples of a can only occur in (G, T ) when p = 2, and then only a/2 can occur. Moreover, in this case k must be imperfect because otherwise pseudoreductive k-groups are reductive and so (G, T ) cannot contain a pair of distinct linearly dependent elements aside from those which are equal up to sign. 2 We call a ∈ (G, T ) multipliable if 2a ∈ (G, T ), so by Theorem 2.3.10 the set of non-multipliable elements of (G, T ) is (G red , Tk ). In §3.2, the k pair (X(T ), (G, T )) will be endowed with a natural structure of root datum (possibly non-reduced when char(k) = 2), compatible with the natural root datum structure on (X(Tk ), (G red , Tk )). k Although the k-groups U(a) for a ∈ (G, T ) were constructed using 1parameter k-subgroups of G, they admit a characterization in terms of the T action on G without any reference to 1-parameter subgroups of G. This will be a consequence of the following result. Proposition 2.3.11 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and let T ⊆ G be a k-split maximal k-torus. For any a ∈ (G, T ) there is a unique T -stable smooth connected k-subgroup Ua ⊆ G such that Lie(Ua ) is the a-root space in Lie(G) when a is non-multipliable and is the span of the a-root space and the 2a-root space when a is multipliable. The k-subgroup Ua is a vector group, and it admits a T -equivariant linear structure. Such a linear structure is unique when a is non-multipliable. If a is multipliable then the quotient Ua /U2a has a unique T -equivariant linear structure (and it makes T act with weight a), but Ua only has a non-canonical T -equivariant linear structure because the quotient map Ua Ua /U2a only has a non-canonical T -equivariant section. Proof. If a is non-divisible then the k-group U(a) settles the existence aspect for Ua as well as the linear structure claims. If 2a ∈ (G, T ) then a is nondivisible and 2a is non-multipliable, and U(a) has a linear structure in which the 2a-weight space settles the existence of U2a , as well as the linear structure claims. Thus, the only issue is to prove the uniqueness of Ua .

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Let Ta = (ker a)0red . The pseudo-reductive k-subgroup ZG (Ta ) in G containing T has Lie algebra whose nontrivial T -weights are precisely the nonzero rational multiples of a that lie in (G, T ). If H is a T -stable smooth connected k-subgroup of G such that Ta acts trivially on Lie(H ) then H ⊆ ZG (Ta ) by Corollary A.8.11. Hence, we lose no generality by replacing G with ZG (Ta ), so by Theorem 2.3.10 the set (G, T ) is equal to either {±a}, {±a/2, ±a}, or {±a, ±2a}, with the latter two only possible when char(k) = 2. Since Ta acts trivially on Lie(G) in all cases, by Corollary A.8.11 we have that Ta is central in G whereas T is non-central. Thus, T -conjugation on G factors through T /Ta , so for any λ : GL1 → T such that a, λ > 0 the map λ is non-central in G and the k-subgroups PG (±λ), UG (±λ), and ZG (λ) are independent of λ by Remark 2.1.7. The quotient ZG (λ) PG (λ)/UG (λ) is a pseudo-reductive k-group in which the maximal k-torus T acts with no nontrivial weights (due to Proposition 2.1.8(1)) and thus is central by Corollary A.8.11. We conclude that ZG (λ) is commutative, by Proposition 1.2.4. Note also that U(a) = UG (λ) by deﬁnition. For the proof of uniqueness of Ua , let U ⊆ G be a smooth connected ksubgroup that is normalized by T and such that Lie(U ) is the a-root space in Lie(G) when a is non-multipliable and is the span of the root spaces for a and 2a when a is multipliable. We ﬁrst claim that U must be unipotent. More generally, if a smooth connected afﬁne k-group H admits a T -action for which Lie(H ) has only nontrivial T -weights then H is unipotent. Indeed, if H is not unipotent then it contains a nontrivial k-torus, so the smooth connected k-group T H contains a maximal k-torus T strictly containing T . But then T = T S for a nontrivial k-torus S of H , so Lie(S) is a nonzero subspace of Lie(H ) with trivial T -action, contrary to our hypothesis on H . Thus, indeed U is unipotent. In particular, T ∩ U = 1. The smooth connected k-subgroup TU = T U in G has Lie algebra which admits as its set of T -weights either {0, a}, {0, a, 2a}. These weights have non-negative pairing against λ, so by Proposition 2.2.4 we therefore have TU ⊆ PG (λ). We claim that U ⊆ UG (λ) = U(a) , or equivalently that U has trivial image in PG (λ)/UG (λ). This quotient is the pseudo-reductive k-group ZG (λ) that we have seen is commutative, so it contains no nontrivial smooth connected unipotent k-subgroup. Hence, indeed U ⊆ UG (λ) = U(a) , and as long as a is non-divisible this inclusion is an equality because the Lie algebras coincide inside of Lie(G). It remains to show that if a/2 ∈ (G, T ) then Ua is unique. Note that in such cases char(k) = 2. The proof of Theorem 2.3.10 shows that

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(G red , Tk ) = {±a}. The smooth connected unipotent k-subgroup UG (λ) has k Lie algebra that is the span of the root spaces for a and a/2 in Lie(G), and UG (λ) is normalized by Ta · λ(GL1 ) = T . The same method as above proves that any smooth connected unipotent k-subgroup U of G that is normalized by T and has Lie algebra whose T -weights are a subset of {a, a/2} necessarily lies in UG (λ). Recall that UG (λ) is a vector group that admits a T -equivariant linear structure, with weights a and a/2. The resulting weight space for a in UG (λ) can be constructed without reference to such a choice of linear structure on UG (λ): Since a/2 is primitive in X (T ) by Theorem 2.3.10 (as a root in a reductive group cannot be divisible by more than 2 in the character lattice), we have Ta = ker(a/2), so the a-weight space in UG (λ) is the centralizer scheme ZUG (λ) (μ) for the action by the 2-torsion group scheme μ := ker(a)/Ta = (T /Ta )[2] μ2 on UG (λ). (See Proposition A.8.10 for a general discussion of such a centralizer scheme, including that it is smooth with Lie algebra equal to the a-weight space in Lie(G).) Consider any T -stable smooth connected unipotent k-subgroup U ⊆ G such that Lie(U ) is the a-root space of Lie(G). We have seen above that U ⊆ U(a) = UG (λ), and by Corollary A.8.11 the μ-action on U is trivial. Thus, U ⊆ ZUG (λ) (μ)0 . This inclusion is an equality on Lie algebras, so it is an equality of k-groups. Hence, the uniqueness of Ua is proved in all cases. 2 Remark 2.3.12 If a is non-divisible in (G, T ) then U(a) = Ua . If a is divisible then U(a) = Ua/2 . Deﬁnition 2.3.13 With hypotheses and notation as in Proposition 2.3.11, for each a ∈ (G, T ) the k-subgroup Ua ⊆ G is the root group associated to a. Remark 2.3.14 If a ∈ (G, T ) is multipliable then U2a ⊆ Ua and Lie(Ua ) is the span of the root spaces for a and 2a in Lie(G), and otherwise it is the root space for a. Since ZG (T ) is smooth, the unique characterization of Ua in Proposition 2.3.11 implies that Ua is normalized by ZG (T ); this reﬁnes Remark 2.3.5. For isotropic connected semisimple groups over a ﬁeld, if the relative root system is non-reduced then the root groups whose Lie algebra supports more than one weight can be non-commutative. In contrast, all root groups in a pseudo-split pseudo-reductive group are commutative, even when the root system is non-reduced (which happens over imperfect ﬁelds of characteristic 2).

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As in the theory of connected reductive groups, the formation of root groups is well-behaved with respect to central quotients in the pseudo-reductive case: Proposition 2.3.15 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and T a k-split maximal k-torus in G. Let G = G/Z be a central quotient of G that is also pseudo-reductive, and let T be the k-split maximal k-torus image of T in G . The inclusion of X(T ) into X(T ) carries (G , T ) bijectively onto (G, T ), and for any a ∈ (G, T ) the quotient map G → G carries Ua isomorphically onto Ua . Proof. The action of G on itself by conjugation factors through an action of the central quotient G on G, so likewise each a ∈ (G, T ) ⊂ X(T ) (which computes the action of T on Ua ) factors through the image T of T in G . for By Remark 2.3.6, the quotient map π : G G carries U(a) onto U(a) all a ∈ = (G, T ), and U(a) = 1. Since the subspace ker(Lie(π )) = Lie(ker π ) = Lie(Z) supports only the trivial weight for T (as Z is central in is injective on Lie algebras. Thus, this map is G), the quotient map U(a) U(a) an étale isogeny. But U(a) admits a T -equivariant linear structure, so there are no nontrivial T -stable ﬁnite étale k-subgroups of U(a) . Hence, the T -equivariant is an isomorphism. In particular, the quotient map carries U map U(a) → U(a) a isomorphically onto Ua for all a ∈ , and each element of = (G , T ) that is a positive rational multiple of an element of actually lies in . It remains to show that every element of is a positive rational multiple of an element of . This is a special case of the ﬁnal part of Remark 2.3.6. 2

2.4 Representability of automorphism functors In the discussion preceding Theorem 1.3.9, we introduced two subgroup functors of the automorphism functor Aut(G) of a smooth afﬁne group G over a ﬁeld k: the functors Aut G,T and Aut G,C of automorphisms respectively restricting to the identity on a maximal torus T and a Cartan subgroup C = ZG (T ). We now prove representability results for these functors, upon which Theorem 1.3.9 was conditionally proved. Theorem 2.4.1 Let G be a smooth connected afﬁne group over a ﬁeld k, and let T be a maximal k-torus in G. Let C = ZG (T ). The functor Aut G,C is represented by an afﬁne k-group scheme Aut G,C of ﬁnite type. If G is pseudo-reductive then the maximal smooth closed k-subgroup scheme ZG,C (in the sense of Lemma C.4.1) of Aut G,C is commutative and has no nontrivial smooth unipotent k0 subgroups. In particular, the identity component ZG,C is pseudo-reductive.

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Proof. By ﬁnite Galois descent (which is effective for afﬁne schemes), it sufﬁces to prove the result after a ﬁnite Galois extension of k. Thus, we can assume that T is k-split. Let denote the set of nontrivial weights for the T -action on Lie(G). (If is empty then G = C by Corollary A.8.11, so Aut G,C = 1 in such cases.) Step 1. For each a ∈ let Ta = (ker a)0red so that, by deﬁnition, U(a) = UZG (Ta ) (λa ) for any λa ∈ X∗ (T ) satisfying a, λa > 0. The k-group U(a) has a natural T -action, so we can deﬁne the automorphism functor Aut U(a) ,T classifying T -equivariant automorphisms of U(a) . Since scheme-theoretic centralizers and the “UH (λ)” construction are characterized by the functors they represent on arbitrary k-algebras (or even k-schemes), for any k-scheme S and S-group automorphism φ of GS that restricts to the identity on TS , we see that φ restricts to an S-group automorphism of each ZG (Ta )S and hence of each UZG (Ta ) (λa )S = (U(a) )S . That is, we get a group functor map Aut G,T → Aut U(a) ,T for each a ∈ . Consider the smooth connected k-subgroup N of G generated by the ksubgroups U(a) . This is normalized by every U(a) and by C (as C normalizes each U(a) ), which collectively generate G (Lemma 2.3.3), so N is normal in G. Clearly G = C · N , which is to say that the map C N → G deﬁned by multiplication is surjective. By [Bo2, 2.2], there is an ordered ﬁnite sequence {ai }i∈I in (with repetitions allowed, and we can assume that every a ∈ occurs at least once) such that the multiplication map of k-schemes i∈I U(ai ) → N is surjective. Thus, we get a surjective map of smooth afﬁne k-schemes Y := C ×

U(ai ) → G.

i∈I

In particular, this identiﬁes the coordinate ring Ø(G) with a k-subalgebra of the coordinate ring Ø(Y ), and Aut G,C with a subfunctor of i Aut U(a ) ,T . i

Step 2. Now the idea is to prove that each Aut U(a) ,T is represented by an afﬁne k-scheme of ﬁnite type, to be denoted by Aut U(a) ,T , and that the subfunctor Aut G,C of i Aut U(a ) ,T is represented by a closed subscheme of the afﬁne i product i Aut U(ai ) ,T . The representability for the Aut U(a) ,T follows from: Lemma 2.4.2 Let U be a smooth connected unipotent k-group equipped with an action by a split k-torus T such that the weights of T on Lie(U ) are nontrivial and are Q× >0 -multiples of each other in X(T )Q . The functor Aut U ,T classifying T -equivariant automorphisms of U is represented by an afﬁne k-group Aut U ,T of ﬁnite type.

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Proof. We may assume U = 1 and can choose a ∈ X(T ) such that all weights a which occur in Lie(U ) have the form a = na a for integers na 1. The k-subgroup ker a acts trivially on Lie(U ), so it acts trivially on U by Corollary A.8.11. Hence, we may replace T with T /(ker a) to reduce to the case when a : T → GL1 is an isomorphism. The coordinate ring k[U ] acquires a k-linear grading ⊕n0 k[U ]n via the T -action (Lemma A.8.8). We claim that k[U ]0 = k. It sufﬁces to check the analogous property for the complete local ring A = ØU ,e at the origin (which contains k[U ] equivariantly). If m denotes the maximal ideal of A, then all weights on m/m2 are positive multiples of a (as this space is equivariantly dual to Lie(U )). Thus, the same holds for its higher tensor powers, and hence for the quotients mi /mi+1 for all i 1. It follows that the space of T -invariant elements of A is k, as desired. Since k[U ] is ﬁnitely generated as a k-algebra, we can choose a ﬁnite set of homogeneous generators, all with degree > 0. Thus, k[U ]n has ﬁnite k-dimension for all n 1. Pick N > 0 such that k[U ] is generated by the k[U ]n for 1 n N . Letting V = ⊕1nN k[U ]n with its evident grading, the functor of graded linear automorphisms of V is represented by a product of general linear groups, and the symmetric algebra Sym(V ) inherits a natural grading so that the quotient map π : Sym(V ) k[U ] respects the gradings. The condition on a graded linear automorphism f of V that Sym( f ) and Sym( f −1 ) preserve ker π is a Zariski-closed condition, so the graded k-algebra automorphism functor of U is represented by an afﬁne k-group of ﬁnite type. This is the functor Aut U ,T , so we are done. 2 Returning to the setting of interest, let R be the coordinate ring of i Aut U(ai ) ,T , so there is a canonical R-scheme automorphism f of YR that is a direct product of the identity on CR and the pullback of the universal T -equivariant automorphism of each U(ai ) . Our aim is to prove that the sub functor Aut G,C of i Aut U(a ) ,T is represented by a quotient of R. It is easier i to think about this in another way: for any k-algebra A and TA -equivariant A-group automorphisms φi of (U(ai ) )A for each i we get a direct product automorphism φ of YA (using the identity on CA ), and we claim that the condition on A-algebras B that φB arises from a (necessarily unique) B-group automorphism of GB is represented by a quotient of A. (Applying this with A = R and the canonical automorphism of YR then gives a quotient of R that represents the subfunctor Aut G,C .)

Step 3. We now proceed in two stages: ﬁnd a quotient of A representing the condition that φB extends to a B-scheme endomorphism of GB , and then impose the conditions of being a B-group endomorphism and ﬁnally automorphism. Now we have a very concrete situation: for an inclusion of k-vector spaces

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V → W (such as Ø(G) → Ø(Y )) and an A-linear endomorphism f of WA , we consider the condition on A-algebras B that fB carries VB into itself. This says that the composite A-linear map f

VA → WA → WA → (W /V )A vanishes after scalar extension to B. Using k-bases of V and W /V to express this composite A-linear map as a “matrix” (with at most ﬁnitely many nonzero entries in each column), the condition is represented by the quotient A/J where J is the ideal generated by the entries of this matrix. Back in the original setting, this proves the existence of a universal quotient of R representing the functor of scheme endomorphisms of G that restrict to T -equivariant automorphisms of the k-subgroups U(a) and restrict to the identity map on C. The condition to be a group scheme endomorphism is a further Zariskiclosed condition (deﬁned by vanishing of entries of a huge matrix expressing the compatibility of a scheme endomorphism with the group scheme structure), so we have constructed an afﬁne k-scheme of ﬁnite type which represents the functor of endomorphisms of the group G that restrict to the identity on C and to automorphisms on the U(a) . But we can likewise construct a universal pair of such endomorphisms and impose the Zariski-closed condition that they are inverse to each other. This represents Aut G,C . (By Remark 2.4.7 below, the ﬁnal step using pairs of endomorphisms is actually not necessary.) Step 4. It remains to analyze the structure of the maximal smooth closed ksubgroup ZG,C of the representing group for Aut G,C : we wish to prove that if G is pseudo-reductive then ZG,C is commutative and contains no nontrivial smooth unipotent k-subgroups. We may assume k = ks , so for commutativity it sufﬁces to prove that the group ZG,C (k) is commutative. By construction ZG,C (k) = Aut G,C (k), so it sufﬁces to show that the group of k-automorphisms of G restricting to the identity on C is commutative. The pseudo-reductivity of G implies (by Proposition 1.2.2) that kautomorphisms of G are faithfully represented by their effect on G red . Since Tk k

is a maximal torus in G red , the problem is reduced to the commutativity of the k

group of automorphisms of G red that are trivial on Tk . The Isogeny Theorem k (Theorem A.4.10) identiﬁes this with the group of points of the image of Tk in the maximal adjoint quotient. In particular, it is commutative. To prove that ZG,C contains no nontrivial smooth unipotent k-subgroups (again assuming that G is pseudo-reductive), we separately treat the cases when char(k) = 0 and char(k) = p > 0. In characteristic 0, the group G is connected reductive, so by Corollary A.4.7(2) the action of T /ZG on G is ﬁnal among

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all smooth connected k-groups equipped with a left action on G ﬁxing C = T pointwise. In particular, the ZG,C -action on G factors through this torus action which visibly restricts to the identity action on C, so by universality it follows that ZG,C = T /ZG . This is connected and reductive, so the case char(k) = 0 is settled. If instead char(k) = p > 0, we observe that a nontrivial smooth unipotent k-group has nontrivial p-torsion k-points (since k = ks ). We saw above that ZG,C (k) is a subgroup of the group of geometric points of the torus image of Tk in the maximal adjoint quotient of G red , and in characteristic p > 0 k there is no nontrivial p-torsion in the geometric points of a torus. 2 The preceding proof used Aut G,C rather than Aut G,T only because the smooth connected normal k-subgroup N generated by the root groups satisﬁes C · N = G, and in general T · N may be strictly smaller (e.g., G may contain a nonreductive smooth connected commutative direct factor). By Proposition A.2.11, if k is inﬁnite then the smooth connected k-subgroup Gt of G generated by the k-tori is normal and satisﬁes (Gt )K = (GK )t for any extension ﬁeld K/k. For completeness we deﬁne Gt for ﬁnite k via Galois descent of (Gk )t . Deﬁnition 2.4.3 A smooth connected afﬁne group G over a ﬁeld k is generated by tori if G = Gt . When G = Gt , the proof of Proposition 2.4.1 works with T in place of C: Corollary 2.4.4 Let G be a smooth connected afﬁne k-group such that G = Gt , and let T be a maximal k-torus of G. The functor Aut G,T is represented by a ﬁnite type afﬁne k-group scheme Aut G,T . The natural map Aut G,C → Aut G,T is a closed immersion, and the maximal smooth closed k-subgroups of Aut G,C and Aut G,T coincide when G is pseudo-reductive. By Proposition A.2.11, the equality G = Gt holds whenever G is perfect. Also, the maximal smooth closed k-subgroups are constructed by Lemma C.4.1. Proof. Using notation from the proof of Theorem 2.4.1, we may again pass to a ﬁnite Galois extension of k so that T is k-split, and deﬁne N as before. To prove the representability claim we just have to check that T · N = G (and then the proof of Theorem 2.4.1 applies without change by using T instead of C everywhere). To verify this equality we may replace G with its smooth quotient G/N to get to the case in which T acts on Lie(G) with no nontrivial weights. Hence, T is central (Corollary A.8.11). But G = Gt , so centrality of T implies G = T . Since the representability is now settled in general, we return to the initial setup and consider properties of Aut G,T . The map Aut G,C → Aut G,T between

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83

afﬁne ﬁnite type k-groups has trivial kernel, so it is a closed immersion (Proposition A.2.1). To compare their maximal smooth closed k-subgroups when G is pseudo-reductive over k, we may extend scalars to ks and then the claim amounts to the equality of sets of k-points. This equality is Example 1.3.8. 2 Example 2.4.5 Over any imperfect ﬁeld k it is easy to give examples of pairs (G, T ) with G perfect pseudo-reductive such that AutG,T is not smooth. Consider k-groups G = R k /k (G ) for a nontrivial purely inseparable ﬁnite extension k /k and a nontrivial split connected semisimple k -group G that is simply connected (e.g., SLn , n 2). In these cases, Theorem 1.3.9 gives an explicit description of the common identity component ZG,C of the maximal smooth closed k-subgroup of Aut G,C and Aut G,T : it is R k /k (T /ZG ) for the maximal k -torus T of G such that C = R k /k (T ). Thus, we just need to exhibit a closed k-subgroup H of Aut G,T with dim H > 0 such that ZG,C ∩ H = 1. Choose C corresponding to a T that is k -split, so (G , T ) (Gk , Tk ) for a k-split pair (G , T ) over k. The algebraic group H of k-algebra automorphisms of k acts naturally on G = R k /k (Gk ) since for any k-algebra A we have G(A) = Gk (A ) = G (A ) for A := k ⊗k A. The H -action clearly restricts to the identity action on T = T . By computing H (k) we see that dim H > 0 since k /k is purely inseparable and nontrivial. Finally, to prove ZG,C ∩ H = 1 we compute A-points for any k-algebra A: (ZG,C ∩ H )(A) = (T /ZG )(A ) ∩ Aut A (A ) ⊆ Aut A (R A /A (G ⊗k A )). This intersection is trivial since the representation of Aut A (A ) on Lie(R A /A (G ⊗k A )) = Lie(G ) ⊗k A is faithful on Lie(T )⊗k A and the action of (T /ZG )(A ) restricts to the identity on this A -submodule. The method of proof of Theorem 2.4.1 and Corollary 2.4.4 gives an interesting result for endomorphisms: Corollary 2.4.6 Let G be a smooth connected afﬁne group over a ﬁeld k, T a maximal k-torus, and C = ZG (T ). For any endomorphism h of G that restricts to the identity on C, the following conditions are equivalent: (i) h is surjective, (ii) h is an automorphism, (iii) Lie(h) is an automorphism. The same holds when h merely restricts to the identity on T provided that G = Gt (e.g., G = D (G)).

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If G is pseudo-reductive then all of these equivalent conditions hold for any endomorphism h such that h restricts to the identity on T . Remark 2.4.7 By using ﬂatness considerations, the same result holds for Sgroup endomorphisms of GS for any k-scheme S. Proof. Since C = ZG (T ) is pseudo-reductive when G is pseudo-reductive, for such G if h|T = id T then h not only induces an endomorphism of C but in fact this endomorphism is the identity (Proposition 1.2.2). Thus, when we later verify that all of the conditions hold in the pseudo-reductive case, it will sufﬁce to verify just one of them. (We will verify (iii).) It is clear that (ii) implies (iii) and that (iii) implies (i). To prove (i) implies (ii) we may and do assume k = k. Now assume h is surjective. For each nontrivial T -weight a on Lie(G), consider the smooth connected unipotent k-subgroup U(a) . Since h is surjective and restricts to the identity on T , by Corollary 2.1.9 it follows that h restricts to a surjective endomorphism of each U(a) . Consider the resulting injective T -equivariant pullback map h∗a on each k[U(a) ]. As we saw in the proof of Lemma 2.4.2, these coordinate rings have ﬁnite-dimensional graded pieces under the grading determined by the T -action. Since h∗a respects the grading and is injective in each degree, it must be an isomorphism in each degree and hence an isomorphism. That is, h restricts to an automorphism of each U(a) . Deﬁne Y as in the proof of Theorem 2.4.1, or if G = Gt and we only assume h to be the identity on T then deﬁne Y similarly except with T in place of C in the deﬁnition of Y as a direct product scheme. In an evident manner, h∗ induces an automorphism f of Ø(Y ) that preserves the subspace Ø(G). The ﬁnite-dimensionality of the graded parts of each k[U(a) ] and the condition that h|C = id C or h|T = id T imply that Ø(Y ) is the directed union of f -stable ﬁnite-dimensional subspaces Vα . Thus, if W is any k-linear subspace of Ø(Y ) that is carried into itself under f (such as Ø(G)) then f restricts to an injective linear endomorphism of every W ∩ Vα . But these are ﬁnite-dimensional, so f is an automorphism on every W ∩ Vα and thus also on W . Taking W = Ø(G) then implies that h is an automorphism. Finally, we check that (iii) holds (so (i) and (ii) hold) when G is pseudoreductive. Equivalently, we will prove that Lie(h) is injective. We may and do assume k = ks , so T is split. Since ker Lie(h) has a weight space decomposition with respect to T , it sufﬁces to check the injectivity of Lie(h) on each T -weight space in Lie(G). The trivial weight space is Lie(ZG (T )), on which Lie(h) is the identity (since an endomorphism of ZG (T ) is uniquely determined by its restriction to T , by Proposition 1.2.2). For each a ∈ (G, T ), h restricts to an endomorphism of ZG (Ta ) (with Ta = (ker a)0red ) and hence also restricts

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85

to an endomorphism of UZG (Ta ) (λa ) = U(a) (with λa ∈ X∗ (T ) satisfying a, λa > 0). Consider the resulting T -equivariant endomorphism ha of Ua . Since Lie(Ua ) contains the a-root space of Lie(G), it sufﬁces to check that ha is an automorphism for all a. If a is not multipliable then Ua has a unique T -equivariant linear structure (Lemma 2.3.8), so ha is a linear endomorphism and thus ker ha is smooth, connected, and unipotent. To deduce that ha is an isomorphism for such a it sufﬁces to show that ker ha = 0, or more generally that ker h contains no nontrivial smooth connected unipotent k-subgroups. If H is a smooth connected k-subgroup of ker h then its G(k)-conjugates generate a smooth connected ksubgroup N ⊆ ker h that is normal in G (as k = ks ). But G is pseudo-reductive, so N is pseudo-reductive. The intersection T ∩ N is a maximal k-torus of N (Corollary A.2.7), yet T ∩ N ⊆ T ∩ ker h = 1 since h restricts to the identity on T , so N is unipotent. In view of the pseudo-reductivity of N it follows that N = 1, so H = 1 as desired. We conclude that ha is a linear isomorphism for non-multipliable a. If a ∈ (G, T ) is multipliable (so char(k) = 2 and all multipliable a are primitive vectors in the lattice X(T )) then ha restricts to the automorphism h2a of U2a , so by the snake lemma for commutative k-group schemes of ﬁnite type we see that the kernel of ha is identiﬁed with the kernel of the induced endomorphism of Ua /U2a . But Ua /U2a admits a unique T -equivariant linear structure, so the induced endomorphism of this quotient is again linear and hence has smooth connected kernel. In other words, ker ha is smooth and connected in the multipliable case as well, so once again it must be trivial (due to the pseudoreductivity of G). In fact, this shows that the linear endomorphism induced on Ua /U2a is an automorphism, so ha : Ua → Ua is also an automorphism. 2

3 Basic structure theory

3.1 Perfect normal subgroups of pseudo-reductive groups In the general study of standardness for pseudo-reductive groups, the most important case turns out to be where G is perfect and the connected semisimple quotient G ss = Gk /R(Gk ) is simple. It is therefore convenient to introduce the k following deﬁnition of Tits [Ti3, II] and a related lemma. Deﬁnition 3.1.1 A pseudo-simple group H over a ﬁeld k is a smooth connected afﬁne k-group that is non-commutative and has no nontrivial smooth connected proper normal k-subgroup. We say that H is absolutely pseudo-simple (over k) if Hks is pseudo-simple over ks . By a specialization argument (as in the proof of Proposition 1.1.9(1)), if K/k is a separable extension ﬁeld then a k-group H is absolutely pseudo-simple if and only if HK is absolutely pseudo-simple over K. Lemma 3.1.2 A group scheme H over a ﬁeld k is pseudo-simple over k if and only if H is pseudo-reductive over k, H = D (H ), and Hkssp is kp -simple, where kp is the perfect closure of k. In particular, H is absolutely pseudo-simple if and only if H is pseudo-reductive, H = D (H ), and H ss is simple. k

This lemma will be used repeatedly, often without comment, when k = ks . Proof. First assume that H is pseudo-simple over k, so D (H ) = H since we require H to be non-commutative (in the deﬁnition of pseudo-simplicity). In particular, R(Hkp ) = Ru (Hkp ) is unipotent. Also, since H cannot be solvable (as D (H ) = H = 1), we cannot have Ru,k (H ) = H , so Ru,k (H ) = 1. That is, H is pseudo-reductive over k. To prove that Hkssp is simple, assume to the contrary. Note that Hkssp = 1 since H is not solvable. Pick a maximal k-torus T ⊆ H , so Tkp is identiﬁed with 86

3.1 Perfect normal subgroups of pseudo-reductive groups

87

a maximal torus in Hkssp . Consider the nontrivial decomposition of Hkssp into an almost direct product of kp -simple smooth connected normal kp -subgroups; this is constructed via Galois descent from the analogous such result for H ss [Bo2, k

14.10] (according to which each smooth connected normal k-subgroup is an almost direct product among a uniquely determined set of the minimal ones). This decomposition over kp induces a nontrivial decomposition of Tkp into an almost direct product of two nontrivial proper kp -subtori S and S that are respectively maximal kp -tori in complementary almost direct products among the kp -simple factors of Hkssp . Since kp /k is purely inseparable, there is a unique k-subtorus T ⊆ T descending S . The normal k-subgroup N of H obtained by Galois-descent of the ks -subgroup of Hks generated by the H (ks )-conjugates of Tks is a nontrivial smooth connected normal k-subgroup of H , and Nkp has proper image in Hkssp , so N = H . This contradicts the pseudo-simplicity of H over k, so Hkssp is simple. Conversely, suppose that H is pseudo-reductive over k, H = D (H ), and Hkssp is simple. Clearly H is non-commutative and Hkred = Hkssp , so R(Hkp ) is p unipotent. Let N ⊆ H be a nontrivial smooth connected normal k-subgroup. This is pseudo-reductive and hence not unipotent, so the image of Nkp in Hkssp must be nontrivial and hence full. It follows that (H /N )ss kp = 1, so H /N is solvable yet it is also equal to its own derived group. We conclude that N = H . 2 Example 3.1.3 As an application of Lemma 3.1.2, we can construct absolutely pseudo-simple groups from certain pseudo-reductive groups, as follows. Let G be a non-commutative pseudo-reductive group over a ﬁeld k, and assume that the nontrivial semisimple group G ss is simple. We claim that D (G) is absolutely k pseudo-simple and that the natural map D (G)ss → G ss is a central isogeny. This k k second claim follows from the ﬁnal part of Proposition A.4.8 (since D (G)ss → k G ss is surjective), so it remains to prove the absolute pseudo-simplicity. k The k-group D (G) is certainly pseudo-reductive, and by Proposition 1.2.6 it is equal to its own derived group. Thus, by Lemma 3.1.2 the k-group D (G) is absolutely pseudo-simple provided that D (G)ss is simple. The k-group D (G)ss k k has a central isogeny to the simple G ss , so we are done. k

(G red , Tk ) k

and (G, T ) when T is k-split (see The relationship between Theorem 2.3.10) enables us to describe derived groups of smooth connected normal k-subgroups in non-commutative pseudo-split pseudo-reductive groups over any ﬁeld k. This requires some preparations in the pseudo-split case (such as over a separably closed ﬁeld), as follows.

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Let G be a pseudo-split non-commutative pseudo-reductive group over a ﬁeld k, and let T be a k-split maximal k-torus in G. The root system := (Gkred , Tk ) in X(Tk ) = X(T ) consists of the non-multipliable elements in (G, T ), and it is not empty since G is not commutative. The structure theory of connected reductive groups implies that under the decomposition = i∈I i into irreducible root systems there is a natural bijection between the non-empty index set I and the set of (pairwise commuting) simple semisimple smooth connected normal subgroups of G red . k Explicitly, for the k-split maximal k-torus S := T ∩ D (G) of D (G) (Corollary A.2.7), the decomposition of corresponds to an isogeny decomposition of S into a product of subtori Si , and the ith simple factor of D (G)ss = D (G red ) (Example A.4.9) contains (Si )k as a maximal torus with k k associated root system i . In general, the smooth connected normal subgroups of D (G)ss = D (G red ) k k are exactly the almost direct products among the simple factors. Thus, we obtain a bijection between the set of such subgroups and the set of subsets of I . We wish to lift this to a description of the derived groups of smooth connected normal k-subgroups of G, and then get an analogous result when G is not assumed to be pseudo-split. Continuing to assume that G is pseudo-split, for i ∈ I let Gi denote the smooth connected k-subgroup of G generated by the U(a) for a ∈ i . For any subset J of I , let J = i∈J i and let GJ denote the subgroup generated by the Gj for j ∈ J . Note that the formation of the Gi and the GJ is compatible with any separable extension of the ground ﬁeld (given that G is pseudo-split). These subgroups will be the building blocks for the derived groups of the smooth connected normal k-subgroups of G, so we ﬁrst prove that they are examples of such subgroups: Lemma 3.1.4 Each GJ is normal in G and is perfect. Proof. By Remark 2.3.5, each GJ is normalized by ZG (T ). Likewise, by Proposition 2.3.9, U(a) commutes with U(a ) for all a ∈ J and a ∈ J . In particular, GJ commutes with U(a ) for all a ∈ J . Since G is generated by ZG (T ) and the k-subgroups U(a) for a ∈ (as their Lie algebras span Lie(G)), it follows that each GJ is normal in G. Hence, GJ is pseudo-reductive over k for all J .

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89

To prove that every k-subgroup GJ in G is perfect, it sufﬁces to prove the stronger claim that for every a ∈ the smooth connected k-subgroup Ha generated by U(a) and U(−a) is perfect. The image of (Ha )k in G red is a simple k semisimple group (of rank 1), so the maximal k-tori of Ha are not central. Since Ha is normalized by T , the k-torus Sa := T ∩ Ha is a maximal k-torus of Ha (Corollary A.2.7), so Sa is 1-dimensional and has nontrivial conjugation action on U(±a) . (In Proposition 3.2.3 we will give a more direct description of Sa .) The T -action on Ha makes the codimension-1 subtorus Ta := (ker a)0red act trivially (as can be checked against the generating subgroups U(±a) ), so T is an almost direct product of Sa and Ta , and a|Sa = 1. Thus, commutators of Sa against U(±a) generate U(±a) . This shows that Ha is perfect for all a ∈ , so GJ is perfect for all J . 2 By Proposition A.4.8 and Lemma 3.1.4, the image of the natural map (GJ )k → G red is identiﬁed with the quotient (GJ )ss of (GJ )k . By construction of k

k

GJ , this image is the smooth connected normal subgroup of D (G red ) = D (G)ss k k generated by the simple factors indexed by elements of J . Lemma 3.1.5 For any J ⊆ I and J := I − J , the k-subgroups GJ and GJ commute with each other and GJ ·GJ = D (G). In particular, GI = D (G); i.e., D (G) is generated by the root groups with respect to any given maximal torus. Proof. Since (U(a) , U(a ) ) = 1 for all a ∈ J and a ∈ J , the groups GJ and GJ commute inside of G. In particular, the product GJ · GJ is a smooth connected k-subgroup of G. It is contained in D (G), so the image of (GJ · GJ )k under the projection Gk G red is equal to D (G)ss (as the image is contained k k in the connected semisimple D (G)ss yet also contains all of its simple factors). k We conclude that the quotient D (G)/(GJ · GJ ) has trivial maximal semisimple quotient over k. This says that D (G)/(GJ · GJ ) is solvable, yet it is its own derived group since D (G) is its own derived group (Proposition 1.2.6). Thus, this quotient is trivial; i.e., GJ · GJ = D (G). 2 The following result uses the concept of a smooth connected afﬁne k-group being “generated by tori”; see Deﬁnition 2.4.3. . For Proposition 3.1.6 Let G be a pseudo-reductive k-group, and G = Gkred p any smooth connected normal k-subgroup N in G, let N denote the image of Nkp in G . Then N → N deﬁnes an inclusion-preserving (in both directions) bijection between the set of smooth connected normal k-subgroups of G generated by tori and the set of smooth connected normal kp -subgroups of G . Under this bijection, N is perfect if and only if N is perfect.

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Proof. By Galois descent we may assume k = ks , so kp = k. Let {Gi }i∈I be the set of simple factors of D (G ), so the set of smooth connected normal ksubgroups of D (G ) is in natural bijection with the set of subsets of I [Bo2, 14.10]. A general smooth connected normal subgroup N in G is reductive, so its center ZN is of multiplicative type. Since ZN is normal in the connected group G , it must be central in G (as groups of multiplicative type have an automorphism scheme that is étale). The resulting inclusion ZN ⊆ ZG implies that the maximal central torus in N is contained in the maximal central torus S of G . Since N is the almost direct product of its maximal central torus and D (N ), we conclude that the set of possible N is in natural bijection with pairs (I , T ) consisting of a subset I ⊆ I and a subtorus T of S . Let Z be the maximal central k-torus in G, so Zk is the maximal central torus in Gk (PropositionA.2.6). The quotient map Gk → G carries Zk isomorphically onto the maximal central torus in G . Indeed, if T is a maximal k-torus of G containing Z then Tk is carried isomorphically onto a maximal torus T in G , so if Zk were not maximal then some subtorus of Tk strictly containing Zk has central image in G . Since k = ks , such a subtorus descends to a subtorus S of T strictly containing Z, and according to Proposition 1.2.2 (and the Zariski-density of S(k) in S) the centrality of Sk in G implies the centrality of S in G. This contradicts the fact that S strictly contains Z. We conclude that every smooth connected normal subgroup N of G lifts to a smooth connected normal ksubgroup of G: a product of a subtorus of Z and some GJ . These speciﬁc liftings N are a product of a central torus and a perfect k-group, so N is generated by tori. This proves the surjectivity of the desired bijection, and it is clear that this reverse construction (going from N to such N ) is inclusion-preserving. It remains to show that N is perfect if and only if N is perfect, and that if N1 and N2 are smooth connected normal k-subgroups of G that are generated by tori and if N2 ⊆ N1 then N2 ⊆ N1 (so in particular, if N1 = N2 then N1 = N2 ). Consider such a pair N1 , N2 . Clearly (N1 N2 ) and N1 coincide in G . But these images of the normal k-subgroups (N1 N2 )k and (N1 )k are the respective maximal reductive quotients (Proposition A.4.8), so N1 N2 /N1 has trivial maximal reductive quotient over k. That is, N1 N2 /N1 is unipotent. But it is generated by tori, so it is trivial. That is, N2 ⊆ N1 , as desired. In general D (N ) = D (N ), and D (N ) is generated by tori (since D (N ) is perfect, by Proposition 1.2.6). Thus, D (N ) = N if and only if D (N ) = N . This yields the equivalence of perfectness for N and N . 2 Now we are ready to describe the possibilities for the derived group D (N ) of a smooth connected normal k-subgroup N in any non-commutative pseudoreductive group G over an arbitrary ﬁeld k. The idea is to construct D (N )

3.1 Perfect normal subgroups of pseudo-reductive groups

91

as a product from among a collection of “explicit” minimal possibilities, much as one does for smooth connected normal subgroups of a nontrivial connected semisimple group over a ﬁeld. Observe that D (N ) is perfect, by Proposition 1.2.6, so it is equivalent to describe those N that are perfect. Remark 3.1.7 It is natural to wonder (based on analogy with the case of connected semisimple groups) if every smooth connected normal k-subgroup N of a nontrivial perfect pseudo-reductive k-group is perfect. This is false over every imperfect ﬁeld, as was shown in Example 1.6.4. Proposition 3.1.8 Let G be a non-commutative pseudo-reductive group over a ﬁeld k. Consider smooth connected normal k-subgroups N ⊆ G that are perfect. The set {Ni }i∈I of minimal such nontrivial k-subgroups of G is ﬁnite, the Ni pairwise commute, and the natural homomorphism π:

Ni → D (G)

(3.1.1)

i∈I

is surjective with central kernel that contains no nontrivial smooth connected k-subgroup. For each subset J ⊆ I , deﬁne NJ to be the subgroup of G generated by the Nj for j ∈ J . Every N has the form NJ for a unique J , and Ni ⊆ N if and only if i ∈ J . Moreover, N = Ni for some i if and only if N is pseudo-simple over k. If G is pseudo-split over k (e.g., k = ks ) then Gkred is kp -split, and the proof p will show that the minimal nontrivial N are precisely the Gi as considered above. Hence, if G is pseudo-split then each N is equal to GJ for some (unique) J , so in particular the Gi are absolutely pseudo-simple over k. Proof. Let G = Gkred . Under the correspondence in Proposition 3.1.6, the Ni p correspond to the simple factors Ni of D (G ). The vanishing of the commutator group (Ni , Nj ) for i = j is a consequence of the vanishing of its image (Ni , Nj ) in G for all i = j. Likewise, the classiﬁcation of all possible N in terms of subsets of I is a consequence of the analogous classiﬁcation of perfect smooth connected normal kp -subgroups of G in terms of products among the kp -simple factors of D (G) = D (G ) (Proposition A.4.8). The surjectivity of (3.1.1) is seen by computing its cokernel. More specif ically, the induced map Ni → D (G) = D (G ) is even an isogeny, so the cokernel of π is a smooth connected afﬁne k-group whose maximal reductive quotient over kp is trivial. That is, the cokernel is unipotent. But D (G) is perfect, so coker π is a unipotent smooth perfect afﬁne group and hence it is trivial.

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Turning our attention to the proof that ker π is central in G = G{i} , this is an instance of the general fact that for a group H generated by a ﬁnite collection of pairwise commuting subgroups {Ni }, the multiplication map from the product Ni has central kernel. Indeed, if (ni ) lies in the kernel then ni = ( j=i nj )−1 centralizes Ni , so ni ∈ ZNi , hence (ni ) is indeed central. Finally, we check that the Ni are pseudo-simple over k. Each Ni is noncommutative, and in general if a non-commutative smooth connected afﬁne k-group H is not pseudo-simple over k then it contains a proper nontrivial smooth connected normal k-subgroup N such that N is either perfect or commutative (take N to be the ﬁnal nontrivial term of the derived series of H ). But in a nontrivial perfect pseudo-reductive k-group H (such as an Ni ), a smooth connected commutative normal k-subgroup N must be trivial. Indeed, perfectness implies H red = H ss , and the image of Nk in here must be trivial (due k k to the structure of connected semisimple groups), so N is unipotent and thus N = 1 since N is pseudo-reductive. We therefore have to show that a nontrivial perfect smooth connected normal k-subgroup in any Ni must equal Ni . Such a k-subgroup is normal in G, by Proposition 1.2.7, so minimality of Ni provides the desired equality. 2 Example 3.1.9 Over any imperfect ﬁeld k there are standard pseudo-reductive G for which the kernel of (3.1.1) has dimension > 0. To make such a G, let k /k be a nontrivial purely inseparable ﬁnite extension and G a connected semisimple k -group that is absolutely simple and simply connected such that the center Z = ZG is not k -étale. (If char(k) = p > 0 then one such G is SLp .) Let T be a maximal k -torus in G and deﬁne C = R k /k (T × T )/R k /k (Z ) using the diagonal embedding of R k /k (Z ). Let j1 : R k /k (T ) → C be the map induced by the inclusion of T into the ﬁrst factor of T × T . Observe that C is pseudo-reductive because it ﬁts into a short exact sequence of k-groups j1

1 → R k /k (T ) → C → R k /k (T )/R k /k (Z ) → 1 in which the outer terms are pseudo-reductive. (For the third term we use that it is naturally a k-subgroup of the commutative pseudo-reductive k-group R k /k (T /Z ).) Since R k /k (Z ) is central in R k /k (G ), the k-group C has a natural left action on R k /k (G × G ) via conjugation by R k /k (T × T ). Thus, it

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93

makes sense to form the semidirect product R k /k (G × G ) C. We embed R k /k (T × T ) as a central k-subgroup of this semidirect product via the anti-diagonal embedding α : x → (x−1 , x mod R k /k (Z )). Consider the k-group G = coker α = (R k /k (G × G ) C)/R k /k (T × T ) = R k /k (G × G )/R k /k (Z ). The k-group G ﬁts into a pushout diagram R k /k (T × T )

/ R k /k (G × G )

C

/ G

of k-groups. Since C is pseudo-reductive over k, this is an instance of the construction of standard pseudo-reductive groups, and so it follows from Proposition 1.4.3 that G is pseudo-reductive over k. Hence, Proposition 3.1.8 can be applied to G. The proof of Proposition 3.1.8 shows that the set {Ni } consists of two copies of R k /k (G ) with the evident embeddings into G, so ker π = R k /k (Z ). This has dimension > 0 since k /k contains an intermediate extension of degree p over k and Zk contains μp , where p = char(k). s

Remark 3.1.10 For the interested reader, we now show how to eliminate the perfectness hypothesis on N in Proposition 1.2.7. Using notation as in that result, but not assuming N to be perfect, since the smooth connected derived group D (N ) is perfect (Proposition 1.2.6 applied to N ) it follows from Proposition 1.2.7 that D (N ) is normal in G. To prove that N is normal in G, we may assume k = ks (so G is pseudo-split) and then deﬁne {Gi }i∈I and GJ (for J ⊆ I ) as above Lemma 3.1.4. It then follows from Proposition 3.1.8 in the pseudo-split case that D (N ) = GJ for some subset J ⊆ I . Hence, by Lemma 1.2.5(i) the subgroups GI −J and N of G commute with each other. Let C be a Cartan k-subgroup of G, so by two applications of Lemma 1.2.5(ii) (to the normal G in G and then the normal N in G ) we see that CN := C ∩ N is a Cartan k-subgroup of N . Thus, by Proposition 1.2.6 and Lemma 3.1.5, we have G = C · D (G) = C · GI = C · GI −J · GJ and N = CN · D (N ) = CN · GJ .

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Basic structure theory

Since GI −J commutes with N , and the commutative C commutes with CN and normalizes GJ , a direct computation gives (G, N ) = (C · GI −J · GJ , CN · GJ ) ⊆ GJ = D (N ) ⊆ N , so N is normal in G.

3.2 Root datum for pseudo-reductive groups To get more insight into the properties of the k-groups Ua in Proposition 2.3.11, we need to introduce a root datum so that we can speak of positive systems of roots. We begin with a preliminary lemma. Lemma 3.2.1 Let f : H → H be a surjective map between smooth connected afﬁne groups over a ﬁeld F, and assume that (ker fF )0red is unipotent. Then for an F-torus T in H and its image T in H , the induced map NH (T ) → NH (T ) is surjective and the induced map NH (T )/ZH (T ) → NH (T )/ZH (T ) is an isomorphism. In particular, ZH (T ) → ZH (T ) is surjective. See Remark C.4.6 for a generalization with weaker hypotheses on H , H , T , and ker f . Proof. We may and do assume that F is algebraically closed. First we check surjectivity for the map between normalizers. Choose h ∈ NH (T )(F) and h ∈ H (F) lifting h via f . Then hTh−1 is a torus of H carried by f onto h T h −1 = T , so hTh−1 is a maximal torus of the smooth connected F-group f −1 (T )0red . But U := (ker f )0red is unipotent and f −1 (T )0red /U → T an isogeny, so f −1 (T )0red is solvable and contains T as a maximal torus. Conjugacy of maximal tori in smooth connected solvable F-groups implies that hTh−1 = uTu−1 for some u ∈ U (F). Thus, replacing h with u−1 h does the job. Now consider the induced map NH (T )/ZH (T ) → NH (T )/ZH (T ). This is a surjection, and both source and target are ﬁnite étale (Lemma A.2.9), so to prove it is an isomorphism we just have to check injectivity on F-points. In other words, if h ∈ H (F) normalizes T and if f (h) centralizes T then we want h to centralize T . Since (ker f )0red is unipotent, the map T → T induced by f is an isogeny. Thus, the desired result is clear. 2 Now let G be a pseudo-split pseudo-reductive group over a ﬁeld k and let T be a split maximal k-torus in G. Deﬁnition 3.2.2 Let R = (G, T ), X = X(T ), and X ∨ = X∗ (T ). The elements of R are called the roots of G relative to T , and the subset R∨ ⊆ X ∨

3.2 Root datum for pseudo-reductive groups

95

of coroots of G relative to T is deﬁned as follows. For any non-multipliable a ∈ R (i.e., a ∈ (G red , Tk )) the coroot a∨ ∈ X ∨ = X∗ (Tk ) is the 1-parameter k

k-subgroup of T corresponding to a viewed as a root for G red relative to Tk . k For a multipliable a ∈ R, deﬁne a∨ = 2 · (2a)∨ . It is trivial to check that a → a∨ is a bijection from R to R∨ , with the non-multipliable (resp. non-divisible) roots corresponding to the nondivisible (resp. non-multipliable) coroots. It is also straightforward to check that a, a∨ = 2 for all a ∈ R, and that the respective reﬂections ra : x → x − x, a∨ a, ra∨ : λ = λ − a, λa∨ ∨ for of X and X ∨ are dual to each other and satisfy ra = r2a and ra∨ = r2a multipliable a. We can describe the image of a∨ : GL1 → T in concrete terms via root groups, just as in the connected reductive case, as follows. Let Ga denote the smooth connected k-subgroup of G generated by Ua and U−a . This is normalized by T , so T ∩Ga is a maximal k-torus of Ga (Corollary A.2.7). In Proposition 3.4.1 we will give a useful alternative description of Ga and its structure. The coroot a∨ is valued in Ga :

Proposition 3.2.3 The k-subgroup a∨ (GL1 ) in T is T ∩ Ga . Proof. Let a = a when a is non-multipliable and a = 2a when a is multipliable. The quotient map Gk → G red carries (Ua )k into the 1-dimensional k

a -root group of (G red , Tk ) (Remark 2.3.6), and the image of (Ua )k is nontrivial k since G is pseudo-reductive (Lemma 1.2.1). Thus, (Ga )k is carried onto the rank-1 semisimple subgroup (G red )a with unipotent kernel. Hence, the maxik

mal torus (T ∩ Ga )k maps isomorphically onto the maximal torus Tk ∩ (G red )a k

that is precisely a ∨ (GL1 ) by the deﬁnition of coroots in connected reductive k

groups over k. It follows that the unique k-torus in T that descends the subtorus a ∨ (GL1 ) in Tk is T ∩ Ga , so a ∨ has image T ∩ Ga . But a∨ ∈ {a ∨ , 2a ∨ } by k

deﬁnition, so a∨ has the same image as a ∨ .

2

We wish to naturally associate a root datum to any pseudo-split pair (G, T ) as above. Recall that a root datum (X , R, X ∨ , R∨ ) implicitly involves the speciﬁcation of a bijection a → a∨ between R and R∨ in order to deﬁne the reﬂections ra and ra∨ in the axioms. This bijection can be omitted from the notation, due to: Lemma 3.2.4 The bijection a → a∨ in a root datum (X , R, X ∨ , R∨ ) is uniquely determined. That is, if f : R → R∨ is a bijection with respect to which the

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axioms are satisﬁed then f (a) = a∨ for all a ∈ R. Moreover, for all a, b ∈ R we have ra (b)∨ = ra∨ (b∨ ). Proof. By [SGA3, XXI, 1.1.4], a coroot c∗ is uniquely determined by the function a → a, c∗ on R. Thus, to prove the uniqueness it sufﬁces to show that the functional ·, c∗ on the Q-vector space V spanned by R is uniquely determined. If c ∈ R is the unique root so that c∗ = c∨ then rc : x → x − x, c∗ c is a reﬂection of V that carries R into itself and negates c, so this reﬂection is uniquely determined by the nonzero c [Bou2, VI, 1.1, Lemma]. Since rc and c determine the functional ·, c∗ , uniqueness is proved. For a ﬁxed a ∈ R, since the respective reﬂections ra and ra∨ on X and X ∨ are dual to each other and preserve R and R∨ , by the preceding uniqueness they must intertwine the bijection b → b∨ with itself. That is, ra (b)∨ = ra∨ (b∨ ) for all b ∈ R. 2 Deﬁnition 3.2.5 The root datum associated to the pseudo-split pair (G, T ) with pseudo-reductive G is the 4-tuple R(G, T ) = (X(T ), (G, T ), X∗ (T ), (G, T )∨ ) equipped with the bijection a → a∨ between (G, T ) and (G, T )∨ as deﬁned above. To justify this terminology, we must verify that the axioms for a root datum are satisﬁed. By Theorem 2.3.10, the root datum for (G red , Tk ) is obtained k by removing the multipliable roots and their associated (divisible) coroots for (G, T ). Thus, since the roots and coroots for (G red , Tk ) satisfy the axioms for k a root datum, we only need to check that the reﬂections ra carry multipliable elements of (G, T ) into (G, T ) and that the reﬂections ra∨ carry divisible elements of (G, T )∨ into (G, T )∨ (for any a ∈ (G, T )). In view of how the reﬂections were deﬁned, this amounts to checking that if a, b ∈ (G red , Tk ) with b divisible in (G, T ) (which can only occur when k is k imperfect with characteristic 2) then ra (b/2) ∈ (G, T ) (i.e., ra (b) is divisible) and ra∨ (2b∨ ) ∈ (G, T )∨ . By Lemma 3.2.4 we have ra∨ (b∨ ) = ra (b)∨ for a, b ∈ (G red , Tk ), so since c∨ was deﬁned to be 2(2c)∨ for multipliable k c ∈ (G, T ), it sufﬁces to check that ra (b/2) ∈ (G, T ) when b is divisible. The reﬂections ra of X(Tk ) are (by deﬁnition) induced by the action of NGred (Tk ) on Tk , so the problem is to show that this action on X(Tk ) = X(T ) = k

X(Tks ) carries (G, T ) = (Gks , Tks ) into itself. This property is an immediate consequence of Lemma 3.2.1 (applied to Gk G red ) because the quotient k

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NG (T )/ZG (T ) is étale (Lemma A.2.9) and hence the set of its geometric points is exactly NG (T )(ks )/ZG (T )(ks ) (since ZG (T ) is smooth). If (X , R, X ∨ , R∨ ) is any root datum then its root lattice Q is the Z-span of R in X and its coroot lattice Q∨ ⊆ X ∨ is the Z-span of R∨ . We call X the character lattice and X ∨ the cocharacter lattice of the root datum. The root datum is semisimple if the root lattice has ﬁnite index in the character lattice, in which case its weight lattice P is the Z-dual of Q∨ inside of QR . For a semisimple root datum we have Q ⊆ X ⊆ P, and the root datum is simply connected when X = P (equivalently, X ∨ = Q∨ ) and is adjoint when X = Q. Example 3.2.6 The root datum R(G, T ) is semisimple if and only if G red is k semisimple, in which case the coroot lattices (and hence the weight lattices) for R(G, T ) and R(G red , Tk ) coincide since removing some divisible coroots k does not affect the coroot lattice. Hence, R(G, T ) is semisimple and simply connected precisely when G red is semisimple and simply connected. k

The Weyl group of the root system (R, QR ) attached to a root datum (X , R, X ∨ , R∨ ) is naturally identiﬁed with a subgroup of Aut(X ) via the canon∨ )⊥ , with (Q ∨ )⊥ denoting the annihilator of ical decomposition XQ = QQ ⊕(QQ Q ∨ QQ in XQ . In the case of the root datum arising from a pseudo-split pair (G, T ) as above, the Weyl group of the associated root system is related to (G, T ) as follows. For any smooth connected afﬁne k-group G and k-torus T in G, the Weyl group W (G, T ) for the pair (G, T ) is the ﬁnite étale quotient k-group NG (T )/ZG (T ). Since ZG (T ) is smooth, we have NG (T )(ks )/ZG (T )(ks ) = W (G, T )(ks ) = W (G, T )(k). When T is maximal and k-split and G is pseudo-reductive, we can do better: Proposition 3.2.7 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and let T be a split maximal k-torus in G. The ﬁnite étale k-group W (G, T ) is constant and the natural map W (G, T )(k) → Aut(X ) carries W (G, T )(k) isomorphically onto the Weyl group of the root system associated to (G, T ). Also, the natural map W (G, T )k → W (G red , Tk ) is an isomorphism. k

In Corollary 3.4.3(1) we will prove that the inclusion NG (T )(k)/ZG (T )(k) → W (G, T )(k) is an equality. In Proposition C.2.10 we will prove this and the constancy of W (G, T ) without a pseudo-reductive hypothesis (by reducing to the pseudo-reductive case). Proof. Since ra = r2a for multipliable a ∈ (G, T ), the Weyl group W of the root system of (G, T ) coincides with that of the root system for (G red , Tk ). The k

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k-rational character group of T naturally coincides with the character groups of Tks and Tk ; denote them all as X , so the diagram / W (G red , T ) W (G, T )(ks ) k k NNN NNN NNN NNN & Aut(X ) commutes, with the right side an isomorphism onto W . The horizontal arrow in the diagram is bijective, by Lemma 3.2.1, so the diagonal arrow is bijective onto W . Since this diagonal arrow is Gal(ks /k)-equivariant yet the Galois action on X is trivial, W (G, T )(ks ) also has trivial Galois action. Thus, W (G, T ) is a constant k-group. 2 The set (G, T ) is empty if and only if G red is a torus, which is to say that k G is solvable, or equivalently commutative. Since (G, T ) is a root system in QR = X(T )R , where Q = Q(G, T ) is the root lattice of R(G, T ), we can speak of a positive system of roots in (G, T ), as for any root system. Example 3.2.8 If the root system (G, T ) is irreducible and non-reduced (of which there is exactly one example, BCr , in each rank r 1) then removing the multipliable roots yields a root system of type C. Thus, if R(G, T ) is a nonreduced root datum then certain simple factors of type C in G red will play a k distinguished role. Remark 3.2.9 Let k be a separably closed ﬁeld. We know that when char(k) = 2, all pseudo-reductive k-groups have a reduced root system (by Theorem 2.3.10). In all characteristics we claim that standard pseudo-reductive k-groups have a reduced root system. To prove this, ﬁrst observe (via Proposition 2.3.15) that reducedness of the root system is insensitive to passage to a pseudo-reductive central quotient. Thus, from the deﬁnition of standardness it sufﬁces to prove that for any separably closed ﬁeld k, nonzero ﬁnite reduced k-algebra k , and reductive k -group G with connected ﬁbers, the Weil restriction R k /k (G ) has a reduced root system. This follows from Example 2.3.2. In Corollary 4.1.6 we will obtain the reducedness of the root system in a more precise form that keeps track of the root spaces. Let G be a nontrivial perfect pseudo-reductive group over a ﬁeld k, and let {Gi }i∈I be the non-empty set of minimal nontrivial perfect smooth connected normal k-subgroups of G. By Proposition 3.1.8, the k-subgroups Gi pairwise

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commute and are pseudo-simple over k, and the natural homomorphism π:

Gi → G

is surjective with central kernel that contains no nontrivial smooth connected k-subgroup (in particular, it contains no nontrivial k-tori). The following reﬁnement relates maximal k-tori of G to collections of maximal k-tori in the k-subgroups Gi , and in the pseudo-split case relates the corresponding root systems inside of a character lattice. Proposition 3.2.10 Using the preceding notation with a nontrivial perfect pseudo-reductive group G over a ﬁeld k, there are mutually inverse bijective correspondences between the set of maximal k-tori T in G and the set of collec tions {Ti } for a maximal k-torus Ti in Gi via T → {T ∩Gi } and {Ti } → π( Ti ). The natural map π : Ti → T := π( Ti ) is an isogeny, and T is k-split if and only if every Ti is k-split. If T is a k-split maximal torus in G, the resulting isomorphism X(T )Q = X(Ti )Q identiﬁes the root system (G, T ) with the direct sum of the root systems (Gi , Ti ). In particular, if G is pseudo-split then it has a non-reduced root system if and only if some Gi has a non-reduced root system. Proof. Given a choice of the Ti , their product Ti is a maximal k-torus in Gi , so it maps onto a maximal k-torus T in the quotient G. Since ker π contains no nontrivial k-tori, the map Ti → T is an isogeny. Each T ∩ Gi is a maximal k-torus in Gi (Corollary A.2.7), and the containment Ti ⊆ T ∩ Gi is an equality by maximality of Ti . In the reverse direction, if T is a maximal k-torus in G and we deﬁne Ti := T ∩ Gi then every Ti is a maximal k-torus in Gi , so π( Ti ) is a maximal k-torus in the image G of π . Thus, the containment π( Ti ) ⊆ T is an equality. In particular, if every Ti is k-split then so is T , and the converse is obvious. Since Gi → G is a central quotient map between pseudo-reductive groups, and it carries Ti onto T , Proposition 2.3.15 reduces the desired relation among root systems in the pseudo-split case to the special case G = Gi , and this case is trivial. 2

3.3 Unipotent groups associated to semigroups of roots For a split connected reductive group G over a ﬁeld k, and a k-split maximal k-torus T , it is useful to know when a set of root groups {Ua }a∈ directly spans (in any order) a smooth connected unipotent k-subgroup U in the sense that mul tiplication deﬁnes a k-scheme isomorphism i Uai U for any enumeration

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{ai } of . There is a sufﬁcient criterion for this to hold [Bo2, 14.5] (expressed in terms of additivity and positivity properties of inside of (G, T )), and we seek an analogue in the pseudo-reductive case. This will be done by a method (inspired by [BoTi2, §2]) which is completely different from the reductive case, and moreover works for arbitrary smooth connected afﬁne k-groups equipped with a left action by a split torus. Rather than proving direct spanning simultaneously with the construction of the ambient unipotent group (as in the reductive case), we will ﬁrst build the unipotent group and only afterwards infer direct spanning properties. We use the following conventions and notation with semigroups. We require semigroups to be associative and commutative, but do not require an identity element or non-emptiness. If and are subsemigroups of a semigroup X then + denotes the subsemigroup of sums γ + γ with γ ∈ and γ ∈ , and , denotes the subsemigroup ∪ ∪ ( + ) generated by and . Finally, if is a subset of X then denotes the subsemigroup of ﬁnite sums of elements of ; this is empty if is empty. Let G be a smooth afﬁne group over a ﬁeld k, S a k-split torus equipped with a left action on G, and (G, S) the set of S-weights that occur in Lie(G). (If S is a split maximal k-torus in G acting by conjugation and G = 1 then (G, S) = (G, S) ∪ {0}.) Let Ø(G) be the coordinate ring of G, and IG = ker(e∗ : Ø(G) → k) the augmentation ideal (so Lie(G) = (IG /IG2 )∗ ). We make S act on the k-algebra Ø(G) by (s.f )(g) = f (s.g), so (G, S) coincides with the set of S-weights on IG /IG2 ; this is consistent with the convention implicit in (2.1.2). Remark 3.3.1 We will need to form centralizers for S-actions. In general, for any locally ﬁnite type k-group G and k-group M of multiplicative type acting on G, PropositionA.8.10(1) provides the existence of a closed subscheme G M ⊆ G representing the functor of M -ﬁxed points in G, and this is smooth when G is smooth. In the special case that G is smooth and afﬁne and M is a k-torus, this can be seen more directly by applying Proposition A.2.5 to M G 0 . For a pair of smooth connected k-subgroups H , H ⊆ G that are normalized by S, we wish to understand the S-weights that appear in the Lie algebra of the k-subgroup they generate, as well as the Lie algebra of their commutator (H , H ). It turns out to be simpler to work with the set of S-weights appearing on the entire augmentation ideal of such groups, rather than focusing on just their Lie algebras, so we will begin with a study of this aspect in a more general setting. (It is essential that we require S-stability for the k-subgroup and not merely for its Lie algebra. The reason for this is illustrated by the following example.

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For a ﬁeld k of characteristic p > 0, let G = Ga × Ga be endowed with an action of a 2-dimensional k-split k-torus T via t.(x, x ) = (χ (t)x, χ (t)x ), where χ , χ ∈ X(T ) are linearly independent. Let H ⊂ G be the ﬁrst Ga -factor and let H ⊂ G be the graph of the relative Frobenius FGa /k , which is to say H = {(x, x p )}. Clearly H is T -stable but H is not. The Lie algebras Lie(H ) and Lie(H ) coincide in Lie(G) with the weight space for χ , and the k-groups H and H generate the k-group G whose Lie algebra supports the weight χ that is linearly independent from χ .) Let V be a geometrically integral S-stable closed subscheme of G that passes through the identity. (A natural example is the closure of the image of the commutator map H × H → G for smooth connected S-stable k-subgroups H and H of G.) Deﬁne IV = ker(e∗ : Ø(V ) → k), and let wt(V ) denote the set of S-weights which occur in IV . Since Ø(V ) is a domain, products of (nonzero) eigenfunctions for S in IV are (nonzero) eigenfunctions, so wt(V ) is a subsemigroup of X(S). (If V = {e} then wt(V ) is empty.) The formation of wt(V ) commutes with any extension of the ground ﬁeld. Example 3.3.2 Let H be an S-stable smooth connected k-subgroup of G. We claim that wt(H ) = (H , S). To prove this, since (H , S) is the set of S-weights occurring in IH /IH2 we certainly have (H , S) ⊆ wt(H ), so (H , S) ⊆ wt(H ). For the reverse inclusion, since Ø(H ) is a noetherian domain it injects into the IH -adic completion ØH ,e of ØH ,e = Ø(H )IH . Thus, wt(H ) is contained in the set of S-weights occurring on the maximal ideal m of ØH ,e . Using the separated IH -adic ﬁltration, every S-weight on m occurs in IHr /IHr+1 for some r 1. But this subquotient is an S-equivariant quotient of (IH /IH2 )⊗r , so the weights obtained all lie in (H , S). Example 3.3.3 Assume char(k) = p > 0, and consider the k-homomorphism FG/k : G → G (p) given by relative Frobenius. This is S-equivariant if we make S act on G (p) via the composition of the natural S (p) -action (deﬁned via base change on k by the p-power map) and the k-homomorphism FS/k : S → S (p) . Since FGL1 /Fp : GL1 → GL1 is the p-power map and S is k-split, for any geometrically integral closed subscheme V ⊆ G passing through e we have wt(V (p) ) = p wt(V ). In particular, if H is an S-stable smooth connected ksubgroup then wt(H (p) ) = p wt(H ). In the special case that H is commutative we have ker FH /k ⊆ H [p], so the S-equivariant p : H → H factors through FH /k and hence wt(pH ) ⊆ p wt(H ). Lemma 3.3.4 Let V , V ⊆ G be S-stable geometrically integral closed subschemes passing through the identity. Let VV denote the closure of the image of the multiplication map V ×V → G, and let (V , V ) denote the closure of the

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image of the commutator map V × V → G deﬁned by (v, v ) → vv v −1 v −1 . Then wt(VV ) = wt(V ), wt(V ) and wt((V , V )) ⊆ wt(V ) + wt(V ). Proof. The dominant map V × V → VV induces an injection Ø(VV ) → Ø(V ) ⊗k Ø(V ), so we get an S-equivariant embedding IVV ⊆ IV ⊕ IV ⊕ (IV ⊗k IV ). Thus, wt(VV ) ⊆ wt(V ), wt(V ). Since wt(VV ) is a subsemigroup of X(S), to prove the reverse inclusion we just need to show that wt(V ) and wt(V ) are contained in wt(VV ), and this follows from the fact that V and V are contained in VV (so IV and IV are S-equivariant quotients of IVV ). The commutator map V × V → (V , V ) restricts to the constant map e on the slices V × {e} and {e} × V , so the same method shows that wt((V , V )) ⊆ wt(V ) + wt(V ). 2 By [Bo2, 2.2], for sufﬁciently large n the images of the multiplication maps V n → G stabilize at a common geometrically integral image V that contains e. Hence, the multiplication map V × V → G factors through V , so V is a closed k-subsemigroup of G passing through the identity. But any closed ksubsemigroup scheme H in G passing through the identity must be a k-subgroup of G. To see this, observe that for any k-algebra R and point h ∈ H (R), the left multiplication mapping h : HR → HR is a monomorphism from HR into itself, so it must be an isomorphism [EGA, IV4 , 17.9.6]. In particular, since H passes through e it follows that H is stable under inversion, so it is indeed a k-subgroup of G. This proves that V is the smooth connected k-subgroup of G generated by V , and repeated application of Lemma 3.3.4 gives that wt(V ) = wt(V ). Since V is a group we also conclude that wt(V −1 ) = wt(V ). Further repeated application of Lemma 3.3.4 yields the following result, which, together with Example 3.3.2, gives us some information about the Sweights on Lie(H ) and Lie((H , H )) in terms of those on Lie(H ) and Lie(H ). Proposition 3.3.5 If H , H ⊆ G are smooth connected k-subgroups normalized by the S-action and H denotes the k-subgroup they generate, then wt(H ) = wt(H ), wt(H ). Also, wt((H , H )) ⊆ wt(H ) + wt(H ). It is natural to ask what constraints are imposed on the subsets wt(H ) ⊆ X(S) beyond the subsemigroup property. There are no further constraints, and there is a canonical way to go in reverse: Proposition 3.3.6 Let G be a smooth afﬁne group over a ﬁeld k, and S a k-split k-torus equipped with an action on G. For every subsemigroup A ⊆ X(S), there is a unique smooth connected S-stable k-subgroup HA (G) in G such that

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Lie(HA (G)) is the span of the weight spaces in Lie(G) for all a ∈ (G, S) ∩ A. The k-subgroups HA (G) and HA (G 0 ) coincide, and if 0 ∈ A then HA (G) is unipotent. The k-subgroup HA (G) is maximal in the sense that a smooth connected S-stable k-subgroup H in G satisﬁes (H , S) ⊆ A if and only if H ⊆ HA (G). When 0 ∈ A we will often write UA (G) instead of HA (G). Proof. If H and H are k-subgroups satisfying the requirements for HA (G) then by Proposition 3.3.5 the k-subgroup H they generate is another, yet its Lie algebra cannot be any larger, so by smoothness and connectedness we have H = H = H . This proves uniqueness. In particular, once existence is proved in general then HA (G 0 ) = HA (G). To prove existence of HA (G), the case A = ∅ is trivial (H = 1), so by Proposition 3.3.5 it sufﬁces to treat the case where A is the cyclic subsemigroup a generated by a single element a ∈ X(S). If a = 0 then (G S )0 works. If a = 0 then Ma := ker a is a (k-split) ksubgroup of S, so G Ma makes sense and is smooth since G is smooth. The 1-dimensional split torus Sa = S/Ma naturally acts on G Ma and is identiﬁed with GL1 via a. By Proposition A.8.10(1), Lie(G Ma ) is the span of the weight spaces for the characters in (G, S) that factor through Sa , which is to say that they are the Z-multiples of a which occur on Lie(G). Letting λ : GL1 Sa be the isomorphism inverse to a, UGMa (λ) is a smooth connected k-subgroup of G by Proposition 2.1.8(3),(4), and its Lie algebra is the span of the weight spaces for the elements of (G, S) ∩ a. This completes the proof of existence. For any A, consider an S-stable smooth connected k-subgroup H in G such that (H , S) ⊆ A. Example 3.3.2 and Proposition 3.3.5 imply that the smooth connected k-subgroup H generated by H and HA (G) satisﬁes wt(H ) ⊆ A, so Lie(H ) has all S-weights in A. But Lie(HA (G)) is the span of all such weight spaces, so the inclusion HA (G) ⊆ H is an equality on Lie algebras and hence is an equality of k-groups. In particular, H ⊆ HA (G). Now assume 0 ∈ A, so S acts on HA (G) with only nontrivial weights on Lie(HA (G)). To prove that HA (G) is unipotent, we use the elementary fact (explained in the proof of Proposition 2.3.11) that if H is a smooth connected afﬁne k-group and a split k-torus S acts on H with only nontrivial weights on Lie(H ) then H is unipotent. 2 Example 3.3.7 Let G be a pseudo-split pseudo-reductive group over k and T a split maximal k-torus. For any a ∈ (G, T ) the k-subgroup Ua (G) is the root group Ua in Proposition 2.3.11 (in view of the constraints on multipliability among roots in Theorem 2.3.10).

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It is immediate from Proposition 3.3.6 that if A ⊆ A then HA (G) ⊆ HA (G), so if A and A are two subsemigroups in X(S) then the smooth connected k-subgroup of G generated by HA (G) and HA (G) is contained in HA,A (G). Likewise, it follows from Proposition 3.3.5 that (HA (G), HA (G)) ⊆ HA+A (G). Since torus centralizers and UG (λ) are both functorial in k-group homomorphisms, it follows from the construction that if f : G → G is an S-equivariant k-homomorphism then f carries HA (G ) into HA (G); the same even holds for S-equivariant R-group homomorphisms GR → GR for any k-algebra R since the theory of UG (λ) works over any ring. In the special case of smooth S-equivariant surjective homomorphisms f : G → G, the induced map fA : HA (G ) → HA (G) is surjective on Lie algebras (as Lie( f ) is an S-equivariant quotient map), so fA is a smooth surjection (due to connectedness). We now focus on the case UA (G) for cyclic A = a with a nonzero a ∈ X(S). There is a natural decreasing ﬁltration on A = a given by the subsemigroups A j consisting of the characters na for n j (with j 1). This gives rise to a decreasing ﬁltration of HA (G) by smooth connected k-subgroups HA j (G). The following result generalizes the structure of root groups in the pseudosplit pseudo-reductive case (including the 2-step ﬁltration {1} ⊂ U2a ⊂ Ua for multipliable roots a), as well as the structure of non-commutative root groups which arise in the theory of relative root systems for non-split connected reductive groups. Lemma 3.3.8 For A = a with a ∈ X(S) − {0}, each UA j (G) is normal in UA (G) and the quotient Vj = UA j (G)/UA j+1 (G) is a vector group such that (Vj , S) = {ja} if Vj = 0. In particular, Vj has a unique S-equivariant linear structure. Proof. The commutator (UA (G), UA j (G)) is contained in UA+A j (G) = UA j+1 (G) ⊆ UA j (G), so UA (G) normalizes UA j (G). Likewise, (UA j (G), UA j (G)) ⊆ UA j +A j (G) = UA2j (G) ⊆ UA j+1 (G), so each Vj := UA j (G)/UA j+1 (G) is commutative. By construction, Vj is also smooth, connected, and unipotent. Since A j − A j+1 = {ja}, Lie(Vj ) clearly has ja as its unique S-weight if Vj = 0. If char(k) = 0 then a smooth connected commutative unipotent k-group is a vector group and admits a unique linear structure, so we are done in such cases. Now assume char(k) = p > 0. By Proposition B.4.1, to show that Vj is a vector group and admits an S-equivariant linear structure, it sufﬁces to show that Vj is p-torsion. The linear structure will then be unique, due to uniqueness of the S-weight on Lie(Vj ) when Vj = 0, exactly as in the proof of Lemma

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2.3.8. We may assume Vj = 0, so by Example 3.3.2, wt(Vj ) = ja. Thus, Example 3.3.3 implies wt(pVj ) ⊆ pja, so Lie(pVj ) has no occurrence of the weight ja. Since ja is the only S-weight on Lie(Vj ), necessarily pVj = 0. 2 Returning to the consideration of general A (not necessarily cyclic), we now study the behavior of HA (G) with respect to S-stable smooth connected ksubgroups G of G, generalizing Remark 2.3.6 (via Remark 2.1.11). We allow for the possibility 0 ∈ A. By functoriality HA (G ) ⊆ G ∩ HA (G), and the Lie algebras coincide. But HA (G ) is smooth and connected, so G ∩ HA (G) has dimension at least as large as that of its Lie algebra. This proves: Lemma 3.3.9 For any S-stable smooth connected k-subgroup G in G, the intersection G ∩ HA (G) is smooth and (G ∩ HA (G))0 = HA (G ). It is natural to ask when G ∩ HA (G) is connected. Connectedness holds if A is a saturated subgroup in X(S). Indeed, in such cases SA := a∈A ker a is a k-subtorus of S (since A is saturated) and it centralizes HA (G) (by Corollary A.8.11), so there is an inclusion G ∩ HA (G) ⊆ G SA which is an equality on Lie algebras (since A is saturated). This must then be an equality of k-groups, due to the connectedness of G SA (which follows from the formula ZG SA (SA ) = G SA × SA ). In general, G ∩ HA (G) can be disconnected. For example, consider the inclusion G = PGL2 → GL(pgl2 ) GL3 =: G deﬁned by the adjoint representation. Let S be a split maximal k-torus in PGL2 , and A the index-2 sublattice in X(S) Z, so HA (G) = ZG (S[2]) and hence G ∩ HA (G) = ZG (S[2]). By inspection this is NG (S), which is disconnected (with identity component S). Disconnectedness never occurs when 0 ∈ A, and more generally we have: Proposition 3.3.10 Let G be a smooth afﬁne k-group, and S a k-split k-torus equipped with an action on G. Let G be an S-stable smooth connected closed k-subgroup of G. We have G ∩ HA (G) = HA (G ) (equivalently, G ∩ HA (G) is connected) in any of the following cases: (a) 0 ∈ A, (b) G is solvable, (c) G is normal in G. Proof. Suppose G ∩ HA (G) is disconnected. We will show that 0 ∈ A, so (a) follows. Consider the idempotent f on G ∩ HA (G) that vanishes on the identity component (and hence at e) but equals 1 elsewhere, so f is S-invariant and lies in the augmentation ideal of Ø(G ∩ HA (G)). The coordinate ring Ø(G ∩ HA (G)) is a quotient of Ø(HA (G)), so f lifts to an S-invariant element in IHA (G) . But the only S-weights in IHA (G) are elements of A, so 0 ∈ A.

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To prove (b) and (c), we may assume 0 ∈ A and k = k. We will prove (c) conditional on (b), and then prove (b). Assuming G is normal in G, let U = Ru (G ) and U = Ru (G), so U = G ∩ U by Proposition A.4.8. (Equivalently, the natural map G red → G red has trivial kernel.) We will reduce to the case where G is reductive, and then to the case where G is also reductive. Applying (b) to U gives that U ∩ HA (G) is connected, so G ∩ HA (G) is connected if and only if the k-group (G ∩ HA (G))/(U ∩ HA (G)) = ((G ∩ HA (G))U )/U is connected. But HA (G)U /U = HA (G/U ), so to prove (c) we may replace G with G /U and replace G with G/U to arrange that G is reductive. In particular, G ∩ U = 1, so G is naturally a k-subgroup of G/U and we have the inclusions HA (G ) → G ∩ HA (G) → G ∩ HA (G/U ). Thus, if the outer terms are equal then so are the ﬁrst two, so we may replace G with G/U to arrange that G is also reductive. With G reductive, it follows from Corollary A.4.7(2) that the S-action on G is given by composing some k-homomorphism S → G/ZG with the natural left action of G/ZG on G. The image of S in G/ZG is contained in a maximal k-torus T , and T = T /ZG for a unique maximal k-torus T in G. Thus, by Lie algebra considerations we can describe HA (G) explicitly (using that 0 ∈ A): it is generated by T = ZG (T ) and the root groups Ua for a ∈ (G, T ) ⊆ X(T ) such that a|S ∈ A (via the restriction map X(T ) → X(S)). An analogous procedure describes HA (G/ZG ), using T and root groups in G/ZG . Comparing these similar descriptions of HA (G) and HA (G/ZG ), we see that HA (G/ZG ) is the image of HA (G) (even though G → G/ZG may not be smooth). But ZG ⊆ T ⊆ HA (G), so HA (G) is the full preimage of HA (G/ZG ) in G. Running through the same argument for the reductive G , we get that HA (G ) is the full preimage of HA (G /ZG ) in G . Our aim is to prove that the containment HA (G ) ⊆ G ∩ HA (G) is an equality. Since G is normal in G, we have ZG ⊆ ZG (Proposition A.1.13). Thus both sides contain ZG , so it is equivalent to check the equality of the images in G /ZG . The induced map G /ZG → G/ZG between maximal adjoint quotients has trivial kernel (Proposition A.4.8), so since ZG ⊆ HA (G) it follows that the image of G ∩ HA (G) in G /ZG is (G /ZG ) ∩ HA (G/ZG ). Hence, we

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can replace G and G with their respective maximal adjoint quotients, in which case G is a direct factor of G and so the desired equality is clear. This proves (c), conditional on (b). Finally, we prove (b). Since HA (G) = HA (G 0 ), we may replace G with G 0 so that G is connected. In particular, G S is connected (by Proposition A.2.5 applied to S G). Let us ﬁx a 1-parameter k-subgroup λ : GL1 → S such that χ , λ = 0 for all nonzero χ ∈ (G, S). Thus, the containment G S ⊆ ZG (λ) is an equality since it induces an equality of Lie algebras and ZG (λ) is smooth and connected (due to the smoothness and connectedness of G). This implies (G ∩ HA (G))S = (G ∩ HA (G)) ∩ G S = (G ∩ HA (G)) ∩ ZG (λ) = ZG ∩HA (G) (λ). But we have reduced to the case 0 ∈ A, so (G ∩ HA (G))S = G S . This is connected since G is connected (Proposition A.2.5), so we conclude that ZG ∩HA (G) (λ) is connected. Applying Proposition 2.1.12(2) to the smooth connected solvable k-group G , its k-subgroup G ∩ HA (G) must therefore be connected. 2 We can now reap the fruit of our labors. Theorem 3.3.11 Let G be a smooth connected solvable k-group equipped with an action by a split k-torus S. Let denote the set of S-weights on Lie(G). Suppose there is a disjoint union decomposition = ni=1 i such that i is disjoint from the subsemigroup Aj := j whenever i = j. The k-scheme map HA1 (G) × · · · × HAn (G) → G

(3.3.1)

deﬁned by multiplication is an isomorphism. Proof. We may assume k = k. Pick a maximal k-torus S in S G containing S. This meets the smooth connected normal k-subgroup G in a maximal k-torus T (Corollary A.2.7), so the maximal k-torus T in G is centralized by the S-action on G. In particular, T normalizes HAi (G) for all i. The ﬁrst step is to reduce to the problem with the solvable G replaced by its unipotent radical U . If 0 ∈ then G is unipotent (as we have already observed at the end of the proof of Proposition 3.3.6), so there is nothing to do. If 0 ∈ then exactly one i contains 0, and 0 ∈ Aj for all j = i. Thus, by Proposition 3.3.6, HAj (G) is unipotent for all j = i. Since G = T U , we then have HAj (G) = HAj (U ) for all j = i. Clearly T ⊆ HAi (G), and U ∩ HAi (G) is

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connected (by Proposition 3.3.10, either (b) or (c)), so inside of G = T U we have HAi (G) = T (U ∩ HAi (G)) = T HAi (U ). This factor of T can be passed across all subgroups HAj (G) (as all are normalized by T ), so (3.3.1) for G is reduced to the same problem for U in place of G. Thus, we may and do assume that G is unipotent. Now we proceed by induction on dim G. We may assume G = 1, so there exists a nontrivial smooth connected central k-subgroup Z ⊆ G that is an S-stable vector group admitting an S-equivariant linear structure. (To construct Z, let N be the last nontrivial term in the descending central series of G, so N is commutative. Set Z = N if char(k) = 0, and set Z = pe N with maximal e 0 such that pe N is nontrivial when char(k) = p > 0, so N is also p-torsion in such cases. The vector group property and S-equivariant linear structure when char(k) > 0 are obtained from Corollary B.2.6 and Theorem B.4.3.) The analogue of (3.3.1) for Z is immediate from the weight space decomposition, and Z ∩ HAi (G) = HAi (Z) for all i, by Proposition 3.3.10 (either (b) or (c)). We can assume G/Z = 1 or else we would be done, and since dim(G/Z) < dim G the analogue of (3.3.1) holds for G/Z by induction. The image of HAi (G) under the smooth surjection G → G/Z is HAi (G/Z) for all i, so our problem is now reduced to a special case of the following general claim: if G is a smooth k-group scheme of ﬁnite type, Z is a central k-subgroup of G, and H1 , . . . , Hn are smooth k-subgroups such that the natural k-scheme maps (Z ∩ H1 ) × · · · × (Z ∩ Hn ) → Z, im(H1 → G/Z) × · · · × im(Hn → G/Z) → G/Z deﬁned by multiplication are isomorphisms then the natural map μ : H1 × · · · × Hn → G deﬁned by multiplication is an isomorphism. If these were ordinary groups then an elementary diagram chase yields the bijectivity (so μ is bijective on geometric points, which is insufﬁcient). Likewise, for the analogous claim with sheaves of groups on a topological space, a straightforward variant of the same diagram chase establishes the isomorphism result. Since we can view the kschemes G, Z, G/Z, Hi , and so on as sheaves of groups for the fppf topology

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on the category of k-schemes, the sheaf version of the diagram chase can be used to establish that μ is an isomorphism of k-schemes. 2 Corollary 3.3.12 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and let T be a split maximal k-torus in G. For any λ ∈ X∗ (T ), UG (λ) is directly spanned in any order by the k-groups Ua for non-divisible a ∈ (G, T ) such that a, λ > 0. Moreover, a root a ∈ (G, T ) occurs as a weight for the T -action on Lie(PG (λ)) if and only if U(a) ⊆ PG (λ). In particular, if + is a positive system of roots in (G, T ) then the Ua for non-divisible a ∈ + directly span (in any order) the unique smooth connected unipotent k-subgroup U+ in G normalized by T whose Lie algebra is the span of the root spaces in Lie(G) for roots in + . Proof. The hypotheses in Theorem 3.3.11 are satisﬁed by taking the sets i to be the singletons {ai } for any enumeration {a1 , . . . } of the set of non-divisible a ∈ (G, T ) such that a, λ > 0. For any non-divisible a ∈ we have Ha (G) = Ua by Example 3.3.7, so the asserted description of UG (λ) follows. By Proposition 2.1.8(1), Lie(PG (λ)) is the span of the weight spaces in Lie(G) for the weights a ∈ (G, T ) ∪ {0} such that a, λ 0. In other words, PG (λ) = HA (G) for the saturated subsemigroup A = X(T )λ0 . For each a ∈ (G, T ) we have U(a) = Hasat (G) where asat denotes the saturation of a in X(T ). But for every T -weight a that occurs on Lie(HA (G)) we have asat ⊆ A since A is saturated, so U(a) ⊆ HA (G) = PG (λ). The assertions concerning U+ (both existence and uniqueness) are obtained by choosing λ (as we may, see Proposition 2.2.8(3)) so that + = (G, T )λ>0 . 2 Corollary 3.3.13 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k and let T be a split maximal k-torus in G. Let + denote a positive system of roots in = (G, T ). (1) Let ⊆ + be a closed set of roots, and let nd be the set of elements of not in 2. The set {Uc }c∈nd directly spans, in any order, a T -stable smooth connected unipotent k-subgroup U of G. This is the unique T -stable smooth connected k-subgroup of G whose Lie algebra is the span of the subspaces Lie(Uc ) for all c ∈ . (2) For any (possibly dependent) a, b ∈ + , let (a, b) denote the closed set of roots (a + b) ∩ + = {c ∈ + | c = ma + nb for some integers m, n 1}. Then (Ua , Ub ) ⊆ U(a,b) . It is understood that U = 1 when is empty.

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Proof. Let A ⊆ X(T ) denote the subsemigroup generated by , so by Proposition 2.2.7 we have = ∩ A (and hence + ∩ A = as well). Since + = λ>0 for some λ ∈ X∗ (T ), we have A ⊆ X(T )λ>0 , so 0 ∈ A. Hence, HA (G) is unipotent. We apply Theorem 3.3.11 to HA (G) and the set nd decomposed into a disjoint union of singletons i = {ai } for an enumeration {a1 , . . . } of nd . Since nd generates the semigroup A and A ∩ = , HA (G) satisﬁes the requirements to be U . The unique characterization of U follows from the uniqueness in Proposition 3.3.6 (since generates A as a semigroup). This proves (1). To prove (2), since Ua = Ha (G) and Ub = Hb (G) we have (Ua , Ub ) ⊆ Ha+b (G) = U(a,b) . 2 Example 3.3.14 Consider the set of non-multipliable roots in := (G, T ), where G is pseudo-reductive over a separably closed ﬁeld k and T is a maximal k-torus of G. The root system is naturally identiﬁed with the root system (G red , Tk ). The construction + → + ∩ deﬁnes a bijection k from the set of positive systems of roots of to those of . For a ﬁxed choice of + we claim that the associated positive system of roots + in is closed in . This is only of interest when is non-reduced (so k is imperfect of characteristic 2), as otherwise we are in the trivial case = . Choose distinct c, c ∈ + , so c and c are linearly independent, and assume c + c ∈ + . We have to show that c + c is not multipliable in . Since c + c ∈ , the linearly independent roots c and c lie in the same irreducible component of . Using the natural bijection between the set of irreducible components of and those of , we may assume that is irreducible. Let r 2 denote its rank. We may also assume that c + c is multipliable, so is non-reduced. By [Bou2, VI, §4.14], there is a unique non-reduced irreducible root system of rank r. Relative to any positive system of roots, it has a basis of simple positive roots {a1 , . . . , ar } such that the positive roots are of two types: the roots of the ﬁrst type are ij=i aj with 1 i i r and the roots of the second type are −1 i

j=i

aj + 2

r

aj

j=i

with 1 i i r. The divisible positive roots are precisely the ones of the second type with i = i; that is, the roots 2 rj=i aj with 1 i r. Hence, the

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multipliable positive roots are rj=i aj with 1 i r. We have to show that if r 2 then there is no pair of distinct positive non-multipliable roots c and c such that c + c is multipliable. (This is also true, and trivial, when r = 1.) Suppose such c, c exist. Since multipliable positive roots have all nonzero coefﬁcients equal to 1 relative to the basis of simple positive roots, necessarily c and c are positive roots of the ﬁrst type. Also, since c + c is assumed to be multipliable, by swapping c and c if necessary we must have c=

−1 i

j=i

aj , c =

r

aj

j=i

with i < i r. But then c is multipliable, a contradiction. Generalizing the relationship between unipotent radicals of Borel subgroups and positive systems of roots in the setting of split connected reductive groups over a ﬁeld, we wish to express Ru,k (P) as a direct span (in any order) of root groups associated to the non-divisible elements in a positive system of roots, with P any minimal pseudo-parabolic k-subgroup of G such that T ⊆ P. As a ﬁrst step, we now show that such P actually exist by proving the equivalence of two senses of minimality: for any pseudo-parabolic k-subgroup P of G containing T such that P is minimal with respect to the property of containing T , necessarily P is minimal as a pseudo-parabolic k-subgroup of G. Indeed, we pick a pseudo-parabolic k-subgroup Q of G contained in such a P and will prove that Q = P. Let S be a maximal k-torus in Q (so it is also maximal in G, as Q contains ZG (μ) for some μ over k), so T and S are maximal k-tori in P. By Proposition A.2.10, some g ∈ P(ks ) conjugates Sks into Tks , so it conjugates Qks to a pseudo-parabolic ks -subgroup Q of Gks contained in Pks and containing Tks . By Proposition 2.2.11, Q = PGks (μ ) for some μ ∈ X∗ (Tks ). But μ is deﬁned over k since T is k-split, so Q descends to a pseudo-parabolic k-subgroup of G that is contained in P and contains T . The minimality hypothesis on P then forces this k-descent of Q to equal P, so Q = Pks and hence Q = P as desired. Now we are in position to relate minimal pseudo-parabolic k-subgroups P containing T with positive systems of roots in (G, T ). In any root system, there are ﬁnitely many choices of a positive system of roots, and the ﬁnite Weyl group of the root system acts simply transitively on this ﬁnite set of positive systems, by [Bou2, IV, §1.5, Thm. 2(i)] and [Bou2, IV, §1.7, Cor. 1]. In a split connected reductive group over a ﬁeld, the set of Borel subgroups containing a given split maximal torus T is in natural bijection with the set of positive systems of roots in the associated (reduced) root system (and correspondingly the Weyl group associated to T acts simply transitively on the ﬁnite set of Borel

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subgroups containing T ). The analogous result for minimal pseudo-parabolic subgroups in pseudo-split pseudo-reductive groups over a ﬁeld is given by: Proposition 3.3.15 Let k be a ﬁeld, and let T be a split maximal k-torus in a non-commutative pseudo-split pseudo-reductive k-group G. (1) The minimal pseudo-parabolic k-subgroups of G that contain T are precisely the subgroups P = PG (λ) for λ ∈ X∗ (T ) such that the set (P, T ) = λ0 := {a ∈ (G, T ) | a, λ 0} of nontrivial T -weights occurring on Lie(P) is a positive system of roots in := (G, T ). (2) Assigning to each such P the set of roots in appearing in Lie(P) deﬁnes a bijection between the set of such P and the set of positive systems of roots in . The group W (G, T )(k) acts simply transitively on the set of such P. (3) For any minimal pseudo-parabolic k-subgroup P of G, the maximal pseudo-reductive quotient P/Ru,k (P) over k is commutative. Proof. By Proposition 2.2.11, the pseudo-parabolic ks -subgroups of Gks containing Tks are precisely the k-subgroups PGks (λ) for λ ∈ X∗ (Tks ). But X∗ (Tks ) = X∗ (T ) since T is k-split, so if P is a pseudo-parabolic k-subgroup of G containing T then P = PG (λ) for some k-rational cocharacter λ of T . For any such λ, the set of roots λ0 is the set of weights for T on PG (λ) (Proposition 2.1.8(1)). By Proposition 2.2.8, it contains a positive system of roots + of (G, T ). Step 1. Fix a positive system of roots + contained in λ0 , so there exists μ ∈ X∗ (T ) such that + = {a ∈ | a, μ > 0}. All other elements of are contained in −+ , so + = μ0 . Consider PG (μ). We claim that PG (μ) ⊆ PG (λ). It sufﬁces to show that ZG (μ) ⊆ ZG (λ) and UG (μ) ⊆ PG (λ). Since ZG (μ) contains T , so it contains the image of λ, the containment ZG (μ) ⊆ ZG (λ) holds provided that ZG (μ) is commutative. The k-group ZG (μ) is pseudo-reductive, so by Proposition 1.2.3 it is commutative provided that T is a central subgroup. By Proposition 2.1.8(1), the T -weights in the Lie algebra of UG (±μ) are ±+ , so the only weight occurring for T in Lie(ZG (μ)) is the trivial weight. Hence, by Corollary A.8.11, T is indeed central in ZG (μ).

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To prove UG (μ) ⊆ PG (λ), it sufﬁces (by Corollary 3.3.12) to show that Ua ⊆ PG (λ) for any non-divisible a ∈ (G, T ) satisfying a, μ > 0 (i.e., a ∈ + ). Such a lie in λ0 , so by Corollary 3.3.12 we have Ua = U(a) ⊆ PG (λ). With the containment PG (μ) ⊆ PG (λ) proved, we see that if PG (λ) is a minimal pseudo-parabolic k-subgroup then this containment is an equality. In particular, λ0 = + when P := PG (λ) is minimal. In this case, UG (λ) = UG (μ) and PG (λ)/UG (λ) = PG (μ)/UG (μ) ZG (μ) is a commutative pseudo-reductive group. This proves (3). Step 2. Conversely, if λ0 is a positive system of roots + of (G, T ) then we claim that a pseudo-parabolic k-subgroup P of G contained in PG (λ) must equal PG (λ). To prove this we may and do assume k = ks . The hypothesis says that the set of nontrivial T -weights that occurs in Lie(PG (λ)) is a positive system of roots in (G, T ), and this is preserved if we change λ and T by conjugation by a common k-point of PG (λ). Hence, we can assume that T is contained in P, since maximal k-tori in a pseudo-parabolic k-subgroup of G are maximal k-tori of G and all maximal k-tori in PG (λ) are PG (λ)(k)-conjugate (as k = ks , so Proposition A.2.10 applies). By Proposition 2.2.11 we may therefore write P = PG (λ ) for some k-homomorphism λ : GL1 → T , so the containment Lie(P) ⊆ Lie(PG (λ)) implies λ 0 ⊆ λ0 = + . But λ 0 is a parabolic set of roots, so it contains a positive system of roots. All positive systems of roots have the same size, so λ 0 = + . Thus, Lie(P) and Lie(PG (λ)) are the spans in Lie(G) for the weight spaces for T associated to a common set of weights (namely + ∪ {0}), so Lie(P) = Lie(PG (λ)). Hence, the containment P ⊆ PG (λ) is an equality, so PG (λ) is indeed minimal. Step 3. Finally, we show that W (G, T )(k) acts simply transitively on the set of minimal parabolic k-subgroups containing T . The Weyl group of any root system acts simply transitively on the set of positive systems of roots, so by Proposition 3.2.7 the group W (G, T )(k) acts simply transitively on the set of positive systems of roots in (G, T ). Since we have shown that every minimal pseudo-parabolic k-subgroup containing T has its Lie algebra equal to the span of the weight spaces for a positive system of roots together with the trivial weight, it remains to show that any minimal pseudo-parabolic k-subgroup P in G containing T is determined by Lie(P) ⊆ Lie(G).

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We saw at the outset (using that T is k-split) that we can choose λ : GL1 → T such that P = PG (λ). The centralizer ZG (λ) P/Ru,k (P) is commutative (since (3) has already been proved), so it equals ZG (T ). But P = ZG (λ) UG (λ) = ZG (T ) Ru,k (P), so we just have to show that UG (λ) is determined by Lie(P). The k-group UG (λ) is generated by the root groups Ua for non-divisible a ∈ λ>0 , and these a are exactly the non-divisible nontrivial T -weights on Lie(P) (since ZG (λ) = ZG (T )), so UG (λ) is determined by Lie(P) and T . 2 Corollary 3.3.16 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and let T be a split maximal k-torus in G. Let + be a positive system of roots in (G, T ), and let {ci } be an enumeration of the non-divisible elements of + . Let P be the minimal pseudo-parabolic k-subgroup of G containing T that corresponds to + . The multiplication map of k-scheme i Uci → G is an isomorphism onto Ru,k (P), and the multiplication map of k-schemes Uci × ZG (T ) × U−ci → G (3.3.2) i

i

is an open immersion. Proof. To prove that the k-groups Uci directly span Ru,k (P) in any order, we apply Corollary 3.3.12 with λ ∈ X∗ (T ) such that + is the set of c ∈ (G, T ) satisfying c, λ > 0. Such a λ is provided by Proposition 2.2.8(3). Combining this with Proposition 2.1.8(2),(3) yields the open immersion claim. 2 We often refer to the open subscheme (3.3.2) in G as the open Bruhat cell of G (determined by T and + ), though sometimes we may put the positive root groups on the right and the negative root groups on the left; the context will make this clear (if it matters). The open Bruhat cell depends on + , but we omit mention of this choice if it is clear from the context (or does not matter). The following application of Proposition 3.3.15 will be useful later. For an analogous result over an arbitrary ﬁeld see Theorem C.2.5. Proposition 3.3.17 Let G be a smooth connected afﬁne group over a separably closed ﬁeld k. All minimal pseudo-parabolic k-subgroups of G are G(k)-conjugate. Proof. Since k = ks , smooth surjections between k-schemes induce surjections on k-points, by Proposition 2.2.10 we may replace G with G/Ru,k (G) to reduce

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115

to the case where G is pseudo-reductive over k. Let P and Q be minimal pseudoparabolic k-subgroups of G. By deﬁnition of pseudo-parabolicity, a maximal k-torus in P is also maximal in G (and likewise for Q). The G(k)-conjugacy of maximal k-tori in G (Proposition A.2.10) thereby reduces us to the case where P and Q contain a common maximal k-torus T . Proposition 3.3.15(2) now completes the argument. 2

3.4 Bruhat decomposition and Levi subgroups If G is a smooth connected afﬁne group over a ﬁeld k, then a Levi k-subgroup of G is a smooth connected k-subgroup L ⊆ G such that for the quotient map π : Gk G := G red , the restriction π |Lk is an isomorphism (i.e., Lk Ru (Gk ) = k Gk ). It is a well-known fact (due to G.D. Mostow) that if k has characteristic 0 then Levi k-subgroups always exist. However, over every algebraically closed ﬁeld with nonzero characteristic there are smooth connected afﬁne groups not admitting a Levi subgroup. For the convenience of the interested reader, a family of counterexamples is given in §A.6 (which will not be used in what follows). The main results of this section are that for a pseudo-split pseudo-reductive group G over a ﬁeld k, Levi k-subgroups always exist and there is a Bruhat decomposition for G(k) relative to a choice of minimal pseudo-parabolic ksubgroup of G. The Bruhat decomposition of G(k) for any smooth connected afﬁne group G over an arbitrary ﬁeld k will be given in Theorem C.2.8, and it rests on the pseudo-reductive case over k = ks as a preliminary step. The existence of Levi k-subgroups in the pseudo-reductive case over k = ks will be especially useful in our classiﬁcation of pseudo-reductive groups in characteristic 2 (and will provide some motivation in our classiﬁcation work in characteristic 3). The hypothesis that G is pseudo-split is necessary for the existence of Levi k-subgroups; in Example 7.2.2 we construct standard pseudo-reductive k-groups having no Levi k-subgroup. The construction of Bruhat decompositions and Levi subgroups in the pseudo-split case rests on the following two results. Proposition 3.4.1 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and let T be a split maximal k-torus in G. Choose a ∈ (G, T ), deﬁne Ta = (ker a)0red , and let U±a be the root groups in G associated to ±a as in Deﬁnition 2.3.13. (1) The smooth connected k-subgroup Ga generated by Ua and U−a is pseudoreductive and perfect with split maximal k-torus S = T ∩ Ga , and (Ga )red k is of type A1 . The roots ±a ∈ (Ga , S) are not divisible. (2) If a is not divisible in (G, T ) then Ga = D (ZG (Ta )).

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Basic structure theory

(3) Assume a is divisible in (G, T ) (so k is imperfect and char(k) = 2). For μ := S[2], the identity component Ha = ZZG (Ta ) (μ)0 is pseudo-split pseudo-reductive with (Ha , T ) = {±a}, and Ha is generated by ZG (T ) and Ga . Moreover, Ga = D (Ha ) for such a. Note that in (3), the k-group scheme Ha is pseudo-reductive; this follows from Proposition A.8.14(2) (taking G and H there to be ZG (Ta ) and μ respectively in the above notation). Proof. Since ZG (Ta ) is pseudo-reductive (Proposition 1.2.4), we can rename it as G to reduce to the case where Ta is central in G. By Corollary A.2.7 the derived group D (G) has (split) maximal k-torus S = T ∩ D (G), and D (G)red = D (G red ) (Proposition A.4.8). The set of roots for (D (G red ), S) k

k

k

coincides with the set of roots for (G red , Tk ), which in turn is the set of nonk multipliable roots in (G, T ) (Theorem 2.3.10). This is {±a} when a is not multipliable, and {±2a} otherwise, so it has rank 1. Hence, dim S = 1. (This also follows from Proposition 3.2.3.) It is clear that a|S = 1, so T = S · Ta . By Proposition 3.2.3, S = a∨ (GL1 ). If a is not divisible in (G, T ) then U±a = U(±a) , so it follows from Lemma 3.1.4 that in such cases Ua and U−a generate D (G). In the root system (D (G), S) of rank 1 the roots ±a are clearly not divisible when a is not divisible in (G, T ). Now suppose that a is divisible in (G, T ), so k is imperfect with characteristic 2 and (G, T ) = {±a, ±a/2}. In this case, the non-multipliable roots ±a are the Tk -roots for G red , and hence the Sk -roots for its derived group D (G)red k

k

(see Proposition A.4.8). But D (G)red is semisimple of type A1 , so the divisk ibility of the root a in its character lattice X(Sk ) implies that either of ±a/2 generates X(S). Hence, by Proposition A.8.10 and Proposition A.8.14(2),(3) it follows that the k-subgroup H := ZG (S[2])0 is pseudo-reductive with T -roots consisting of those for (G, T ) that are trivial on S[2]. Thus, (H , T ) = {±a}, so Lie algebra considerations imply that H is generated by ZG (T ), Ua , and U−a . Applying Lemma 3.1.4 and Lemma 3.1.5 to H gives that D (H ) is generated by Ua and U−a . In particular, (D (H ), S) = {±a}, so ±a are not divisible in (D (H ), S). 2 Proposition 3.4.2 (Tits) Let k be a ﬁeld, G a pseudo-split pseudo-reductive k-group, and T a split maximal k-torus of G. For a ∈ := (G, T ) let the root groups U±a be as in Deﬁnition 2.3.13. Fix a and u ∈ Ua (k) − {1}. There exist unique u , u ∈ U−a (k) such that u uu normalizes T , and moreover u , u = 1. The induced action of m(u) :=

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117

u uu on the character group X(T ) is the reﬂection ra associated to the root a, u = m(u)um(u)−1 = u , and m(u)2 = a∨ (−1) ∈ T (k). For z ∈ ZG (T )(k), if zuz −1 ∈ Ua (k) then zu z −1 ∈ U−a (k) and m(zuz −1 ) = zm(u)z −1 . Proof. By arguing separately with Proposition 3.4.1 depending on whether or not a is divisible, the smooth connected k-subgroup of G generated by C := ZG (T ), Ua , and U−a is pseudo-reductive, and by replacing G with this ksubgroup (as we may) we gain the properties that G red has a root system of rank k 1 with root groups U±a = U(±a) (so ±a are now not divisible). Let U = Ua and U = U−a , so Proposition 3.4.1(2) shows that D (G) is generated by U and U . For the Cartan k-subgroup C, the multiplication map U × C × U → G is an open immersion (by Proposition 2.1.8(2),(3)). By Proposition 3.2.7, the normalizer N = NG (T ) has two connected components: C and N − C = nC for a ﬁxed n ∈ N (k) − C(k). Let Y be the preimage of N −C under the multiplication map U ×U ×U → G. Note that C acts on Y through conjugation on all factors. The condition u uu = nc can be rewritten as n−1 u = (n−1 u −1 n)cu −1 . Since U × C × U is open in G via multiplication and U = nUn−1 , the C-equivariant projection Y → U is an isomorphism onto the open subscheme ⊆ U consisting of points u of U such that n−1 u lies in the open Bruhat cell U × C × U of (G, T , {a}). Thus, if (v , v, v ) ∈ Y (k) and v ∈ U (k) then v and v are uniquely determined and lie in U (k). Letting R = Ru (Gk ), we claim that k = Uk ∩ (Gk − R). The pseudoreductive derived group H = D (G) contains U±a and has maximal k-torus S := T ∩ H with dimension 1, so a|S = 1. Since T = S · Ta for the central codimension-1 torus Ta = (ker a)0red , a point in H (k) normalizes (resp. centralizes) T if and only if it normalizes (resp. centralizes) S. Thus, to justify the description of and to prove all other claims we may replace G with H so that G is perfect with G red semisimple of type A1 (i.e., SL2 or k PGL2 ). The equality k = Uk ∩ (Gk − R) is a trivial calculation when k = k for the groups SL2 and PGL2 (with R = {1}). For the general case, it sufﬁces to check that the open cell (UCU )k in Gk is the full preimage of its image under the projection π : Gk → Gk /R = G red . Equivalently, if we choose k λ : GL1 → T such that a, λ < 0 and deﬁne P := PG (λ) = CU then we want Uk P Rk = Uk P . Let μ : GL1 × R → R denote the left action induced k k by conjugation on R by λk . By Remark 2.1.11 and Proposition 2.1.12(1), the multiplication map UR (−μ)×PR (μ) → R is an isomorphism, with UR (−μ) = UG (−λ)k ∩ R = Uk ∩ R and PR (μ) = PG (λ)k ∩ R = P ∩ R both smooth k

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connected k-subgroups of R. From this decomposition of R (and the normality of R in Gk ) we see that Uk P R = Uk RP = Uk P , as desired. k k k Since ker(π|U (ks ) ) and ker(π |U (ks ) ) are normalized by T (ks ), either would be inﬁnite if nontrivial, due to the existence of linear structures on U and U normalizing the T -action. But R(k)∩G(ks ) is ﬁnite since G is pseudo-reductive, so π is injective on U (ks ) and U (ks ). Thus, U (k) − {1} ⊆ (k) and so for each u ∈ U (k) − {1} there exist unique u , u ∈ U (k) such that m(u) := u uu ∈ N (k) − C(k). To prove uniqueness only requiring u uu ∈ N (k), we need u uu ∈ C(k) for any u , u ∈ U (k). This is trivial for G red (i.e., SL2 or k PGL2 ), so it holds for G since π(u) = 1. Likewise, since π |U (k) is injective, the identities u = u = 1 and u = m(u)um(u)−1 in U (k) follow from the trivial case of G red . This argument also shows π(m(u)) = m(π(u)), and the k

C-action on Y ⊆ U implies that if c ∈ C(k) and cuc−1 ∈ U (k) then cu c−1 ∈ U (k) and m(cuc−1 ) = cm(u)c−1 . Finally, we prove m(u)2 = a∨ (−1). If char(k) = 2 then u and u = u have order 2, so clearly m(u)2 = 1. Now assume char(k) = 2, so = {±a}. We may and do assume k = ks , so we can choose t ∈ T (k) such that a(t) = −1; this uses that char(k) = 2 and either a or a/2 generates X(T ). Conjugation by t carries the points u ∈ U (k) and u = u ∈ U (k) to their inverses, so tm(u)−1 t −1 = m(u) and hence m(u)2 = m(u)tm(u)−1 t −1 . This lies in T (k) ? since m(u) ∈ N (k). The desired identity m(u)2 = a∨ (−1) therefore compares two elements of T (k). Since π carries Tk isomorphically onto a maximal torus in G red , and π identiﬁes with (G red , Tk ), it sufﬁces to check the equality k

k

for G red (i.e., SL2 or PGL2 ), which is trivial. k

2

Corollary 3.4.3 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and T a split maximal k-torus in G. (1) The inclusion NG (T )(k)/ZG (T )(k) → W (G, T )(k) is bijective. (2) If G is also absolutely pseudo-simple then for any two roots a, a ∈ (G, T ) with the same length there exists n ∈ NG (T )(k) whose conjugation action on G carries a to a . Proof. The group W (G, T )(k) coincides with the Weyl group of (G, T ) (Proposition 3.2.7), and so it is generated by the reﬂections ra associated to the roots. But for each such a we can choose a nontrivial u ∈ Ua (k) since Ua is a nonzero vector group over k, and so the element m(u) ∈ NG (T )(k) as in Proposition 3.4.2 maps to ra . This proves (1). Assertion (2) follows from (1) and the fact that (G, T ) is an irreducible root system when G is absolutely pseudo-simple since the Weyl group of an

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irreducible (possibly non-reduced) root system acts transitively on the set of roots of a given length. 2 Example 3.4.4 To appreciate how remarkable it is that NG (T )(k) → W (G, T )(k) is surjective when G is pseudo-split, we explain why naive Galois cohomology considerations cannot yield the result. The obstructions to surjectivity lie in H1 (k, ZG (T )), and this cohomology group can be nonzero, even when G is perfect and standard. To give some examples, let k be an imperfect ﬁeld of characteristic p > 0, and choose purely inseparable nontrivial ﬁnite extensions k /k /k so that k = k 1/p ∩ k and [k : k ] = p. Let G = R k /k (SLp )/R k /k (μp ). Letting D denote the diagonal k -torus of SLp , the k-subgroup C := R k /k (D)/R k /k (μp ) in G is a Cartan k-subgroup (by Proposition A.2.8) and it clearly contains a k-split maximal k-torus. The pseudo-reductivity of G is a special case of the self-contained argument in Example 5.3.7. The k-group C admits as a direct factor (i,N)

Q := R k /k (GL1 )/R k /k (μp ) coker(R k /k (GL1 ) −→ R k /k (GL1 ) × GL1 ), where i is the natural map and N = Nk /k is the norm, so it sufﬁces that H1 (k, Q) = 0. The smooth presentation of Q yields an exact sequence in Galois cohomology H1 (k, Q) → Br(k ) → Br(k ) × Br(k), so if Br(k ) → Br(k ) × Br(k) is not injective then we are done. This happens for the rational function ﬁeld k = k0 (x, y) over an algebraically closed ﬁeld k0 2 with k = k(x1/p ) and k = k(x1/p ). (Setting u = x1/p and v = y, this is the natural map Br(k) → Br(k(u1/p )) × Br(k(v 1/p )) for k = k0 (u, v).) Here is the Bruhat decomposition in the pseudo-split case (e.g., k = ks ). Theorem 3.4.5 (Borel–Tits) Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, T a split maximal k-torus, and P a minimal pseudo-parabolic k-subgroup of G containing T . The natural map NG (T )(k)/ZG (T )(k) → P(k)\G(k)/P(k) is bijective. Proof. Let N = NG (T ), Z = ZG (T ), and = (G, T ). By Corollary 3.4.3(1), W := N (k)/Z(k) is equal to W (G, T )(k) via the natural map. For a ∈ , let the root group Ua be as in Proposition 2.3.11, and deﬁne Ma to be the set of elements of N (k) whose image in the Weyl group W (G, T )(k) is the reﬂection ra in a. Let + be the positive system of roots (P, T ) (see Proposition 3.3.15(2)),

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and − = −+ . Let ⊆ + be the basis of given by the simple positive roots. Choose a 1-parameter k-subgroup λ : GL1 → T such that P = PG (λ) (Proposition 2.2.11). We claim that the datum (Z(k), (Ua (k), Ma )a∈ ) is a generating root datum (donnée radicielle génératrice) of type in G(k) in the sense of [BrTi, (6.1.1)]. This requires verifying a list of six axioms (DR1)–(DR6) and a group generation claim, as follows. Axiom (DR1) is the obvious fact that Z(k) is a subgroup of G(k) and each Ua (k) is a nontrivial subgroup of G(k). Corollary 3.3.13(2) is precisely (DR2), and (DR3) is the obvious fact that U2a (k) is a proper subgroup of Ua (k) when a and 2a lie in . By Proposition 3.4.2, for all a ∈ we have U−a (k) − {1} ⊆ Ua (k)Ma Ua (k) with Ma a right Z(k)-coset in G(k); this is precisely (DR4). The deﬁnition of Ma implies that nUb n−1 = Ura (b) for all a, b ∈ and n ∈ Ma , which is (DR5). Finally, by Corollary 3.3.12, the subgroup U + ⊆ G(k) generated by the groups Ua (k) for a ∈ + is UG (λ)(k) and the subgroup U − ⊆ G(k) generated by the groups Ua (k) for a ∈ − is UG (−λ)(k). Hence, Z(k)U + = P(k), so Z(k)U + ∩ U − = {1} by Proposition 2.1.8(3). The triviality of this intersection is (DR6). Having veriﬁed the axioms, the remaining condition to verify is that Z(k) and the subgroups Ua (k) generate G(k). We certainly get the k-points of the open Bruhat cell = UG (−λ)ZUG (λ) (with (k) = U − Z(k)U + ). Thus, it sufﬁces to show that the dense open V := g ∩ has a k-point for each g ∈ G(k). If k = ks (the only case we need) this is obvious. In general, since V is stable under right translation by P = PG (λ) = ZUG (λ), it is the pullback of the dense open V /P in G/P that is contained in /P. By the selfcontained Lemma C.2.1 (whose proof simpliﬁes a bit in the pseudo-reductive case) we have (G/P)(k) = G(k)/P(k), so (V /P)(k) = V (k)/P(k). Hence, it sufﬁces to show that (V /P)(k) is non-empty. The coset space /P is isomorphic to UG (−λ), which is an afﬁne space over k, so V /P is a dense open in an afﬁne space over k and hence has a k-point if k is inﬁnite. This veriﬁes that (Z(k), (Ua (k), Ma )a∈ ) is a generating root datum when k is inﬁnite. If k is ﬁnite then (by the perfectness of ﬁnite ﬁelds) the pseudo-split k-group G is (connected) reductive, so the desired Bruhat decomposition is classical [Bo2, 21.15] (which even formally implies the group generation claim in such cases). It now follows from [BrTi, Cor. 6.1.11(ii), Prop. 6.1.12] that if we deﬁne R = {ra }a∈ to be the indexed set of images in W of the sets Ma for a ∈ then (G(k), P(k), N (k), R) is a (saturated) Tits system with Weyl group W in the sense of [Bou1, Ch. IV, §2.1, Def. 1]. There is a Bruhat decomposition for any group equipped with a Tits system [Bou1, Ch. IV, §2.3, Thm. 1]. For the Tits system just constructed in G(k), the Bruhat decomposition is precisely the desired bijectivity of the map W → P(k)\G(k)/P(k). 2

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The rest of this section is devoted to the proof of the existence of Levi k-subgroups in any pseudo-split pseudo-reductive group G over a ﬁeld. Our existence result will also include a uniqueness aspect, so we ﬁrst set up some notation. Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, T a split maximal k-torus of G, = (G, T ), and the set of non-multipliable roots in . We will view T := Tk as a maximal torus of G := G red , so the identiﬁcation k

X(T ) = X(T ) carries over to (G , T ). Let + be a positive system of roots in (equivalently, the set of non-multipliable elements in a positive system of roots in ), and let be the set of simple roots in + . (Example 3.3.14 shows that the set + of non-multipliable positive roots is closed.) By Proposition 2.3.11, for each a ∈ the root group Ua in G is a vector group which admits a unique linear structure relative to which the conjugation action of T is linear, and this linear action is through the character a. A k-linear subgroup of Ua is a k-subgroup which corresponds to a k-vector subspace of Ua (k). Since a = 1, the k-linear subgroups of Ua are precisely the smooth connected k-subgroups of Ua that are normalized by T . Our main result on Levi subgroups is:

Theorem 3.4.6 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, T a split maximal torus, and + a positive system of roots in the set of non-multipliable roots in = (G, T ). For each simple positive a ∈ + , let Ea be a smooth connected 1-dimensional k-subgroup of Ua that is normalized by T (i.e., Ea is a k-linear subgroup of Ua of dimension 1). There is a unique Levi k-subgroup L of G containing T and every Ea . Moreover, every Levi k-subgroup of G containing T arises in this way. Proof. If L is to be a Levi k-subgroup of G containing T then the isomorphism Lk G red carrying Tk onto a maximal k-torus implies that inside of k

X(T ) = X(Tk ) we have (L, T ) = (G red , Tk ) = . For each a ∈ the k

a-root group of (L, T ) lies in ZG (Ta ) for Ta := (ker a)0red , and in fact it is contained in UZG (Ta ) (a∨ ) =: U(a) (as we can compute the limiting behavior via the isomorphism Lk G red ). But Ua is the subgroup of U(a) on which ker(a) k

acts trivially, so the a-root group of (L, T ) for each simple positive a ∈ + can be taken as a choice for Ea . Hence, the proposed recipe will have to yield all possible L containing T . Let be the set of simple positive roots in + , and for each a ∈ make a choice of Ea and a nontrivial ua ∈ Ea (k). (Such a k-point ua exists since Ea Ga .) By Proposition 3.4.2, there is a unique va ∈ U−a (k) such that m(ua ) := va ua va normalizes T , and the induced action of m(ua ) on X(T )

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is the reﬂection ra associated to a. The Weyl group W (G red , Tk ) = W ( ) k is generated by the ra , and each Weyl group orbit in meets , so the subgroups (m(ua1 ) · · · m(uaq ))Ea (m(ua1 ) · · · m(uaq ))−1 ⊂ G for varying a ∈ , q 0, and a1 , . . . , aq ∈ (allowing repetitions) must vary through precisely the root groups of (L, T ) for any possible L. But the connected reductive k-group L is generated by its k-split maximal k-torus T and its root groups with respect to T , so this establishes the uniqueness of L. The real work is to prove that such an L exists for any speciﬁcation of the k-subgroups Ea . Let k /k be the minimal ﬁeld of deﬁnition for the geometric unipotent radical of G; k /k is a ﬁnite purely inseparable extension, and Ru,k (Gk ) is a k -descent of Ru (Gk ). Thus, the quotient π : Gk Gk /Ru,k (Gk ) =: G is a k -descent of the maximal reductive quotient of Gk . By deﬁnition, the khomomorphism G → R k /k (G ) corresponding to π (via the universal property of Weil restriction) is iG . The kernel of iG cannot contain any nontrivial smooth connected k-subgroups (due to Lemma 1.2.1). Thus, iG is nontrivial on each Ea , so π is nontrivial on each (Ea )k . We identify Tk with a k -split maximal k -torus T in G in the evident manner. Consider the Tk -equivariant composite map (Ea )k → Gk G . This lands in the a-root group Ua for (G , T ), and the resulting Tk -equivariant k -homomorphism (Ea )k → Ua must be linear. It has already been shown that this map is nontrivial, so it is an isomorphism. In particular, ker(iG |Ea ) = 1. We will use the resulting k-structure Ea on the k -group Ua for each a ∈ , together with the k-structure T on T , to specify a k-structure (G0 , T0 ) on (G , T ). This rests on the general concept of pinning for split reductive pairs, which is discussed in Deﬁnition A.4.12 and Theorem A.4.13, and the procedure goes as follows. For each a ∈ , let Ga denote the k -subgroup of G generated by Ua and : this subgroup is connected semisimple of rank 1, with T ∩ G as a k U−a a split maximal k -torus. By Example A.4.1, the pair (Ga , T ∩ Ga ) is isomorphic to either (SL2 , D) or (PGL2 , D/μ2 ) where D is the diagonal torus. Thus, we can choose a central k -isogeny ϕa : SL2 → Ga which carries the diagonal

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k -torus to T ∩ Ga and the upper triangular unipotent subgroup U + of SL2 onto Ua . By the construction of coroots, on the diagonal torus this map must be ϕa (diag(y, 1/y)) = ak∨ (y). The collection {ϕa }a∈ is a pinning in the sense of Deﬁnition A.4.12. Using the parameterization x : Ga U + deﬁned by x(u) = 10 u1 , for a ∈ consider the composite k -isomorphism Ga U + Ua (Ea )k . This might not be deﬁned over k, but we can ﬁx that problem as follows. The k -automorphism of SL2 deﬁned by any diag(c , 1) ∈ PGL2 (k ) restricts to the identity on the diagonal k -torus and induces multiplication by c ∈ k × on U + Ga . Composing ϕa with such an automorphism for a suitable c allows us to impose the additional requirement that ϕa : Ga (Ea )k is deﬁned over k for each a ∈ . Applying Theorem A.4.13, the pinning {ϕa }a∈ determines a canonical kstructure (G0 , T0 ) on (G , T ), with T0 a k-split maximal k-torus of G0 . The resulting isomorphism (T0 )k T = Tk must (uniquely) descend to a kisomorphism θ : T0 T since these k-tori are k-split. Hence, the k-subgroup inclusion j0 : G0 → R k /k ((G0 )k ) R k /k (G ) carries T0 isomorphically onto the maximal k-torus T in R k /k (T ) = R k /k (Tk ) via θ . Using the resulting equality (G0 , T0 ) = (G , T ) = inside of X(T0 ) = X(T ) = X(T ), the characterization of the k-structure (G0 , T0 ) in terms of the pinning (in Theorem A.4.13) implies that for each a ∈ , the composite isomorphism of root groups (UaG0 )k Ua (Ea )k is deﬁned over k. The conceptual signiﬁcance of this fact is explained by the following lemma. Lemma 3.4.7 Using notation as above, consider the diagram G

G0

j0

iG

/ R k /k (G )

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The restriction of j0 to T0 and to each root group UaG0 (a ∈ ) uniquely factors through iG via a homomorphism to G, with T0 carried isomorphically onto T via θ . Proof. Since ker iG contains no nontrivial smooth connected k-subgroups, its only smooth connected closed subscheme through the identity is {1}. (This follows from Lemma C.4.1, or arguments using subgroups generated by connected subvarieties passing through the identity.) Hence, the desired factorizations are unique if they exist. The map iG carries T isomorphically onto the maximal k-torus in R k /k (T ) = R k /k (Tk ) via the canonical map (due to how iG and T are deﬁned), so j0 carries T0 isomorphically onto iG (T ). We have seen in the discussion preceding this proof that this latter isomorphism is iG ◦ θ, so the factorization of j0 |T0 is settled. Consider a ∈ . The restriction of j0 to UaG0 lands inside of R k /k (Ga ). We claim that its image is exactly iG (Ea ), so since iG |Ea is a k-subgroup inclusion we would get the desired lifting on UaG0 for each a ∈ . More generally, consider a k -group H of ﬁnite type and a pair of k-subgroups H1 , H2 → R k /k (H ) such that the induced k -homomorphisms (Hi )k → H are isomorphisms. These latter isomorphisms say exactly that each Hi is a k-structure on H . (The equivalence follows from the explicit descriptions in Proposition A.5.7.) Thus, the images of the Hi coincide (i.e., the Hi are k-isomorphic respecting their inclusions into R k /k (H )) if and only if the composite isomorphism (H1 )k H (H2 )k is deﬁned over k. Applying this with H1 = UaG0 , H2 = Ea , H = Ua , and the inclusions j0 | G0 and iG |Ea , we are done when a ∈ . Ua

Now consider an arbitrary a ∈ . Let W = W ( ), so a = w.c for some w ∈ W and c ∈ . We have equalities NG0 (T0 )(k)/T0 (k) = W (G0 , T0 )(k) = W = W (G, T )(k) = NG (T )(k)/ZG (T )(k) since the pairs (G0 , T0 ) and (G, T ) are pseudo-split over k (see Corollary 3.4.3(1)). This identiﬁcation of Weyl groups of pseudo-split pairs matches their identiﬁcation with the Weyl group W (R k /k (G ), T )(k). Thus, if we can ﬁnd representatives g0 ∈ NG0 (T0 )(k) and g ∈ NG (T )(k) of w whose images in G (k ) coincide then by composing the lifted map UcG0 Ec (for c ∈ ) with conjugation by g0−1 on the source and by g on the target we get a k-homomorphism UaG0 Ea that is the desired lift.

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To ﬁnd such g0 and g, by writing w as a word in reﬂections ra for a ∈ it sufﬁces to treat the case w = ra . In this case we choose a nontrivial u0 ∈ UaG 0 (k) and let u ∈ Ea (k) be the corresponding nontrivial element via the lift UaG 0 Ea constructed above (so j0 (u0 ) = iG (u) as nontrivial elements in Ua (k ); denote G0 it as u ). Proposition 3.4.2 provides unique elements v0 ∈ U−a (k) and v ∈ U−a (k) such that m(u0 ) := v0 u0 v0 and m(u) := vuv respectively normalize (k ) is either of T0 and T and each represent ra . It follows that if v ∈ U−a the elements j0 (v0 ) or iG (v) then v u v normalizes T and represents ra . Thus, the uniqueness aspect of Proposition 3.4.2 (applied to (G , T )) implies that j0 (v0 ) = iG (v), so the elements g0 = v0 u0 v0 and g = vuv have the same image in G (k ), as required. 2 Lemma 3.4.7 now yields the desired Levi k-subgroup, as follows. Choose a positive system of roots + in , and ﬁx an enumeration {ai } of + . We then have the open cell :=

G0 U−a × T0 × i

i

UaGi 0 → G0

i

deﬁned via multiplication, so the restriction of j0 to this open cell factors through iG by forming the corresponding product (in the same order) of the unique lifts of j0 |T0 and j0 | G0 (a ∈ ). The resulting factorization diagram of pointed kUa schemes G w; ww w iG ww ww w w / R k /k (G ) f

j0

uniquely determines f because ker iG contains no nontrivial smooth connected closed subscheme passing through the identity. Applying the same uniqueness consideration to the product map iG × iG , it follows from the k-homomorphism property of j0 (and iG ) that f is a krational homomorphism. Hence, f uniquely extends to a k-homomorphism F : G0 → G (as we see via Galois descent by using the covering of (G0 )ks by (ks )-translates of ks ). This must be a k-subgroup inclusion, since iG ◦F0 = j0 is a k-subgroup inclusion. By construction L := F(G0 ) contains T and every Ea (a ∈ ). Since ( j0 )k and (iG )k induce isomorphisms after composition with the quotient map R k /k (G )k G over k , it follows that (F0 )k induces

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an isomorphism after composing it with the quotient map Gk G . This says that L is a Levi k-subgroup of G. 2 Corollary 3.4.8 Let T be a maximal k-torus in a pseudo-reductive group G over a ﬁeld k, and let = (Gks , Tks ). Assume G red is semisimple of adjoint k type, and for each a ∈ let Ua denote the a-root group of (Gks , Tks ). If the (ZG (T ))(ks )-action on a∈ (Ua (ks ) − {0}) makes it a principal homogeneous space then there is a cohomology class cT , ∈ H1 (k, ZG (T )/T ) whose vanishing is necessary and sufﬁcient for the existence of a Levi k-subgroup of G containing T . The formation of cT , commutes with separable extension on k. This cohomological criterion will be used in Proposition 7.3.7 to prove the existence of Levi k-subgroups in certain “exotic” pseudo-reductive groups G over arbitrary imperfect ﬁelds k of characteristic 3. Proof. The assumption that G red is semisimple of adjoint type has two useful k consequences: it implies that there is no simple factor of simply connected type C, so is reduced (even in characteristic 2), and is a Z-basis of X(Tks ). Thus, the principal homogeneous space hypothesis implies that the space of choices {a }a∈ of ks -lines a ⊆ Ua (ks ) is a principal homogeneous space for (ZG (T )/T )(ks ). Theorem 3.4.6 identiﬁes this space of choices of such lines with the set of Levi ks -subgroups L of Gks containing Tks . Concretely, the action of (ZG (T )/T )(ks ) on the set of such Levi ks -subgroups is g.L = g L g −1 for g ∈ ZG (T )(ks ) representing g. Thus, if we choose some L over ks then for each γ ∈ Gal(ks /k) we have γ (L) = gγ .L for a unique gγ ∈ (ZG (T )/T )(ks ). It is easy to check that γ → gγ is a continuous 1-cocycle whose cohomology class is independent of the initial choice of L, independent of the choice of ks /k, and vanishes precisely when there is a Gal(ks /k)-stable choice of L (i.e., there exists a Levi k-subgroup of G containing T ). This class is cT , , and by the construction its formation is compatible with separable extension of k. 2 As a consequence of the existence of Levi subgroups in pseudo-split pseudoreductive groups, we get a better understanding of the relationship between pseudo-reductivity and reductivity, as well as between pseudo-parabolicity and parabolicity: Proposition 3.4.9 Let G be a perfect pseudo-reductive group over a ﬁeld k. (1) The root groups of Gks are all 1-dimensional if and only if G is reductive. (2) Let P be a pseudo-parabolic k-subgroup of G that does not contain any nontrivial perfect smooth connected normal k-subgroup of G. The coset

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space G/P is complete (i.e., P is a parabolic subgroup) if and only if G is reductive. Proof. The structure theory of connected reductive groups [Bo2, 13.18] implies that G has 1-dimensional root groups if it is reductive. Likewise, by Proposition 2.2.9, the pseudo-parabolic k-subgroups of G are precisely the parabolic ksubgroups when G is reductive. This settles the “if” direction in both parts. For the converse implication in (1), we may assume k = ks , G = 1, and that G has 1-dimensional root groups. Let T be a maximal k-torus of G. The root system := (G, T ) is reduced since any multipliable root a has root group Ua = U(a) with dimension at least 2. Since k = ks , by Theorem 3.4.6 there is a Levi k-subgroup L of G which contains T and has the same root system (with respect to T ) as does G. But the root groups Ua of G are 1-dimensional, so L contains Ua for all a ∈ . It follows that L is the smooth connected k-subgroup of G generated by T and the k-groups Ua for a ∈ . Since G is generated by ZG (T ) and the k-groups Ua for a ∈ (Lemma 2.3.3), and ZG (T ) normalizes each root group Ua , we conclude that L is a normal k-subgroup of G. Thus, the quotient G/L makes sense as a k-group and, being a quotient of the commutative group ZG (T ), it is commutative. But we assumed that G is perfect, so commutativity of G/L forces triviality of G/L; i.e., G = L is reductive. This proves (1). Now consider the converse implication in (2). We may assume G = 1, so P = G. Fix a maximal k-torus T in P, so T is also maximal in G. It sufﬁces to prove that all root groups for Tks in Gks are 1-dimensional. Since P = G, G is non-commutative. The ks -subgroup Pks ⊆ Gks is pseudo-parabolic, and it contains no nontrivial perfect smooth connected normal ks -subgroup N of Gks . (Indeed, otherwise Pks would contain the nontrivial perfect smooth connected normal ks -subgroup of Gks generated by the Gal(ks /k)-orbit of N . This descends to a nontrivial perfect smooth connected normal k-subgroup of G contained in P, contradicting our hypothesis on P.) Thus, we can assume k = ks . Let {Gi }i∈I be the set of minimal nontrivial perfect smooth connected normal k-subgroups of G. By Proposition 3.1.8 this is a ﬁnite set of pairwise commuting k-subgroups, the Gi have no nontrivial smooth connected normal k-subgroups, and the natural multiplication homomorphism π : Gi → G is surjective with kernel containing no nontrivial smooth connected k-subgroup. Also, by the construction of the Gi (and the condition k = ks ), the semisimple quotients (Gi )ss are naturally identiﬁed (without repetition) with the simple connected k

semisimple normal k-subgroups of G red = G ss . k k By normality of Gi in G, Ti = T ∩ Gi is a maximal k-torus in Gi , so Ti is carried by π onto a maximal k-torus in G. But π carries Ti into T , so π

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restricts to a surjection of k-tori Ti → T with kernel containing no nontrivial k-tori. Thus, this surjection of k-tori is an isogeny. By Proposition 2.2.11, we can choose a 1-parameter k-subgroup λ : GL1 → T such that P = PG (λ), Moreover, we may replace λ with λn for any n 1 (Remark 2.1.7). Thus, taking n to be the degree of the isogeny Ti → T allows us to arrange that there exist 1-parameter k-subgroups λi : GL1 → Ti such that λ = π ◦ where : GL1 → Ti has ith component equal to λi for all i. Consider the pseudo-parabolic k-subgroup Pi = PGi (λi ) in Gi containing Ti . Thus, Pi = P Gi () is carried by π onto PG (π ◦ ) = PG (λ) = P (see Corollary 2.1.9). In particular, Pi ⊆ Gi ∩ P for all i. The GL1 -action on Gi through λ-conjugation coincides with the action deﬁned by λi -conjugation because λ = π ◦ and Gi centralizes Gi for all i = i. Hence, Pi = Gi ∩ P by functorial considerations. Since no Gi is contained in P by hypothesis, it follows that Pi = Gi for all i and that Gi /Pi = (Gi P)/P. But (Gi P)/P is closed in G/P since Gi P is closed in G (as it is a k-subgroup), so we conclude that Gi /Pi is complete. Thus, each triple (Gi , Pi , Ti ) satisﬁes the same hypotheses as on (G, P, T ). The surjectivity of π implies that G is reductive if each Gi is reductive, so we may now treat the triples (Gi , Pi , Ti ) separately. That is, we can assume that G has no smooth connected proper normal k-subgroup. In particular, the semisimple quotient G red = G ss is simple. k k Fix a 1-parameter k-subgroup λ : GL1 → T such that P = PG (λ). Let G = G red (so G is semisimple and simple), so the natural projection π : k

Gk G carries Tk isomorphically onto a maximal k-torus T of G . The set of non-multipliable roots in is a root system and is naturally identiﬁed with the root system (G , T ). In particular, is reduced and irreducible, so is irreducible. Since G/P is complete, it follows from the Borel ﬁxed point theorem that Pk contains a conjugate of the smooth connected solvable subgroup Ru (Gk ) in Gk . Hence, by normality of the unipotent radical, Ru (Gk ) ⊆ Pk . By hypothesis P = G, so Lie(G) is strictly larger than Lie(P). In other words, since P = PG (λ), λ<0 := {a ∈ | a, λ < 0} is non-empty. Since Ru (Gk ) ⊆ Pk , it follows that no positive rational multiple of a ∈ λ<0 occurs as a T -weight on Lie(Ru (Gk )). But the exact sequence of smooth connected k-groups 1 → Ru (Gk ) → Gk → G → 1

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induces a Tk -equivariant exact sequence of Lie algebras, so the span of the root spaces in Lie(G) for the roots that are positive rational multiples of a is 1-dimensional over k since the same holds in Lie(G ) over k. The k-subspace Lie(U(a) ) ⊆ Lie(G) is the span of the root spaces for the positive rational multiples of a in , so we conclude that U(a) is 1-dimensional when a ∈ λ<0 = ∩ λ<0 . In particular, such an a is also not divisible, and Ua = U(a) . Since the Weyl group of an irreducible root system acts transitively on the set of roots of a common length [Bou2, VI, §1.3, Prop. 11], and NG (T )(k) maps onto the common Weyl group of and (Proposition 3.2.7), conjugation by NG (T )(k) is transitive on the set of elements of with a common length. We are aiming to prove that all roots groups for (G, T ) are 1-dimensional, so it now sufﬁces to show that if is any reduced and irreducible root system then λ<0 contains roots of all lengths for any nonzero λ ∈ X∗ (T ). Let W = W () be the Weyl group of . It sufﬁces to show that the W -orbit of any a ∈ meets the locus where λ does not vanish, as the orbit is stable under negation. Assuming to the contrary, the hyperplane ker λQ in X∗ (T )Q would contain a nonzero W -stable subspace, and this contradicts the irreducibility of the W -action on X∗ (T )Q [Bou2, VI, §1.2, Cor.]. 2 The perfectness hypothesis on G in Proposition 3.4.9 is reasonable because the criteria in (1) and (2) cannot distinguish between the pseudo-reductive G and its perfect derived group D (G), and over any imperfect ﬁeld k there are non-reductive standard pseudo-reductive G such that D (G) is nontrivial and reductive. To be precise, for (1) the root groups of D (G)ks and Gks coincide (using a maximal k-torus T of G and the maximal k-torus T ∩ D (G) of D (G)). For (2), the results we will prove in §3.5 imply that P → D (G) ∩ P is a bijective correspondence between the sets of pseudo-parabolic k-subgroups in G and D (G), with D (G)/(D (G) ∩ P) → G/P an isomorphism. Moreover, the perfect smooth connected normal k-subgroups of G and D (G) coincide, by Proposition 3.1.8. Here is an example of a non-reductive standard pseudo-reductive G over any imperfect ﬁeld k of characteristic p > 0 such that D (G) is reductive. Consider a nontrivial ﬁnite extension k /k with k p ⊆ k and expand SLp by expanding a GL1 -factor in a split maximal k-torus T to be R k /k (GL1 ). That is, we identify p−1 T with GL1 so that the center μ = ZSLp μp is in the ﬁrst factor, and deﬁne p−2

C = R k /k (GL1 ) × GL1 , so T → T /Z factors through the natural inclusion T → C by using Nk /k : R k /k (GL1 ) → GL1 . Then G = (SLp C)/T is a standard pseudo-reductive k-group with Cartan k-subgroup C that is not a torus (so G is not reductive) but D (G) = SLp .

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In general there is a natural intermediate group between G and D (G) whose reductivity is equivalent to that of D (G) and whose derived group coincides with D (G): the k-subgroup Gt generated by the tori of G. (See Proposition A.2.11 for the fact that Gt is generated by the ks -tori of Gks when k is inﬁnite. For ﬁnite k it can be deﬁned by Galois descent from ks .) The perfect group D (G) is contained in Gt , by Proposition A.2.11, so Gt /D (G) is the maximal k-torus in G/D (G). Thus, in Proposition 3.4.9 it sufﬁces to impose the slightly weaker hypothesis G = Gt .

3.5 Classiﬁcation of pseudo-parabolic subgroups Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and T a split maximal k-torus in G. An interesting application of Corollary 3.3.13, to be discussed below, is to parameterize all pseudo-parabolic k-subgroups of G that contain T , generalizing Proposition 3.3.15(2). This involves the notion of a parabolic set of roots in a root system , as discussed in Deﬁnition 2.2.6 and Proposition 2.2.8. In particular, for any such , the set of non-divisible a ∈ with −a ∈ (so a ∈ ) is a closed set of roots contained in a suitable positive system of roots + ⊆ (so in the case = (G, T ), U makes sense). Proposition 3.5.1 Let G be a pseudo-reductive group over a separably closed ﬁeld k, and let T be a maximal k-torus in G. (1) For every pseudo-parabolic k-subgroup P ⊆ G containing T , Lie(P) is the span of Lie(T ) and some root spaces for T acting on Lie(G), and Lie(P) uniquely determines P. (2) The possibilities for the set (P, T ) of nontrivial T -weights on Lie(P) are precisely the parabolic subsets of (G, T ), and for each P we have Ru,k (P) = U , where is the set of non-divisible roots in (G, T ) whose negative is not in (P, T ). (3) If P and P are pseudo-parabolic k-subgroups containing T then the following are equivalent: P ⊆ P , Lie(P) ⊆ Lie(P ), and (P, T ) ⊆ (P , T ). (4) A pseudo-parabolic k-subgroup P in G is minimal if and only if P/Ru,k (P) is commutative. Since (P, T ) in (2) is a parabolic subset of (G, T ), the set in (2) is contained in (P, T ). We also note that Proposition 3.5.1 is valid for any k if we assume the pair (G, T ) is pseudo-split over k, but we need Proposition 3.5.2(1) to justify this; see Remark 3.5.3.

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Proof. By deﬁnition, P = PG (λ) for some k-homomorphism λ : GL1 → G. By Proposition 2.2.11, we can arrange that λ(GL1 ) ⊆ T (which conversely forces T ⊆ ZG (λ) ⊆ PG (λ)). It then follows from Proposition 2.1.8(1) that (P, T ) = λ0 := {a ∈ (G, T ) | a, λ 0}. By Proposition 2.2.8, these are exactly the parabolic sets of roots (as we vary λ) and the set = λ>0 is a closed set of roots contained in a positive system of roots, so by Corollary 3.3.13(1) the k-group U makes sense. The quotient P/UG (λ) ZG (λ) is pseudo-reductive over k, so Ru,k (P) = UG (λ) has Lie algebra that is the span of the root spaces for the roots in λ>0 . By construction we have U ⊆ UG (λ) (due to the limiting property that characterizes points of UG (λ)), so the equality of their Lie algebras yields equality of groups. In other words, Ru,k (P) = U . We shall now prove that if Lie(P) ⊆ Lie(P ) then P ⊆ P , from which everything else remaining to be proved for (1), (2), and (3) will follow. Consideration of weight spaces gives (P, T ) ⊆ (P , T ). Choose λ : GL1 → T such that P = PG (λ ). Thus, every a ∈ (P, T ) is contained in (P , T ) = λ 0 . That is, a, λ 0. In particular, all points of U(a) satisfy the limiting property to belong to PG (λ ) = P . (To do the limiting calculation, choose a T -equivariant linear structure on the vector group U(a) , and work one root space at a time.) Clearly ZG (T ) ⊆ P for the same reason (as λ is valued in T ). Hence, it sufﬁces to show that the k-group P = PG (λ) is generated by ZG (T ) and the k-groups U(a) for all a ∈ (P, T ) = λ0 . Certainly ZG (T ) and all such k-groups U(a) are contained in P (by the limiting property that characterizes points of P), and the Lie algebras of ZG (T ) and the k-groups U(a) span Lie(P), so we are done. Finally, we prove (4). If P is minimal then P/Ru,k (P) is commutative by Proposition 3.3.15(3). Conversely, suppose that P = PG (λ) is a pseudoparabolic k-subgroup such that P/Ru,k (P) = ZG (λ) is commutative. We wish to prove that P is minimal. Suppose P strictly contains a pseudo-parabolic ksubgroup Q, and let T ⊆ Q be a maximal k-torus of G. We may and do arrange that λ takes values in T . In particular, T ⊆ ZG (λ). By parts (1), (2), and (3), (P, T ) must strictly contain a positive system of roots and ZG (λ) must contain a nontrivial T -weight on its Lie algebra. But this is impossible since ZG (λ) is commutative and contains T . 2 Proposition 3.5.2 Let G be a smooth connected afﬁne group over a ﬁeld k, and P a smooth connected k-subgroup containing a maximal k-torus T in G. (1) Let K/k be a separable extension of ﬁelds. The k-subgroup P in G is pseudoparabolic if and only if the K-subgroup PK in GK is pseudo-parabolic.

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(2) The pseudo-parabolic k-subgroups of G containing T are in natural bijective correspondence with Gal(ks /k)-stable parabolic subsets of ((G/Ru,k (G))ks , Tks ). The separability hypothesis in (1) can be dropped for the “if” direction; see Corollary 3.5.9. Proof. Once (1) is proved, (2) is immediate from Proposition 2.2.10 and Proposition 3.5.1. To prove (1), since the formation of Ru,k (G) commutes with separable extension on k (Proposition 1.1.9(1)), we may assume G is pseudoreductive over k (see Proposition 2.2.10). The “converse” implication is trivial, so to prove the other direction it sufﬁces to treat two cases: K = ks , and k separably closed. First we carry out the descent from K = ks down to k. We have to show that if Pks = PGks (λ ) for a 1-parameter ks -subgroup λ : (GL1 )ks → Gks then we can change the choice of λ if necessary so that it is deﬁned over k. By Proposition 2.2.11 we may arrange that λ takes values in Tks . We pick a ﬁnite Galois extension k /k contained in ks such that Tk is k -split and λ descends to a k -homomorphism. In other words, we have Pk = PGk (λ ) for a 1-parameter k -subgroup λ : (GL1 )k → Tk ⊆ Gk . Exactly as in the proof of Proposition 2.2.9, the 1-parameter k-subgroup γ ∗ (λ ) ∈ Homk (GL1 , T ) λ= γ ∈Gal(k /k)

has the property that the pseudo-parabolic k -subgroups Pk = PGk (λ ) and PG (λ)k share the same sets of Tk -weights on their Lie algebras over k . By Proposition 3.5.1 (applied over ks ), it follows that Pks = PG (λ)ks , so P = PG (λ). It remains to prove that P is pseudo-parabolic when PK is pseudo-parabolic provided that k = ks . We may assume K/k is ﬁnitely generated, so by separability the ﬁeld K is the fraction ﬁeld of a smooth k-algebra domain A. Since GK is pseudo-reductive, PK = PGK (λ) for a K-homomorphism λ : GL1 → GK . By localizing A, we can “spread out” λ to an A-homomorphism μ : GL1 → GA . The theory of 1-parameter limits in §2.1 was developed over any base ring, so PGA (μ) makes sense. The closed A-subgroups PA and PGA (μ) in GA coincide over K, so by localizing A we can arrange that PA = PGA (μ) inside GA . Specializing at a k-point of the k-smooth A then gives the required description of P as a pseudo-parabolic k-subgroup of G. 2 Remark 3.5.3 We can now show that Proposition 3.5.1 is valid over any ﬁeld k if we require (G, T ) to be pseudo-split, and moreover assertion (4) can be

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improved: for any separable extension K/k, P is minimal in G if and only if PK is minimal in GK . The idea is to reduce the proof to the settled case k = ks (for which (4) clearly gives a criterion insensitive to separable extension to another separably closed ﬁeld). The implication “⇒” in (4) was shown in Proposition 3.3.15(3), and for the converse it certainly sufﬁces to work over ks . For (1) and (3) it sufﬁces to work over ks , and once (3) is proved in general (over any k) it follows by Galois descent that to prove (2) it sufﬁces to work over ks . Proposition 3.5.4 Let G be a pseudo-reductive group over a ﬁeld k, and T a maximal k-torus of G. There is a natural inclusion-preserving bijection between the set of pseudo-parabolic k-subgroups P of G containing T and the set of parabolic k-subgroups P of G := G red containing the (isomorphic) image T k of Tk in G . Explicitly, P is the image of Pk under the quotient map Gk G , and if T is k-split then (P , T ) = (P, T ) ∩ (G , T ). For such T , the k-group P is a Borel subgroup of G if and only if P is a minimal pseudo-parabolic k-subgroup of G. For k-split T , an equivalent way to formulate the comparison of sets of nontrivial T -weights occurring on the Lie algebras of the pseudo-parabolic subgroups containing T is that (P , T ) is the set of non-multipliable roots of (P, T ). In Corollary 3.5.11 we will strengthen Proposition 3.5.4 by showing that T ⊆ P if and only if T ⊆ P (with P denoting the image of Pk in G ). Proof. By Galois descent and Remark 3.5.3, we may assume k = ks . Let = (G, T ) and = (G , T ), so is identiﬁed with the set of non-multipliable elements of (Theorem 2.3.10). By Proposition 2.2.11, the possibilities for P are precisely the k-subgroups PG (λ) for λ ∈ X∗ (T ). Hence, by Corollary 2.1.9, the image P of Pk = PGk (λk ) in G is a pseudo-parabolic subgroup of G containing T . Proposition 2.2.9 ensures that the pseudo-parabolic subgroups of G are the same as its parabolic subgroups, so the proposed inclusion-preserving correspondence at least makes sense. We have to prove the following: (i) P determines P, (ii) every parabolic subgroup of G containing T has the form P for some pseudo-parabolic k-subgroup P of G containing T , (iii) (P , T ) = (P, T ) ∩ for all such P (from which it follows via Proposition A.2.10, Proposition 3.3.17, and Proposition 3.5.1 that P is minimal in G if and only if P is minimal in G , which is to say that P is a Borel subgroup).

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The proof of (ii) is straightforward, as follows. Every parabolic subgroup in G containing T has the form PG (λ ) for some λ ∈ X∗ (T ). But k is separably closed, so λ = λk for a unique λ ∈ X∗ (T ). Thus, P := PG (λ) is carried onto P , establishing (ii). In view of the characterization of parabolic subsets of a root system (Proposition 2.2.8), the correspondence → ∩ is an inclusion-preserving bijection between the set of parabolic subsets of and the set of parabolic subsets of (as is the set of nonmultipliable roots of ). Hence, if we prove (iii) then (i) follows. We will now prove (iii). Choose any P containing T , so P = PG (λ) for some λ ∈ X∗ (T ). By Corollary 3.3.12, P is generated by ZG (T ) and the k-subgroups U(a) for all a ∈ (P, T ). The formation of U(a) is unaffected by replacing a with any positive rational multiple that is also a root, so P is also generated by ZG (T ) and the k-subgroups U(a) for a ∈ (P, T ) ∩ . The compatibility of quotient maps with the formation of torus centralizers in smooth connected afﬁne groups (Proposition A.2.8) implies that the quotient map π : Gk G carries ZG (T )k = ZGk (Tk ) onto T (viewed as a maximal torus of G ). Hence, if Ua denotes the root group in G associated to any a ∈ , we will be done if (U(a) )k maps onto Ua for all a ∈ . Letting Ta = (ker a)0red and Ta be the image of (Ta )k in G , the quotient map π carries ZG (Ta )k onto ZG (Ta ) by Proposition A.2.8. Hence, it follows from the deﬁnition of U(a) and Corollary 2.1.9 that π . But is a reduced root system, so U = U by carries (U(a) )k onto U(a) a (a) Proposition 2.3.11. 2 P

Let G be a smooth connected afﬁne group over a ﬁeld k. Due to the analogy with parabolic subgroups in connected reductive groups, it is natural to wonder if a smooth closed k-subgroup Q of G containing a pseudo-parabolic k-subgroup P of G is automatically pseudo-parabolic (and in particular, connected), and to wonder if P = NG (P). To settle both questions (in the afﬁrmative), we ﬁrst handle the normalizer question, using the following lemma of independent interest. Lemma 3.5.5 Let G be a smooth connected afﬁne group over a ﬁeld k, T a maximal k-torus of G, and P a pseudo-parabolic k-subgroup of G containing T . Let U = Ru,k (P), and let T be the image of T in the maximal pseudo-reductive quotient P = P/U . The formation of images and preimages under the quotient map π : P P deﬁnes a bijection between the set of pseudo-parabolic k-subgroups Q of G contained in P and the set of pseudo-parabolic k-subgroups of P. Moreover, Q contains T if and only if π(Q) contains T .

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Proof. Observe that Ru,k (G) ⊆ P by deﬁnition of pseudo-parabolicity, so Ru,k (G) ⊆ U . Hence, by Proposition 2.2.10 we may replace G with G/Ru,k (G) and replace T and P with their respective images in this quotient to reduce to the case where G is pseudo-reductive over k. The proposed bijectivity also allows us to assume k = ks (by Galois descent and Proposition 3.5.2(1)). The pseudo-reductivity of G implies that any pseudo-parabolic k-subgroup Q of G containing T has the form PG (μ) for a 1-parameter k-subgroup μ : GL1 → T (see Proposition 2.2.11). Assume Q is contained in P, so Q = PG (μ) ∩ P = PP (μ). Thus, the image Q of Q in P is PP (π ◦ μ) (Corollary 2.1.9). This is a pseudo-parabolic k-subgroup of the pseudo-reductive k-group P, and it contains T since π ◦ μ takes values in T . The key point is that Q is the preimage of Q, which is to say that Q contains U = Ru,k (P). To prove this, recall from Proposition 3.5.1(2) that U is directly spanned by the T -root groups Ua for a in the set of non-divisible roots in (G, T ) such that −a is not a T -weight on Lie(P). This applies just as well to similarly describe Ru,k (Q) in terms of T -weights on Lie(Q), and since Lie(Q) ⊆ Lie(P) we see that −a is not a T -weight on Lie(Q) for each a ∈ . Hence, such Ua are contained in Ru,k (Q), so Ru,k (P) = U ⊆ Ru,k (Q). This proves that Q = π −1 (π(Q)). It remains to show that every pseudo-parabolic k-subgroup of P is the image of a pseudo-parabolic k-subgroup of G contained in P. Since every pseudoparabolic k-subgroup of G contains a maximal k-torus of G (and likewise for P) and every maximal k-torus of P = P/U is the image of a maximal k-torus of P (due to smoothness of U ), by varying the choice of T , it sufﬁces to restrict attention to a pseudo-parabolic k-subgroup Q in P containing T . We will use root system considerations to construct a pseudo-parabolic k-subgroup Q of G contained in P such that T U ⊆ Q and the pseudo-parabolic image Q of Q in P satisﬁes (Q , T ) = (Q, T ). Proposition 3.5.1(3) applied to (P, T ) will then force Q = Q, so π −1 (Q) = Q is a pseudo-parabolic k-subgroup of G containing T , as desired. Let = (G, T ). Since P is pseudo-parabolic and contains T , P = PG (λ) for a 1-parameter k-subgroup λ : GL1 → T (Proposition 2.2.11). Hence, P = ZG (λ) U with U directly spanned by the root groups Ua for a ∈ λ>0 , and Lie(ZG (λ)) is the span of the T -weight spaces in Lie(G) for the trivial weight and the roots in λ=0 . By Proposition 2.2.8, there is a positive system of roots + in such that the parabolic set (P, T ) = λ0 has the form + ∪[I ] for a (unique) subset I of the set of simple positive roots in + (with [I ] denoting the set of Z-linear combinations of I that lie in ). In particular, [I ] is the set of elements in (P, T ) whose negative is also in (P, T ), so [I ] is equal to the set λ=0 of T -weights on Lie(ZG (λ)).

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The map ZG (λ) → P = P/U induced by π is an isomorphism, so [I ] = λ=0 is identiﬁed with the root system (P, T ) in which (Q, T ) is a parabolic set of roots. That is, I is identiﬁed with a basis of simple positive roots for (P, T ) and there is a linear form μ on the Q-linear span of [I ] such that the parabolic subset (Q, T ) ⊆ [I ] is the locus in [I ] at which μ 0. Now we note that λ vanishes on I and its values on − I are > 0. Since is a Q-basis for X(T )Q , we can deﬁne a unique linear form on X(T )Q by requiring it to agree with λ on − I and with εμ on I , where ε ∈ Q>0 is so small that λ<0 ⊆ <0 . Then the parabolic set of roots 0 is contained in λ0 = (P, T ). By Proposition 3.5.1 there is a unique pseudo-parabolic k-subgroup Q ⊆ G containing T with (Q , T ) = 0 , and Q ⊆ P because (Q , T ) ⊆ (P, T ). Hence, by the unique characterization of root groups in terms of the T -action on G, for each a ∈ λ>0 we have Ua ⊆ Q . The collection of such Ua directly spans U = Ru,k (P), so U ⊆ Q . Hence, T U ⊆ Q . It remains to prove that the image Q of Q in P satisﬁes (Q , T ) = (Q, T ). By deﬁnition, Q = Q /U . Since U is directly spanned by root groups associated to certain non-divisible roots, Lie(U ) is spanned by the root spaces in Lie(G) for all roots in (U , T ). Hence, (Q , T ) consists of the roots in (Q , T ) = 0 not in (U , T ) = λ>0 . By our choice of we have λ<0 ⊆ <0 , so the complement of (U , T ) in (Q , T ) consists of the locus in (Q , T ) = 0 at which λ vanishes. But λ=0 = [I ], and coin cides with εμ on [I ]. Hence, (Q , T ) is the locus in [I ] at which μ 0, and this is (Q, T ) due to how μ was initially chosen. 2 Remark 3.5.6 Consider a smooth connected afﬁne group G over a ﬁeld k and a pseudo-parabolic k-subgroup P. Since every maximal k-torus in P is maximal in G, by applying Proposition 2.2.10 to P it follows from Lemma 3.5.5 that every pseudo-parabolic k-subgroup P of P is a pseudo-parabolic k-subgroup of G. (Proposition 3.5.8 gives a converse.) Proposition 3.5.7 Let G be a smooth connected afﬁne group over a ﬁeld k. A pseudo-parabolic k-subgroup P in G is its own scheme-theoretic normalizer in G, as is PG (λ) for any 1-parameter k-subgroup λ of G. Proof. We may assume k = ks . The proof that PG (λ) is its own normalizer will be given at the end. As a ﬁrst step in the study of NG (P), we show that Lie(NG (P)) = Lie(P), so NG (P) is smooth with identity component P. This will then reduce the problem for NG (P) to a comparison of k-points. Our argument will rest on a general description of Lie algebras of scheme-theoretic normalizers and centralizers of smooth k-subgroups H of G:

3.5 Classiﬁcation of pseudo-parabolic subgroups Lie(ZG (H )) =

h∈H (k) ker(Ad(h) − 1)

137

and

Lie(NG (H )) = {X ∈ Lie(G) | Ad(h)(X ) − X ∈ Lie(H ) for all h ∈ H (k)}. (In nonzero characteristic these are generally strictly smaller than the corresponding Lie-theoretic centralizer and Lie-theoretic normalizer of Lie(H ) in Lie(G).) For the convenience of the reader, let us now recall the proofs of these formulas. By Zariski-density of H (k) in H we have ZG (H ) = h∈H (k) ZG (h), so Lie(ZG (H )) is the intersection of the Lie algebras Lie(ZG (h)). The centralizer ZG (h) is the ﬁber over the identity of the k-scheme map g → (hgh−1 )g −1 , so passing to tangent spaces at the identity yields Lie(ZG (h)) = ker(Ad(h)−1). This establishes the formula for Lie(ZG (H )). To handle NG (H ), for each h ∈ H (k) let fh : G → G denote x → hxh−1 x−1 (so fh (1) = 1). By the Zariski-density of H (k) in H , we have NG (H ) = h∈H (k) fh−1 (H ). Passing to tangent spaces at the identity yields the desired description of Lie(NG (H )). By deﬁnition of pseudo-parabolicity, P = PG (λ)Ru,k (G) for a khomomorphism λ : GL1 → G. Let T be a maximal k-torus in G containing λ(GL1 ), so ZG (T ) ⊆ ZG (λ) ⊆ PG (λ) ⊆ P. Suppose there exists X ∈ Lie(NG (P)) not contained in Lie(P). Both of these Lie subalgebras of g := Lie(G) are stable under the adjoint action of T (as T is contained in NG (P)), so we can ﬁnd such an X that is an eigenvector for the T -action on g. But Lie(ZG (T )) ⊆ Lie(P) (due to the corresponding group containment inside of G), so X ∈ Lie(ZG (T )) =

ker(Ad(t) − 1).

t∈T (k)

It follows that if a : T → GL1 is the eigencharacter for X then a = 1. Choose t0 ∈ T (k) such that a(t0 ) = 1, so Ad(t0 )(X )−X = (a(t0 )−1)X ∈ Lie(P). But X ∈ Lie(NG (P)) by hypothesis, so since t0 ∈ P(k) we have Ad(t0 )(X ) − X ∈ Lie(P). This is a contradiction. We have shown that Lie(P) = Lie(NG (P)), so P = NG (P)0 as subschemes of G. In particular, NG (P) is smooth. It remains to show that if Q is a smooth k-subgroup of G containing P and normalizing P then Q = P. (The point is that we do not assume Q is connected.) Since Q is smooth, its conjugation action on G preserves Ru,k (G). But Ru,k (G) ⊆ P by deﬁnition of pseudo-parabolicity, so we can pass to the quotients G/Ru,k (G) and P/Ru,k (G) to reduce to the case where G is pseudo-reductive. By smoothness, to prove Q = P it is equivalent to show that Q(k) = P(k). For any q ∈ Q(k), qTq−1 is a maximal k-torus in the smooth connected k-group

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P. Since all maximal k-tori in P are P(k)-conjugate (as k = ks ), there exists h ∈ P(k) such that qTq−1 = hTh−1 . Hence, h−1 q ∈ NG (T )(k), so Q(k) = P(k) · (Q(k) ∩ NG (T )(k)). Thus, it sufﬁces to show that Q(k) ∩ NG (T )(k) ⊆ P(k). Let U = Ru,k (P), so H = P/U is a pseudo-reductive k-group and T maps isomorphically onto a maximal k-torus T in H . Fix a positive system of roots + in (H , T ). For q ∈ Q(k) ∩ NG (T )(k), conjugation by q on P induces an automorphism of H that preserves T , so q likewise acts on (H , T ). In particular, q.+ is a positive system of roots in (H , T ). Since W (H , T )(k) acts transitively on the set of positive systems of roots in (H , T ), there exists w ∈ W (H , T )(k) such that w.(q.+ ) = + . Let nw ∈ NH (T )(k) be a representative of w. Using Lemma 3.2.1 we now choose a lift g of nw in P(k) which normalizes T . We can replace q with gq, so q.+ = + . The set of roots + corresponds to a unique minimal pseudo-parabolic k-subgroup P in H containing T (Proposition 3.3.15), and since q normalizes T and q.+ = + it follows that the q-action on H carries P back onto itself. Hence, the preimage P of P in P is normalized by q. By Proposition 3.5.5, P is a minimal pseudo-parabolic k-subgroup of G, so the set (P , T ) of nontrivial T -weights on Lie(P ) is a positive system of roots in (G, T ). The action on (G, T ) by the element q ∈ NG (T )(k) preserves this positive system of roots (since it preserves P ). Thus, by simple transitivity of the action of W (G, T )(k) on the set of positive systems of roots in (G, T ) (by Proposition 3.2.7), we conclude that q has trivial image in W (G, T )(k). That is, q ∈ ZG (T )(k) ⊆ P(k), so we have proved that NG (P) = P. To prove that P := PG (λ) is its own scheme-theoretic normalizer for any λ, we can assume k = k. Let U = Ru (G), so the image of P in G red = G/U is a parabolic subgroup. Parabolic subgroups in connected reductive groups are their own scheme-theoretic normalizers (either as a special case of what we just proved above, or by [SGA3, XXVI, 1.2]), so we are reduced to showing that U ∩ NG (P) = U ∩ P. Since λ deﬁnes a GL1 -action on the normal subgroup U of G, by applying Proposition 2.1.12 to the solvable U we have UU (−λ)×PU (λ) = U via multiplication. But PU (λ) = U ∩PG (λ) = U ∩P, so the problem is to prove that UU (−λ)∩NG (P) = 1. More generally, we claim that UG (−λ) ∩ NG (P) = 1. Let R be a k-algebra and choose u ∈ UG (−λ)(R) that normalizes PR = PGR (λR ). We will show that u ∈ ZG (λ)(R) and hence u = 1. Since λ normalizes UG (−λ), for any R-algebra R and t ∈ R × the commutator (u, λ(t)) lies in UG (−λ)(R ). But it also lies in PG (λ)(R ) since u

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139

normalizes PGR (λR ) (in which λR takes its values). Hence, this commutator is trivial, so u ∈ ZG (λ)(R). 2 Using Proposition 3.5.7, we can now analyze subgroups containing pseudoparabolic subgroups: Proposition 3.5.8 Let G be a smooth connected afﬁne group over a ﬁeld k, and let P be a pseudo-parabolic k-subgroup of G. Any smooth k-subgroup Q of G containing P is pseudo-parabolic in G. (In particular, Q is connected.) Moreover, P is a pseudo-parabolic k-subgroup of Q. Proof. By deﬁnition of pseudo-parabolicity, P contains Ru,k (G). Using Proposition 2.2.10, we may pass to images in G/Ru,k (G) to reduce to the case where G is pseudo-reductive over k. By Proposition 3.5.2, we can assume k = ks . Connectedness of P implies that Q0 contains P. If we can prove that Q0 is pseudo-parabolic in G then Q0 is its own normalizer in G by Proposition 3.5.7, yet Q normalizes Q0 so then Q = Q0 and we would be done. Hence, we may replace Q with Q0 to reduce to the case where Q is connected. Moreover, we can assume that P is a minimal pseudo-parabolic k-subgroup of G. Fix a maximal k-torus T of G contained in P, so ZG (T ) ⊆ P (Proposition 2.2.11). We may and do identify Tk with a maximal torus T in G := G red . Let k = (G, T ), and = (G , T ), so is the set of non-multipliable roots in (Theorem 2.3.10). The set (P, T ) of roots in occurring as T -weights on Lie(P) is a positive system of roots in (Proposition 3.3.15), and by Proposition 3.5.4 the image P of Pk under the quotient map π : Gk G is the Borel subgroup of G containing T such that (P , T ) = (P, T )∩ is the positive system of roots in corresponding to (P, T ). Consider the image Q = π(Qk ) in G . This is a smooth connected subgroup of G containing the Borel subgroup P , so it is a parabolic subgroup containing T . In particular, it is a pseudo-parabolic subgroup containing T (by Proposition 2.2.9), so Q = PG (λ ) for some λ ∈ X∗ (T ) (by Proposition 2.2.11). Since k = ks we have X∗ (T ) = X∗ (T ) via the isomorphism Tk T , so λ = λk for a unique λ ∈ X∗ (T ). We will prove that PG (λ) ⊆ Q and that the Lie algebras coincide inside of Lie(G), whence the groups coincide inside of G (by smoothness and connectedness) and we will be done with the proof that Q is pseudo-parabolic in G. By Corollary 3.3.12, the k-group PG (λ) is generated by ZG (T ) and the kgroups U(a) for non-multipliable a ∈ (PG (λ), T ), which is to say a ∈ such that a, λ 0. But π(PG (λ)) = PG (λ ) = Q , so by Proposition 3.5.4 we can equivalently say that PG (λ) is generated by ZG (T ) and the k-groups

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U(a) for a ∈ (Q , T ) (viewed as characters of T in the natural way). Since ZG (T ) ⊆ P ⊆ Q, it sufﬁces to prove that for all a ∈ (Q, T ), (i) the unique non-multipliable positive rational multiple a of a in occurs as a T -weight on Lie(Q ), (ii) U(a) ⊆ Q (as then PG (λ) ⊆ Q with a reverse inclusion of Lie algebras inside of Lie(G)). The root a lies in , so by Remark 2.3.6 (applied to the quotient map Gk G and the subgroup H = Q) it lies in (Q , T ). This establishes (i). To prove (ii), choose a ∈ (Q, T ). If a lies in the positive system of roots + := (P, T ) ⊆ (Q, T ) inside of then U(a) ⊆ P ⊆ Q. Thus, if in general a is carried by NQ(k) (T ) into + then we are done. It is equivalent to say that the non-multipliable root a ∈ is carried by NQ(k) (T ) into + . To construct enough such elements of NQ(k) (T ) requires some preliminary considerations with ﬁnite étale groups. Since ZG (T ) ⊆ P ⊆ Q, we have ZQ (T ) = ZG (T ). By Lemma A.2.9, NQ (T ) is smooth and W (Q, T ) := NQ (T )/ZQ (T ) is ﬁnite étale over k. Likewise, ZQ (T ) = ZG (T ) = T , NQ (T ) is smooth, and W (Q , T ) := NQ (T )/T is ﬁnite étale over k. There is a natural map of ﬁnite étale k-groups W (Q, T )k → W (Q , T ) induced by the surjective map Qk Q that carries Tk isomorphically onto T . This latter isomorphism of tori implies that a point of NQ (T ) valued in any k-algebra R centralizes TR inside of QR if and only if the corresponding Rk -point of NQ (T ) centralizes TR inside of QR = (QR )k . Hence, the k

k

map W (Q, T )k → W (Q , T ) between ﬁnite étale k-groups has trivial kernel. By Lemma 3.2.1, the map π : NQ (T )k = NQk (Tk ) → NQ (T ) is surjective. In particular, W (Q, T )k → W (Q , T ) is an isomorphism. Since W (Q, T ) is étale and k = ks , we have W (Q, T )(k) = W (Q, T )(k). But it was just proved that W (Q, T )k → W (Q , T ) is an isomorphism of k-groups, so W (Q, T )(k) → W (Q , T )(k) is an isomorphism. The formation of coset spaces for smooth k-groups commutes with the formation of k-points since k = ks , so the group NQ (T )(k) = NQ(k) (T ) maps onto W (Q , T )(k). Hence, the natural map NQ(k) (T ) → W (Q , T )(k) is surjective. In particular, the NQ(k) (T )-action on restricts to the W (Q , T )(k)-action on via the surjective map NQ(k) (T ) W (Q , T )(k). The classiﬁcation of parabolic subgroups of connected reductive k-groups (as “standard” parabolic subgroups containing a given Borel subgroup) shows that the W (Q , T )(k)-action on carries each element of (Q , T ) into the positive system of roots (P , T ) = ∩ + . Applying this to a ∈ (Q , T ), we have proved that Q is pseudo-parabolic in G.

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141

Finally, we prove that P is pseudo-parabolic in Q. Since G is pseudoreductive, P = PG (μ) for some μ : GL1 → G. But μ takes values in ZG (μ) ⊆ P ⊆ Q, so PQ (μ) makes sense and PQ (μ) = PG (μ) ∩ Q = P ∩ Q = P. Hence, P is pseudo-parabolic in Q provided that P contains Ru,k (Q). The kgroup Ru,k (Q) is directly spanned by the groups U(a) for those a ∈ (Q, T ) such that −a ∈ (Q, T ). Since Lie(P) ⊆ Lie(Q), we have −a ∈ (P, T ). Also, by pseudo-parabolicity of P, some positive system of roots in (G, T ) is contained in (P, T ). Hence, a ∈ (P, T ), so U(a) ⊆ P by Corollary 3.3.12. Varying over all such a, indeed Ru,k (Q) ⊆ P. 2 Corollary 3.5.9 Let G be a smooth connected afﬁne group over a ﬁeld k, P a k-subgroup of G, and K/k an extension ﬁeld. If PK is pseudo-parabolic in GK then P is pseudo-parabolic in G. This is a reﬁnement of Proposition 3.5.2(1). Proof. We may and do assume k = ks . Since PK is smooth and connected, so is P. Since Ru,k (G)K ⊆ Ru,K (GK ) ⊆ PK , necessarily Ru,k (G) ⊆ P. Thus, by Proposition 2.2.10, we can pass to quotients by Ru,k (G) so that G is pseudoreductive over k. Let T ⊆ P be a maximal k-torus, so TK is a maximal K-torus in PK and hence in GK . Let π : GK → G = GK /Ru,K (GK ) be the natural projection onto the maximal pseudo-reductive quotient over K. The pseudo-parabolic K-subgroup PK of GK must contain Ru,K (GK ), so the quotient P = π(PK ) is a pseudo-parabolic K-subgroup of G in which TK is a maximal K-torus. Hence, by Proposition 2.2.11, there exists a 1-parameter K-subgroup λ : GL1 → TK such that P = PG (λ ). Since k = ks , λ descends uniquely to a 1-parameter k-subgroup λ : GL1 → T , so PG (λ)K = PGK (λ ). Thus, π(PG (λ)K ) = PG (λ ) = P = π(PK ), so PK = PG (λ)K Ru,K (GK ). This implies that P contains the pseudo-parabolic k-subgroup PG (λ) of G, and so P is a pseudo-parabolic k-subgroup of G by Proposition 3.5.8. 2 Corollary 3.5.10 Let G be a smooth connected afﬁne group over a ﬁeld k, and let P be a pseudo-parabolic k-subgroup of G. For any separable ﬁeld extension K/k, the normalizers in G(K) of Ru,k (P)K = Ru,K (PK ) and Rus,k (P)K = Rus,K (PK ) both equal P(K). Proof. Since PK is pseudo-parabolic in GK (Proposition 3.5.2(1)), we can rename K as k. By Galois descent we can also assume k = ks . The k-subgroup Rus,k (P) in Ru,k (P) is stable under k-automorphisms, so its normalizer in G(k) contains the normalizer of Ru,k (P) (which in turn clearly contains P(k)). Thus, letting Q denote the Zariski closure of NG(k) (Rus,k (P)) in G, it sufﬁces to prove that Q = P. By its deﬁnition, Q is a smooth k-subgroup of G containing P (since

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k = ks ), so it is pseudo-parabolic in G by Proposition 3.5.8. Choose a maximal k-torus T in P, so T is also maximal in G and lies in Q. Let G = G red and let k T denote the (isomorphic) image of Tk in G . By Proposition 3.5.4, to prove P = Q it sufﬁces to prove the equality of the corresponding parabolic image subgroups P and Q in G that contain T . Since Q(k) has dense image in Q , as k = ks , it follows that Q normalizes the image of Rus,k (P)k in G . We claim that the image of Rus,k (P)k in G is exactly Ru (P ). By Corollary 2.2.5, this image is the same as that of Ru,k (P)k . Hence, we may ﬁrst replace G with G/Ru,k (G) so that G is pseudo-reductive over k. It follows that P = PG (λ) for some λ : GL1 → T , so Ru,k (P) = UG (λ). Let λ : GL1 → T correspond to λk , so the quotient map Gk G carries UG (λ)k = UGk (λk ) onto UG (λ ) and carries Pk onto PG (λ ) (Corollary 2.1.9). We conclude that P = PG (λ ), so the k-group Ru (P ) = UG (λ ) is indeed the image of Rus,k (P)k . The containments P ⊆ Q and Q (k) ⊆ NG (k) (Ru (P )) now allow us to replace G with G so as to reduce to checking that in a connected reductive group G over an algebraically closed ﬁeld, any parabolic subgroup P is the normalizer in G of Ru (P) at the level of geometric points. This is an immediate consequence of the explicit description [Bo2, 21.12] of the parabolic subgroups containing a ﬁxed Borel subgroup. (The point is that if Q is any parabolic subgroup strictly containing P then relative to a maximal k-torus of P the Lie algebra of Q supports a pair of opposite roots one of which is supported on Lie(Ru (P)). A suitable Weyl element coming from Q(k) switches these roots, so Q cannot normalize Ru (P).) 2 Here is a strengthening of the correspondence in Proposition 3.5.4: Corollary 3.5.11 Let G be a smooth connected afﬁne group over a ﬁeld k, and let G = G red . Choose a maximal k-torus T in G, and let T be the (isomorphic) k image of Tk in G . For a pseudo-parabolic k-subgroup P in G and the corresponding parabolic image P of Pk in G , T ⊆ P if and only if T ⊆ P . In particular, P determines P without reference to a choice of T . P

Note that we do not claim that every parabolic k-subgroup of G has the form for some pseudo-parabolic k-subgroup P in G; this is generally not true.

Proof. Choose P such that T ⊆ P . By Proposition 3.5.4, there is a unique pseudo-parabolic k-subgroup Q in G containing T such that Q = P inside of G . It is necessary and sufﬁcient to prove P = Q (so then T ⊆ P). We can replace Q with the smooth connected k-subgroup of G generated by P and Q, as this subgroup is pseudo-parabolic by Proposition 3.5.8, so we can assume P ⊆ Q.

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143

Let S be a maximal k-torus in P, so it is maximal in G and contained in Q. The bijectivity in Proposition 3.5.4 for pseudo-parabolic subgroups containing S implies P = Q since P = Q . 2 As an application of our understanding of pseudo-parabolic subgroups in smooth connected afﬁne groups G over a general ﬁeld k, we can now prove G(k)-conjugacy results due to Borel and Tits for maximal k-split tori and minimal pseudo-parabolic k-subgroups of G. We do not use these facts in this monograph, so the development is given in Appendix C. The following result concerning intersections of pseudo-parabolic subgroups is used in §C.2 in the proof of the Bruhat decomposition for G(k) with any (smooth connected afﬁne) G and any ﬁeld k. Proposition 3.5.12 Let G be a smooth connected afﬁne group over a ﬁeld k. Let P and Q be pseudo-parabolic k-subgroups of G. Then: (1) The scheme-theoretic intersection P ∩ Q is k-smooth and connected, and it contains a maximal k-torus of G. (2) The k-subgroup (P ∩ Q)Ru,k (P) in G is pseudo-parabolic. (3) If P is G(k)-conjugate to Q and if the smooth connected k-subgroup P ∩ Q in G is pseudo-parabolic then P = Q. Proof. We may and do assume k = ks . By Proposition 2.2.10 we can replace G with G/Ru,k (G) and replace P and Q with their respective images in G/Ru,k (G) to assume that G is pseudo-reductive. To prove that P ∩ Q contains a maximal k-torus of G, we may assume that P and Q are minimal pseudoparabolic k-subgroups. By conjugacy of such subgroups (Proposition 3.3.17), there exists g ∈ G(k) such that Q = gPg −1 . Choose a maximal k-torus S of P, so S is maximal in G and hence the Bruhat decomposition of G(k) in Theorem 3.4.5 yields g = pnp for some p, p ∈ P(k) and n ∈ NG (S)(k). Hence, Q = gPg −1 = pnPn−1 p−1 , and so P ∩ Q = p(P ∩ nPn−1 )p−1 . Since n normalizes S, we conclude that P ∩Q contains the maximal k-torus pSp−1 in G. We now drop minimality hypotheses on P and Q, and we ﬁx a maximal k-torus T of G contained in P ∩ Q. Let = (G, T ), Z = ZG (T ), and z = Lie(Z). For each a ∈ , let U(a) denote the k-subgroup as in Deﬁnition 2.3.4, and let ua be the a-root space in the Lie algebra g of G. By Proposition 2.2.11 there exist 1-parameter k-subgroups λ, μ : GL1 ⇒ T such that P = PG (λ) and Q = PG (μ), so Z ⊆ P ∩ Q. By Corollary 3.3.12, for a ∈ , U(a) ⊆ P if and only if a ∈ (P, T ), and likewise for Q. Hence, U(a) ⊆ P ∩ Q for every a ∈ (P, T )∩(Q, T ), so the smooth connected k-subgroup H of G generated by Z and the k-groups U(a) with a ∈ (P, T ) ∩ (Q, T ) is contained in P ∩ Q.

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The T -weight space decomposition of Lie(G) yields the T -equivariant decompositions

ua ), Lie(Q) = z ⊕ ( ua ), Lie(P) = z ⊕ ( a∈(P,T )

a∈(Q,T )

so Lie(H ) ⊆ Lie(P ∩ Q) = Lie(P) ∩ Lie(Q)

=z⊕(

ua )

a∈(P,T )∩(Q,T )

⊆ Lie(H ). Thus, the inclusion H ⊆ P ∩Q of k-subgroup schemes of G induces an equality on Lie algebras, so since H is smooth and connected it follows that H = (P ∩ Q)0 . In particular, (P ∩ Q)0 is smooth and hence P ∩ Q is smooth. This concludes the proof of (1) apart from the connectedness claim. According to Corollary 2.2.5, Rus,k (P) = UG (λ) = Ru,k (P). By Proposition 3.5.8, any smooth k-subgroup of G containing a pseudo-parabolic k-subgroup is pseudo-parabolic, and in particular is connected. Thus, to prove (2) it sufﬁces to show that (P ∩ Q)0 Rus,k (P) is a pseudo-parabolic k-subgroup of G, as then (P ∩ Q)Rus,k (P) = (P ∩ Q)0 Rus,k (P).

(3.5.1)

We claim that the intersection J := (P ∩ Q)0 ∩ Rus,k (P) is smooth. Its Lie algebra is the intersection of the Lie algebras of P, Q, and Rus,k (P), so Lie(J ) is spanned by the ua for a ∈ such that U(a) ⊆ J (see Proposition 3.5.1(2)). Hence, as in our proof of smoothness of P ∩ Q, the smooth connected k-subgroup of J 0 generated by the U(a) that are contained in J must coincide with J 0 . In particular, J 0 is smooth, so J is smooth. It follows that the surjective homomorphism (P ∩ Q)0 Rus,k (P) → (P ∩ Q)0 Rus,k (P) has smooth kernel, so Lie((P ∩ Q)0 Rus,k (P)) = Lie((P ∩ Q)0 ) + Lie(Rus,k (P)) = Lie(P) ∩ Lie(Q) + Lie(UG (λ)). For any a ∈ , we have a ∈ (P, T ) if and only if a, λ 0, and moreover a occurs as a T -weight on the Lie algebra of Rus,k (P) = UG (λ) if and only if

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a, λ > 0. Hence, if a ∈ satisﬁes a, λ = 0, then exactly one b ∈ {±a} occurs as a T -weight in Lie(Rus,k (P)), with the corresponding U(b) contained in Rus,k (P), whereas if a, λ = 0 then both a and −a lie in (P, T ) and at least one of them lies in (Q, T ) (since (Q, T ) contains a positive system of roots in ). But U(b) ⊆ (P ∩ Q)0 for each b ∈ (P, T ) ∩ (Q, T ). Thus, the pseudoparabolicity of (P ∩ Q)0 Rus,k (P) (and so the proof of (2)) is a consequence of the following surprising general lemma. Lemma 3.5.13 Let G be a pseudo-reductive group over a ﬁeld k, and T a maximal k-torus in G. Let H be a smooth k-subgroup of G such that: (i) ZG (T ) ⊆ H , (ii) (Gks , Tks ) = (Hks , Tks ) ∪ −(Hks , Tks ), and (iii) a ∈ (Hks , Tks ) implies U(a) ⊆ Hks . Then H is pseudo-parabolic in G. Proof. We may and do assume k = ks . Observe also that by (iii), if an element a ∈ (G, T ) lies in (H , T ) then so does every positive rational multiple of a in (G, T ). Let H denote the image of Hk in G = G red and let T be the maximal torus k image of Tk in G , so H contains T . The quotient map Hk → H cannot kill any (U(a) )k for a ∈ (H , T ) (due to the pseudo-reductivity of G over k), so (U(a) )k is carried T -equivariantly onto a root group of G contained in H . Thus, this root group must correspond to a positive rational multiple of a, so it follows from Theorem 2.3.10 that (H , T ) consists of the non-multipliable roots in (H , T ). This argument also shows that H satisﬁes the analogues of (i), (ii), and (iii) relative to (G , T ). Assume H is parabolic in G , so there is a unique pseudo-parabolic ksubgroup P in G such that T ⊆ P and Pk maps onto H (Proposition 3.5.4), and (H , T ) is the set of non-multipliable roots in (P, T ). Hence, (P, T ) and (H , T ) have the same non-multipliable elements, so (P, T ) = (H , T ). By (i) and (iii), we would then have H 0 = P, so H = P by Proposition 3.5.7. It remains to show that H is parabolic in G , which is to say that we can assume k is algebraically closed and G is reductive. Since H contains the maximal torus T of G, this is also a maximal torus of H and so there is a Borel subgroup B in H containing T . The containment Ru (H ) ⊆ B and the reductivity of H /Ru (H ) imply that (H , T ) is the union of (B, T ) and −(B, T ), so likewise for (G, T ) by (ii). In particular, the size of (B, T ) is at least as large as (1/2)#(G, T ), so dim B = dim T + #(B, T ) dim T + (1/2)#(G, T ). This lower bound is the common dimension of the Borel subgroups of the connected reductive G, so for dimension reasons B is also a Borel subgroup of G. Hence, H is parabolic in G. 2

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Basic structure theory

Returning to the proof of Proposition 3.5.12, to complete the proof of (1) it remains to prove that P ∩ Q is connected. By (3.5.1), we have P ∩ Q ⊆ (P ∩ Q)0 Rus,k (P), so by consideration of geometric points it sufﬁces to show that the k-points of the triple intersection I := P ∩ Q ∩ Rus,k (P) are contained in (P ∩ Q)0 . In other words, we need to show that every g ∈ I (k) is contained in a connected subset of P ∩ Q that contains the identity. Consider the k-scheme map φ : GL1 → Gk deﬁned by t → λ(t)gλ(t)−1 . This map lands inside of the closed subscheme Ik since T normalizes each of P, Q, and Rus,k (P). Moreover, by deﬁnition of Rus,k (P) = UG (λ) it follows that φ extends to a k-scheme map : A1 → G carrying 0 to 1. But the image of φ must remain inside of I , and φ k k k = g and φ(0) = 1. it is connected and contains φ(1) It remains to prove (3), so we assume that Q = gPg −1 for some g ∈ G(k) and that the smooth connected k-subgroup P ∩ Q in G is pseudo-parabolic. In particular, P ∩ Q contains a minimal pseudo-parabolic k-subgroup P of G. Thus, g −1 P g is a minimal pseudo-parabolic k-subgroup of G, yet it is also contained in g −1 Qg = P, so by Remark 3.5.6 and Proposition 3.5.8 both P and g −1 P g are minimal pseudo-parabolic k-subgroups of P. Hence, by the P(k)conjugacy of such k-subgroups (Proposition 3.3.17), there exists p ∈ P(k) such that p−1 g −1 P gp = P . This says that gp ∈ NG (P )(k) = P (k) (see Proposition 3.5.7), so Q = gPg −1 = P as desired. 2

PART II Standard presentations and their applications

4 Variation of (G , k /k, T , C)

4.1 Absolutely simple and simply connected ﬁbers Let k be a ﬁeld and G a standard pseudo-reductive k-group in the sense of Deﬁnition 1.4.4. By deﬁnition of being standard, G arises from a 4-tuple (G , k /k, T , C) (and a factorization diagram involving C as in (1.4.1), which we suppress from the notation), but several such 4-tuples can give rise to the same standard k-group G. We now focus our efforts on ﬁnding a canonical way to choose the 4-tuple (if it exists), as that will make it easier to carry out Galois descent arguments. Observe that the k-group G is solvable if and only if G = T , or equivalently (by Proposition 1.2.3) if and only if G is commutative. In such cases it is natural to use the choices k = k, G = T = {1}, and C = G, so we shall now focus on the non-commutative case. It is proved below that a non-commutative standard pseudo-reductive k-group G can always be obtained from a 4-tuple (G , k /k, T , C) such that the ﬁbers of G over the factor ﬁelds ki of k are absolutely simple and simply connected. Thus, there is no loss of generality in the deﬁnition of standardness for a non-commutative pseudo-reductive k-group if we require that G → Spec k has absolutely simple and simply connected ﬁbers. In §4.2 we will show that with these ﬁner requirements, the pair (G , k /k) (equipped with certain additional structure) is uniquely determined up to unique k-isomorphism by the non-commutative k-group G. There is complete ﬂexibility in the choice of maximal k -torus T ⊆ G appearing in the standard construction of G, in the following sense. Such choices are generally not G (k )-conjugate to each other (when k is not separably closed), but if we make another choice of maximal k -torus S ⊆ G , then Proposition 4.1.4(2), (3) will show that the factorization diagram of k-groups ψ

φ

R k /k (T ) → C → R k /k (T ) 149

(4.1.1)

Variation of (G , k /k, T , C)

150

used to construct G can be changed by replacing C with an étale twist (and replacing T and T with S and S = S /ZG respectively) so that the resulting standard pseudo-reductive k-group is k-isomorphic to the given k-group G. Theorem 4.1.1 Let k be a ﬁeld, and G a non-commutative standard pseudoreductive k-group. The k-group G arises from a 4-tuple (G , k /k, T , C) as in Deﬁnition 1.4.4 such that G has absolutely simple and simply connected ﬁbers over Spec k . Proof. Pick a 4-tuple (G , k /k, T , C) as in the standard construction so that there is an isomorphism of k-groups (R k /k (G ) C)/R k /k (T ) G. Here it is only assumed that the ﬁbers of G → Spec k are connected reductive. We ﬁx such an isomorphism, and try to improve the choice of the 4-tuple. Let Gi and Ti respectively denote the ﬁber of G and ﬁber of the maximal k -torus T ⊆ G over the factor ﬁeld ki of k . If Gi0 = Ti0 for some i0 then we can drop the i0 -factor from k and G as well as from the ﬂanking terms in (4.1.1) (leaving C unchanged) without affecting G. (Since G is non-commutative, we cannot have Gi = Ti for all i.) Thus, we can assume that all Gi are noncommutative, so the connected semisimple derived group D (Gi ) is nontrivial for all i. → D (G ) be the (ﬁberwise) simply connected central cover of D (G ) Let G over Spec k , and let Z ⊆ T be the maximal central k -torus in G . Under the × Z → G , the preimage of T is central k -isogeny G T × Z for a maximal . Thus, T viewed as a quotient of k -torus T ⊆ G T × Z inherits a natural left action on G × Z induced by conjugation. This yields a push-out presentation × Z ) T )/( G ((G T × Z ) involving the quotient by a smooth k -group, so the formation of this presentation commutes with R k /k (Corollary A.5.4). We therefore arrive at a commutative diagram T × Z ) R k /k (

/ R k /k (T )

(G × Z )

/ R k /k (G )

R

k /k

φ

/ C / G

ψ

/ R (T ) k /k

4.1 Absolutely simple and simply connected ﬁbers

151

in which T = T /ZG and the two small rectangles are non-commutative pushouts in the sense of Remark 1.4.5, resting on the natural left action of that is trivial on Z . Hence, the larger rectangle is such a pushout. T on G The k -torus T is the quotient of T × Z modulo the scheme-theoretic center × Z , so (G × Z , k /k, of G T × Z , C) gives rise to G via the standard , k /k, construction in Deﬁnition 1.4.4. The 4-tuple (G T , C) also makes sense as input into the standard construction (with T /ZG = T ), and it also yields G. : R k /k ( Remark 4.1.2 Beware that the map φ T ) → C induced by φ can fail to be surjective, even if φ is surjective. For example, suppose G = R k /k (G ) with G → Spec k having connected semisimple ﬁbers. If the central isogeny → G is non-étale on the ﬁber over some factor ﬁeld k that is not k-étale, G i is not surjective. then φ and Continuing with the proof of Theorem 4.1.1, rename G T as G and T respectively, so the ﬁbers of G over the factor ﬁelds of k are semisimple and simply connected. By applying Proposition A.5.14 over the factor ﬁelds of k , there is a k -group isomorphism G R k /k (G ) for a faithfully ﬂat ﬁnite étale k -algebra k and a semisimple k -group G whose ﬁbers are connected, absolutely simple, and simply connected. The behavior of maximal tori with respect to Weil restriction (Proposition A.5.15(2)) implies that T is the maximal k -torus in R k /k (T ) for a unique maximal k -torus T in G . But since k is k -étale, R k /k (T ) is a k -torus, so it is equal to T . We can therefore replace k and (G , T ) with k and (G , T ) respectively to recover the same output G without changing (1.4.1) but achieving absolute simplicity for the simply connected ﬁbers of G over k . This completes the proof of Theorem 4.1.1. 2 Deﬁnition 4.1.3 Let G be a non-commutative standard pseudo-reductive group over a ﬁeld k. A standard presentation of G is a 4-tuple (G , k /k, T , C) as in Deﬁnition 1.4.4 such that G is a semisimple k -group whose ﬁbers are connected, absolutely simple, and simply connected, along with a choice of isomorphism (1.4.3). We also call this a standard presentation of G resting on (G , k /k). For later purposes we need to understand the effect on a standard presentation when we change the choice of maximal k -torus T in G . This is addressed in part (3) of: Proposition 4.1.4 Let G be a non-commutative standard pseudo-reductive k-group, and ﬁx a standard presentation (G , k /k, T , C) of G. (1) Let j : R k /k (G ) → G be the natural map, and let T be the maximal k-torus in R k /k (T ). The image of j is D (G), and if T is the maximal

152

Variation of (G , k /k, T , C)

k-torus in G that is an almost direct product of the maximal central k-torus Z of G and j(T ), then C = ZG (T ). (2) The natural map T → j(T ) is an isogeny, and there is a bijective correspondence between the set of maximal k -tori S in G and the set of maximal k-tori in G deﬁned by S → j(S ) · Z, where S is the maximal k-torus in R k /k (S ). (3) Let S be a maximal k-torus in G, S the associated maximal k -torus in G , and S the maximal k-torus in R k /k (S ). The conjugation action of ZG (S) on D (G) = j(R k /k (G )) uniquely lifts to an action of ZG (S) on R k /k (G ) ﬁxing the maximal k -torus S . For the standard pseudo-reductive k-group associated to (G , k /k, S , ZG (S)) and the resulting factorization diagram R k /k (S ) −→ ZG (S) −→ R k /k (S /ZG ) via Theorem 1.3.9 for (G , k /k, S ), the natural k-homomorphism γ : (R k /k (G ) ZG (S))/R k /k (S ) → G is an isomorphism. In particular, there is a natural bijective correspondence between the set of standard presentations of G resting on (G , k /k) and the set of maximal k-tori of G. Proof. Since the formation of the maximal central torus in a smooth connected afﬁne group over a ﬁeld commutes with any extension of the ground ﬁeld, by Galois descent we may and do assume k = ks . The k-group R k /k (G ) is perfect (Corollary A.7.11), so the image of j is D (G) since C is commutative. The image of R k /k (G ) in G meets C in the image of the Cartan k-subgroup R k /k (T ) of R k /k (G ). Thus, C is the image under the quotient map R k /k (G ) C G of the Cartan k-subgroup R k /k (T ) C. Hence, C is a Cartan k-subgroup of G, and in fact it is ZG (T ) where T is the maximal k-torus in the commutative C. Since the composite map in the factorization diagram (1.4.1) underlying the standard presentation is an isogeny between maximal k-tori (as we can check on maximal reductive quotients over k), this diagram canonically expresses T as an almost direct product of the image j(T ) of the maximal k-torus T in R k /k (T ) and the maximal k-torus in ker ψ. This latter k-torus is the maximal central k-torus Z in G, due to C being a Cartan k-subgroup, so (1) is proved. The common kernel ker j = ker φ contains no nontrivial k-tori (it is even contained in the torsion group ker(ψ ◦ φ) = R k /k (ZG )), so T → j(T ) is an

4.1 Absolutely simple and simply connected ﬁbers

153

isogeny. By rational conjugacy of maximal tori over a separably closed ﬁeld and the identiﬁcation of C with ZG (T ), (2) now follows by applying G(k)conjugacy throughout. The same rational conjugacy considerations prove (3), provided we check that the ZG (S)-action on R k /k (G ) is uniquely determined by the conjugation action on the quotient R k /k (G )/ ker φ = im(j) = D (G). By Theorem 1.3.9, the ZG (S)-action on R k /k (G ) is classiﬁed by a khomomorphism R k /k (G ) → R k /k (S /ZG ) to a commutative target. Thus, we just have to check that if the action on R k /k (G )/ ker φ is trivial then the action on R k /k (G ) is trivial. But ker φ ⊆ R k /k (ZG ), so by considering k-points it sufﬁces to prove that if f is a k-automorphism of R k /k (G ) that induces the trivial automorphism on the quotient R k /k (G )/R k /k (ZG ) then f is the identity. This condition says that g → f (g)g −1 is a pointed k-scheme map from R k /k (G ) into R k /k (ZG ), or equivalently a pointed k -scheme map R k /k (G )k → ZG . Such a map is trivial since the source has smooth geometrically connected ﬁbers over k whereas ZG has ﬁbers that are ﬁnite. 2 Remark 4.1.5 By Proposition 4.1.4, it makes sense to say that a standard presentation of a non-commutative pseudo-reductive k-group G is compatible (or not) with a uniquely determined maximal k-torus T of G (which can be chosen arbitrarily) when the pair (G , k /k) is ﬁxed and the ﬁbers of G over Spec k are connected, absolutely simple, and simply connected. In §4.2 we will prove that G canonically determines (G , k /k), so choosing a standard presentation of G is “the same” as choosing a maximal k-torus of G. Corollary 4.1.6 Let G be a non-commutative standard pseudo-reductive group over a separably closed ﬁeld k, and (G , k /k, T , C) a standard presentation. Let T be the associated maximal k-torus of G. Let g = Lie(G) and g = Lie(G ). Let {ki }i∈I be the set of factor ﬁelds of k , and (Gi , Ti ) the ki -ﬁber of (G , T ). Let gi = Lie(Gi ), so g = gi . There is a canonical decomposition of root systems (G, T ) =

(Gi , Ti )

(i.e., the root systems (Gi , Ti ) are the irreducible components of (G, T )), and if a ∈ (G, T ) goes over to ai ∈ (Gi , Ti ) for a unique i ∈ I then the

154

Variation of (G , k /k, T , C)

natural map j

g = Lie(R k /k (G )) → Lie(G) = g carries (gi )ai isomorphically onto ga . In particular, (G, T ) is always a reduced root system when G is standard. Proof. Let Z be the maximal k-torus in C that is central in G (i.e., the maximal k-torus in the kernel of ψ : C → R k /k (T /ZG )), and let T be the maximal k-torus in R k /k (T ). Thus, T → j(T ) is an isogeny and T is the almost direct product of j(T ) and Z. In particular, all elements of (G, T ) factor through the quotient T /Z that is an isogenous quotient of T . Since ker j ⊆ R k /k (ZG ) is central in R k /k (G ), the root spaces for T acting on Lie(R k /k (G )) map isomorphically onto the root spaces for T acting on g, so (G, T ) = (R k /k (G ), T ). This reduces the problem to the special case G = R k /k (G ) and T the maximal k-torus in R k /k (T ). In particular, g = gi . Decomposing according to the factor ﬁelds of k reduces us to the case where k is a ﬁeld. Under the canonical identiﬁcation of X(T ) and X(T ), the distinct weight spaces for T acting k -linearly on g correspond to the distinct weight spaces for T acting k-linearly on g . 2

4.2 Uniqueness of (G , k /k) We have shown in §4.1 that for any non-commutative standard pseudo-reductive k-group G there is a nonzero ﬁnite reduced k-algebra k such that G can be obtained (via the standard construction in Deﬁnition 1.4.4) from data (G , k /k, T , C) with a k -group G whose ﬁbers over Spec k are connected, semisimple, absolutely simple, and simply connected. We wish to reconstruct the pair (G , k /k) (up to k-isomorphism) from the non-commutative (hence non-solvable) standard pseudo-reductive k-group G by a procedure that will make sense for an arbitrary non-solvable smooth connected afﬁne k-group (perhaps not even pseudo-reductive). This rests on a certain category that is associated to any smooth connected afﬁne group G over a ﬁeld k, as follows. Consider pairs (F/k, q : GF H ) with F a reduced artinian k-algebra (i.e., a product of ﬁnitely many extension ﬁelds of k, not necessarily of ﬁnite kdegree), H an adjoint semisimple F-group with absolutely simple and connected ﬁbers, and q a quotient homomorphism over F that is a maximal simple adjoint

4.2 Uniqueness of (G , k /k)

155

semisimple quotient on geometric ﬁbers. Deﬁne a morphism (F/k, q : GF H ) → (F /k, q : GF H ) to consist of a k-algebra map α : F → F and commutative diagram of F -group maps GF

GF

α ∗ (q)

α ∗ (H )

/ H

q

(4.2.1)

Note that the bottom arrow in this diagram is uniquely determined by α (if it exists). Let CG/k denote the category just deﬁned. There is a ﬁnal object in CG/k , corresponding to F = 0 and H the empty group scheme over Spec(0), and if G is solvable then this is the only object in the category. If k/k is a separable extension of ﬁelds then there is a natural “base change” functor CG/k → CGk /k deﬁned by (F/k, q) (F/k, q)k := (Fk /k, q ⊗F Fk ) where Fk = k ⊗k F (a ﬁnite product of ﬁelds since k/k is separable). This functor is left adjoint to the forgetful functor CGk /k → CG/k when k/k is separable. Proposition 4.2.1 There exists an initial object (K/k, πG : GK G ) in the category CG/k . The structure map Spec K → Spec k is ﬁnite, and the formation of (K/k, πG ) commutes with any separable extension k/k in the sense that (K/k, πG )k is initial in CGk /k . We call (K/k, πG ) the universal maximal absolutely simple adjoint quotient of G. Before we prove it exists, let us brieﬂy explain why a certain possible generalization can fail to exist. Consider the larger category CG/k in which we allow ﬁberwise (over SpecF) absolutely simple adjoint semisimple quotients q : GF H that may not be maximal on geometric ﬁbers over Spec F (and we drop the requirement that the bottom arrow in (4.2.1) is an isomorphism). The initial object in Proposition 4.2.1 can fail to be initial in CG/k when k is imperfect because composing an adjoint semisimple quotient of Gk with an iterated Frobenius isogeny can decrease its ﬁeld of deﬁnition over k. Proof. For any separable extension k/k, the adjointness between the forgetful functor CGk /k → CG/k and the base change functor (F/k, q) (F/k, q)k implies that such base change carries an initial object of CG/k to an initial object of CGk /k .

Variation of (G , k /k, T , C)

156

Now we construct an object (K/k, πG : GK G ) in CG/k so that for any (F/k, q : GF H ) in CG/k there exists a unique k-algebra map α : K → F such that α ∗ (πG ) : GF α ∗ (G ) is isomorphic to H (necessarily uniquely) as quotients of GF . The case of solvable G is trivial (as then CG/k consists of only the empty object), so we now assume G is non-solvable. As a ﬁrst step, we show that if an object (K/k, πG ) has been constructed so that there exists a unique α as above for any (F/k, q) with F a separably closed ﬁeld then the existence and uniqueness of α holds in general. Consider a general pair (F/k, q), say with F = Fi for ﬁelds Fi /k and qi : GFi Hi the Fi -ﬁber of q. To give a k-algebra map α : K → F = Fi such that the quotient map α ∗ (πG ) over F is (necessarily uniquely) identiﬁed with H as a quotient of GF is equivalent to giving k-algebra maps αi : K → Fi such that the quotient map αi∗ (πG ) over Fi is identiﬁed with qi as a quotient of GFi for each i (the link is that αi is the Fi -component of α). Hence, there is no loss of generality in restricting attention to the case when F is a ﬁeld. Applying the assumption that existence and uniqueness is known for (Fs /k, qFs ), there exists a unique k-algebra map αs : K → Fs such that there is a (necessarily unique) identiﬁcation θs : αs∗ (πG ) = qFs as quotients of GFs . But then for every σ ∈ Gal(Fs /F) the map σ ◦ αs works as well, using the composition of the identiﬁcations σ ∗ (θs ) and σ ∗ (qFs ) = qFs of quotients of σ ∗ (GFs ) = GFs . Hence, uniqueness for (Fs /k, q) implies that σ ◦ αs = αs for all σ , which is to say that αs (K) ⊆ F. Letting α : K → F be the k-algebra map induced by αs , the identiﬁcation θs of quotients α ∗ (πG )Fs = αs∗ (πG ) and qFs of GFs is unique (as for identiﬁcations of quotients in general), so it is automatically Gal(Fs /F)-equivariant and thus descends to an identiﬁcation θ : α ∗ (πG ) = qF of quotients of GF . This proves existence of the desired α for (F/k, q), and its uniqueness is immediate from the assumed uniqueness for (Fs /k, qFs ). Now we may and do only consider the mapping property to pairs (F/k, q) for which F is a separably closed ﬁeld. We next wish to reduce to the construction of the desired (K/k, πG ) to the case k = ks . Assume there is an initial object (Ks /ks , πGks ) in CGks /ks with Ks a ﬁnite ks -algebra. By expressing ks as a direct limit of ﬁnite Galois extensions k/k, for a sufﬁciently large k/k we can write Ks = K ⊗k ks for a ﬁnite k-algebra K and descend πGks to a quotient map

π : GK G in CGk /k . For any pair (F/k, q) with F a separably closed ﬁeld, we may choose a k-embedding of ks into F, so there is a unique morphism (K /k, π )ks = (Ks /ks , πGks ) → (F/ks , q) in CGks /ks . Thus, by Galois descent with respect to ks /k, there is a unique morphism (K /k, π ) → (F/k, q) in CGk /k . It follows that (K /k, π ) is initial in CGk /k .

4.2 Uniqueness of (G , k /k)

157

The universal property of (K /k, π ) provides a natural action by Gal(k/k) on (K /k, π ) covering its natural action on k and respecting the k-structure on Gk . Since G is afﬁne, this Galois descent datum is effective. That is, (K /k, π ) descends to an object (K/k, πG ) in CG/k , and the same argument that descended from ks to k (now using pairs (F/k, q) with F a separably closed ﬁeld) also shows that (K/k, πG ) is initial. Now we may assume k = ks , and we will construct an object (K/k, πG ) in CG/k for which the desired existence and uniqueness of α holds relative to all objects (F/k, q) with F a separably closed ﬁeld. As we have seen above, such a (K/k, πG ) would then be an initial object in CG/k . Let K = ki where {ki } is the set of ﬁelds of deﬁnition over k for the kernels of the projections πi : Gk G i onto the distinct maximal simple adjoint semisimple quotients. Let πG : GK G have ki -ﬁber Gki G i that is the unique descent of πi to its minimal ﬁeld of deﬁnition ki over k. Since k = ks , k has a unique k-embedding into the algebraic closure F of F = Fs . The maximal simple adjoint semisimple quotient map q : GF H over F = Fs identiﬁes HF with a unique simple factor of GFss /ZGss . F Since R(GF ) = R(Gk )F inside of GF , there is a unique isomorphism GFss /ZGss (Gkss /ZGss )F F

k

as quotients of GF . This isomorphism respects the formation of simple factors over k and F, so we conclude that there is a unique i such that qF is identiﬁed with (πi ) ⊗k F. In other words, the kernel of the projection from GF onto its ith maximal simple adjoint semisimple quotient is deﬁned over ki /k and over F/k. Hence, it is deﬁned over ki ∩ F (as an extension of k). But ki /k is the minimal ﬁeld of deﬁnition of this quotient viewed over k, so in particular this quotient cannot be descended over a proper subﬁeld of ki /k. It follows that ki ∩ F = ki , so ki ⊆ F inside of F. This inclusion of ﬁelds over k is the required unique α. 2 Proposition 4.2.2 For any short exact sequence j

p

1 → G → G → G → 1 of smooth connected afﬁne groups over a ﬁeld k, the universal maximal absolutely simple adjoint quotient (K/k, πG ) for G is canonically identiﬁed with the “disjoint union” of the corresponding objects (K /k, πG ) and (K /k, πG ) for G and G . That is, Spec K is identiﬁed with Spec K Spec K over which πG is the disjoint union of πG and πG .

Variation of (G , k /k, T , C)

158

Proof. Consider the following two assertions: (i) for any (F /k, q ) in CG /k , the pair (F /k, q ◦ p) is an object in CG/k , (ii) for any (F /k, q ) in CG /k , there is a unique (F /k, q) in CG/k such that (F /k, q ◦ j) = (F /k, q ). Once (i) and (ii) are proved, applying them to initial objects deﬁnes a canonical map (K/k, πG ) → ((K × K )/k, πG πG ) in CG/k whose formation commutes with any separable extension on k, and we will show that this canonical map is an isomorphism. In view of the uniqueness assertions, by Galois descent we can assume k = ks . To prove (i) and the existence in (ii), it is easy to reduce to the case of the initial objects attached to G and G . By the ﬁnal assertion in Proposition A.4.8, there is an induced short exact sequence among the maximal adjoint semisimple quotients of G , Gk and G . Since connected semisimple groups of adjoint k

k

type over k are canonically a direct product of their simple factors, the maximal adjoint semisimple quotient of Gk is canonically the product of those for G k and G . Thus, (i) is clear. In view of how the universal maximal absolutely k simple adjoint quotients (i.e., initial objects in CG /k , CG/k and CG /k ) are constructed, it remains to (a) prove the uniqueness in (ii), and (b) show that for each maximal simple adjoint semisimple quotient q : Gk H the ﬁeld of deﬁnition F/k for the kernel is the same as for the corresponding such quotient for G or G . k k Consider (b). The case of q factoring through G is trivial, so we just have k to check that if q is nontrivial on G then the ﬁeld of deﬁnition F /k for the k kernel of the resulting maximal simple adjoint semisimple quotient map q = q ◦ jk : G H is equal to F. Clearly F ⊆ F over k = ks , and we need to k prove this is an equality. We can rename F as k, so it sufﬁces to prove that any adjoint semisimple quotient q : G H over k that is maximal as such over k lifts uniquely to a quotient map q : G H over k. This would also settle (a) letting k vary through the factor ﬁelds of F . The uniqueness of q follows from the structure of G ss /ZGss . For the existence, k

k

since G is smooth its conjugation action on G induces an action of Gk on G ss , k and hence likewise on the maximal adjoint semisimple quotient G ss /ZG ss . By k k connectedness of Gk , this latter action preserves each of the simple factors, so q is a G-equivariant quotient (as that can be checked over k). We can therefore form the “non-commutative pushout” of G along the quotient map q as in Remark 1.4.5 to reduce to the case where G is a simple adjoint semisimple k-group.

4.2 Uniqueness of (G , k /k)

159

By Corollary A.4.7(2), the G-action on G is conjugation through a khomomorphism f : G → G . But this action restricts to the inner action of G on itself, so f (g )-conjugation on G coincides with g -conjugation on G for all points g of G . Since ZG = 1, it follows that f |G = id G , so f expresses G as a direct factor of G over k. 2 Using the universal property of Weil restriction, the canonical map πG in Proposition 4.2.1 corresponds to a map of k-groups fG : G → R K/k (G ) =

R ki /k (G i ),

(4.2.2)

where {ki } is the set of factor ﬁelds of K and G i is the ki -ﬁber of G . By construction, the formation of fG commutes with any separable extension on the ground ﬁeld. Deﬁnition 4.2.3 Let G be a non-solvable smooth connected afﬁne group over a ﬁeld k, and consider the initial object (K/k, πG : GK → G ) as in Proposi, K/k, fG ) tion 4.2.1. The simply connected datum associated to G is the triple (G where fG is as in (4.2.2) and G → G is the simply connected central cover of G over K. , K/k, fG ) is compatible The formation of the simply connected datum (G with any separable extension on k. Our main interest in the preceding considerations is due to the fact that we can describe (K/k, πG ), or equivalently (K/k, fG ), when G is a non-commutative (hence non-solvable) standard pseudo-reductive group over a ﬁeld k: Proposition 4.2.4 Let k be a ﬁeld and G a non-commutative standard pseudoreductive k-group arising from a 4-tuple (G , k /k, T , C) such that G → Spec k has absolutely simple and simply connected ﬁbers. The simply connected , K/k, fG ) associated to G is naturally isomorphic to (G , k /k, f ), datum (G where f : G → R k /k (G ) is the displayed map in Remark 1.4.8. Proof. By two applications of Proposition 4.2.2, we are reduced to proving that if G = R k /k (G ) then πG is the natural quotient map q : R k /k (G )k G G . That is, we claim that q is a maximal absolutely simple adjoint quotient of Gk on geometric ﬁbers over Spec k and that the resulting natural map (K/k, πG ) → (k /k, q) is an isomorphism. We can assume k = ks , and then another application of Proposition 4.2.2 (for split exact sequences arising from the direct product decomposition of G according to the factor ﬁelds of k ) reduces us to the case where k is a ﬁeld. In this case q is a k -descent of the maximal adjoint semisimple quotient of

160

Variation of (G , k /k, T , C)

R k /k (G )k , by PropositionA.5.11(2), so (due to how (K/k, πG ) was constructed in the proof of Proposition 4.2.1) we just have to check that k /k is the minimal ﬁeld of deﬁnition over k for ker q. This follows from Theorem 1.6.2(1). 2 The identiﬁcation of fG with f in Proposition 4.2.4 implies that if G is a non-commutative standard pseudo-reductive group then ker fG is central in G (Remark 1.4.8). The image of fG is mysterious in general, but it is always pseudo-reductive. This is a special case of the following result that will be useful in our work in characteristics 2 and 3: Proposition 4.2.5 Let G be a smooth connected afﬁne group over a ﬁeld k, k a nonzero ﬁnite reduced k-algebra, and π : Gk G a quotient over k such that the ﬁbers of G over k are pseudo-reductive. The associated map of k-groups f : G → R k /k (G ) has pseudo-reductive image. The case of most interest to us is when G has reductive ﬁbers over k . Proof. Let H = f (G), so H is a smooth connected afﬁne k-group. Let i : H → R k /k (G ) be the inclusion. We ﬁrst wish to replace (G, f ) with (H , i). By applying functoriality of the universal property of Weil restriction to the quotient map G → H , π factors as ik

q

Gk Hk → R k /k (G )k → G where the ﬁnal map is the natural one as in Proposition A.5.7. Moreover, the uniqueness aspect of the universal property of Weil restriction ensures that i corresponds to q ◦ ik . Since surjectivity of π implies that q ◦ ik is surjective, we can therefore replace (G, π, f ) with (H , q◦ik , i) to reduce to the case ker f = 1. By surjectivity, the map π carries Ru,k (G)k into a smooth connected unipotent normal k -subgroup of G . But G is pseudo-reductive over k , so Ru,k (G)k has trivial image in G . Equivalently, Ru,k (G) is killed by the khomomorphism f : G → R k /k (G ) corresponding to π . Since ker f = 1, we have Ru,k (G) = 1. 2 , K/k, fG ) associated to a nonThe concept of simply connected datum (G solvable smooth connected afﬁne k-group G is closely related to the universal maximal absolutely simple adjoint quotient as in Proposition 4.2.1. In particular, by construction, the factor ﬁelds of K/k are identiﬁed with the ﬁelds of deﬁnition over k for representatives of the Aut(k/k)-orbits of maximal simple adjoint semisimple quotients of Gk . It is essential that we work with the maximal simple adjoint semisimple quotients of Gk rather than with the maximal simple semisimple quotients of Gk , as otherwise the corresponding ﬁelds of deﬁnition

4.2 Uniqueness of (G , k /k)

161

over k for the quotients could increase and so there would be no analogue of Proposition 4.2.4 that reconstructs k /k from G in the non-commutative standard pseudo-reductive case. This is best understood with an example: Example 4.2.6 Let k be a ﬁeld, G be a connected semisimple k-group, and T ⊆ G a maximal k-torus. Let T → C → T /ZG be a factorization of the canonical quotient map through a commutative pseudo-reductive k-group C such that T maps onto the maximal k-torus in C. Assume that C is not a torus. (We will soon exhibit some examples satisfying these conditions.) These hypotheses force ZG = 1. Indeed, if ZG = 1 then the hypotheses yield a direct product decomposition of the k-group C as T ×C , with T the maximal k-torus of C and C a smooth connected commutative k-group. But C is not a torus, so C is nontrivial and unipotent, contrary to the pseudo-reductivity of C over k. Under these assumptions, G := (G C)/T is a standard pseudo-reductive k-group whose maximal adjoint semisimple quotient G /ZG is deﬁned over k (as a quotient of G). The initial object in CG/k is (k/k, G G /ZG ). However, R(Gk ) is the subgroup Ru (Ck ) in Gk whose ﬁeld of deﬁnition K over k cannot equal k (as C is pseudo-reductive over k but not a torus). Thus, the maximal semisimple quotient of Gk has a strictly larger ﬁeld of deﬁnition over k than the maximal adjoint semisimple quotient, and these quotients are both simple if G is absolutely simple over k. (In these examples, R(Gk ) is unipotent but D (G) = G because G has a nontrivial commutative quotient given by the quotient of C modulo the image of T in C. Failure of perfectness of G is a crucial feature of these examples, as we will see in Proposition 5.3.3.) The conditions for this construction can be satisﬁed over any imperfect ﬁeld k of characteristic p > 0, as follows. Take G to be SLp and T to be the split diagonal torus. Let K/k be any purely inseparable extension of degree p and let p−2 p−1 C = R K/k (GL1 ) × GL1 . Consider the k-isomorphism T GL1 deﬁned by the map diag(t1 , . . . , tp−1 , 1/(t1 · · · tp−1 )) → (t1 , t2 /t1 , . . . , tp−1 /tp−2 ). This carries the center ZG = μp over to the canonical inclusion of μp into the p−1 ﬁrst factor of GL1 . Deﬁne the maps T → C and C → T /ZG to each be the identity on the last p−2 factors of T and C, and to respectively be the inclusion GL1 → R K/k (GL1 ) and the norm map R K/k (GL1 ) → GL1 GL1 /μp on the ﬁrst factor. This satisﬁes all of the requirements.

5 Ubiquity of the standard construction

5.1 Main theorem and central extensions Although it is difﬁcult to say much about the general structure of commutative pseudo-reductive groups, the commutative case is essentially the only mystery. This follows from the ubiquity of standard pseudo-reductive groups, as we now make precise over arbitrary ground ﬁelds k, assuming [k : k 2 ] 2 when char(k) = 2. Theorem 5.1.1 Let G be a pseudo-reductive group over a ﬁeld k. (1) The k-group G is standard over k in the sense of Deﬁnition 1.4.4 except possibly when char(k) = p ∈ {2, 3}, k is imperfect, and the Dynkin diagram of the maximal semisimple quotient G ss contains an edge with multiplicity k p or has an isolated vertex when p = 2. (2) Assume that k is imperfect with char(k) = p ∈ {2, 3}, and [k : k 2 ] 2 if p = 2. The pseudo-reductive k-group G is standard if and only if the root system of Gks is reduced and all roots in a common irreducible component of have root spaces in Lie(Gks ) with the same dimension. (3) If Gks has a non-reduced root system (so char(k) = 2) and [k : k 2 ] = 2 then there is a unique decomposition G = G1 × G2 where (G1 )ks has a reduced root system and G2 is totally non-reduced in the sense of Deﬁnition 10.1.1. (4) If char(k) ∈ {2, 3}, with [k : k 2 ] 2 when char(k) = 2, and Gks has a reduced root system then G is generalized standard in the sense of Deﬁnition 10.1.9. Remark 5.1.2 Every connected reductive k-group G over any ﬁeld k is trivially a standard pseudo-reductive k-group by taking k = k and G = G in Deﬁnition 1.4.4. Also, by Proposition 1.2.3, if a pseudo-reductive k-group G is solvable then it is commutative and hence is standard over k. Over a perfect ﬁeld every 162

5.1 Main theorem and central extensions

163

pseudo-reductive group is reductive. Thus, Theorem 5.1.1 is only interesting when k is imperfect and G is non-solvable. We also note that the totally nonreduced factor in part (3) is classiﬁed by (the proof of) Theorem 9.4.3(1) and Proposition 10.1.4. Before we take up the task of proving that standardness always holds, apart from some exceptional situations in characteristics 2 and 3, we wish to prove over an arbitrary ground ﬁeld k that standardness is well behaved with respect to basic operations such as formation of torus centralizers, passage to derived groups or smooth connected normal k-subgroups, and scalar extension to ks . We begin with an important criterion for the splitting of certain central extensions. It is the key to controlling connectedness properties of kernels and the size of Cartan subgroups in the classiﬁcation of pseudo-reductive groups. Proposition 5.1.3 Let k be a ﬁeld and G a nontrivial perfect pseudo-reductive k-group. Let T be a maximal k-torus in G, and assume that a dense open locus in ZG (T )ks can be expressed group-theoretically in terms of root groups: for some ﬁnite sequence of (not necessarily distinct) roots a1 , . . . , an ∈ (Gks , Tks ) there / Uai to the corresponding exist rational ks -scheme maps hi : ZG (T )ks root groups such that their product

hi : ZG (T )ks

/ Gks is the natural

inclusion. For any commutative k-group scheme Z of ﬁnite type that contains no nontrivial smooth connected k-subgroup, every central extension 1→Z →E→G→1

(5.1.1)

over k is uniquely split. In Proposition 5.1.3 we do not require E (or equivalently, Z) to be afﬁne, though we reduce to this case in the proof. The result is only used with afﬁne E, so the reader can restrict to that case. There will be several classes of G for which the hypotheses of Proposition 5.1.3 will be veriﬁed: Weil restrictions of simply connected semisimple groups, and later exotic constructions in characteristics 2 and 3. Before we take up the proof of Proposition 5.1.3, we now verify its assumptions in the ﬁrst of these classes of G. Example 5.1.4 Consider G = R k /k (G ) for a nonzero ﬁnite reduced k-algebra k and a nontrivial semisimple k -group G whose ﬁbers are connected and simply connected. To verify the assumptions in Proposition 5.1.3 for this nontrivial G, we can assume k = ks and that k is a ﬁeld.

164

Ubiquity of the standard construction

By Corollary A.7.11, G is perfect. There is a unique maximal k -torus T in G such that T is the maximal k-torus in R k /k (T ) (Proposition A.5.15(2)). Explicitly, the natural map Gk → G is a k -descent of the geometric maximal reductive quotient (Proposition A.5.11(2)) and this carries Tk isomorphically onto T , thereby identifying X(T ) and X(T ). Hence, it follows from the natural identiﬁcation of Lie(G) with the Lie algebra over k underlying Lie(G ) (CorollaryA.7.6) that the root data R(G, T ) and R(G , T ) naturally coincide. Likewise, by the unique characterization of root groups in Proposition 2.3.11 we have Ua = R k /k (Ua ) for corresponding roots a ∈ (G, T ) and a ∈ (G , T ). Since ZG (T ) is equal to R k /k (ZG (T )) = R k /k (T ) (Proposition A.5.15(1)), / G over k it sufﬁces to construct the required rational maps hi : ZG (T ) to do so for (G , T ) over k and then apply Weil restriction. Hence, we may and do assume k = k and G = G. Since G is simply connected, if we choose a positive system of roots in (G, T ) then the coroots associated to the simple positive roots are a basis for the cocharacter group of T , and for each positive root a the root groups Ua and U−a generate a copy of SL2 in G with a∨ (GL1 ) as a maximal k-torus. Thus, it sufﬁces to treat the case G = SL2 with T equal to the diagonal k-torus D. In this case, the standard isomorphisms x±a : Ga U±a and the isomorphism a∨ : GL1 D deﬁned by t → diag(t, 1/t) satisfy a∨ (t) = xa (t)x−a (−1/t)xa (t − 1)x−a (1)xa (−1), from which we see the evident choices for h1 , . . . , h5 as rational maps D o

a∨

GL1

/ Uai .

Remark 5.1.5 The hypothesis in Proposition 5.1.3 that Z contains no nontrivial smooth connected k-subgroup appears to be necessary. For example, suppose that k has characteristic p > 0, K/k is an extension with [K : k] = p2 , and K p ⊆ k (which forces K/k to be non-primitive); such a K can be found when k is ﬁnitely generated with transcendence degree > 1 over Fp . A construction using a nonzero k-linear map φ : 2K/k → k appears to give rise to a non-split central extension of R K/k (SLn ) by Ga for any n > 2. (To make this extension, argue as follows. The universal central extension of SLn (K) = R K/k (SLn )(k) is by K2 (K) since n > 2. Form the pushout by the map K2 (K) → k deﬁned by {f , g} → φ((df /f )∧(dg/g)). This is classiﬁed by a 2-cocycle given generically over R K/k (SLn )2 by a rational function. Such a pushout seems likely to give rise to a non-split central extension of R K/k (SLn ) by Ga as k-groups.)

5.1 Main theorem and central extensions

165

The imprimitivity of K/k in such examples is essential: it can be proved that if k is a ﬁeld of arbitrary characteristic and K is a nonzero primitively generated (with ﬁnite k-algebra then for any simply connected semisimple K-group G by a k-group scheme of connected ﬁbers), any central extension of R K/k (G) ﬁnite type is split. Proof of Proposition 5.1.3 Since G = D (G) by hypothesis, the splitting of (5.1.1) is unique if it exists. Hence, by Galois descent it sufﬁces to treat the case where k is separably closed. (The scalar extension from k to ks does not ruin the property of Z not containing a nontrivial smooth connected subgroup over the ground ﬁeld.) The commutator map of k-schemes cE : E ×E → E uniquely factors through a k-scheme map cE : G × G (E × E)/(Z × Z) → E lifting the commutator map of E/Z = G since (5.1.1) is a central extension, and clearly cE (g1 , g2 ) ∈ E(k) lifts g1 g2 g1−1 g2−1 for gi ∈ G(k). Since G = D (G) and k = ks , the commutator subgroup of G(k) is Zariski-dense in G. We just showed that all commutators in G(k) lift to E(k), so it follows that E(k) has image in G(k) that is Zariski-dense in G. Hence, the Zariski closure H ⊆ E of E(k) maps onto G, and so does the identity component H 0 (as G is connected). Our goal is to ﬁnd a section G → E as k-groups, so we can replace E with the k-subgroup H 0 and replace Z with Z ∩ H 0 to reduce to the case where E is smooth and connected. Obviously D (E) also maps onto G, so we can replace E with D (E). This reduces us to the case where E is afﬁne (or equivalently, Z is afﬁne), since the derived group of a smooth connected group over a ﬁeld is always afﬁne. (Indeed, it sufﬁces to prove such afﬁneness over the perfect closure, and by Theorem A.3.7 every smooth connected group over a perfect ﬁeld is an extension of an abelian variety by a smooth connected afﬁne group. Thus, the derived group is afﬁne.) Consider the surjective map π : E G between smooth connected afﬁne k-groups. If it admits a k-rational section over some dense open subset ⊆ G then π −1 () ×Z as k-schemes. But π −1 () is smooth and connected since E is smooth and connected, so Z would have to be smooth and connected. Since Z has no nontrivial smooth connected k-subgroup, this would force Z = 1, so π would be an isomorphism and hence we would have the required splitting over k.

166

Ubiquity of the standard construction

We have reduced the proof of Proposition 5.1.3 to showing that any central extension π

1→Z →E→G→1

(5.1.2)

of k-group schemes with E a smooth connected afﬁne k-group admits a krational section over a dense open in G. We will construct such a section with E an arbitrary afﬁne k-group scheme of ﬁnite type. Let T ⊆ G be a maximal k-torus, and = (G, T ). Fix a positive system of roots + ⊆ , and let U + denote the k-unipotent radical of the minimal pseudo-parabolic k-subgroup P ⊆ G corresponding to + , so U + is directly spanned (in any order) by the root groups associated to the non-divisible positive roots. Likewise deﬁne U − using − := −+ , so the direct product scheme := U − × ZG (T ) × U + is a dense open subscheme of G via multiplication (Corollary 3.3.16). We will construct a section to π over a dense open locus in . By hypothesis, there is a dense open subscheme V ⊆ ZG (T ) and a ﬁnite set of k-morphisms hi : V → Uai for a sequence of (not necessarily distinct) roots a1 , . . . , an such that v = hi (v) for all v ∈ V . We can take the ai to be non-divisible, since Ua ⊆ U(a) = Ua/2 for each divisible root a. Thus, if we can ﬁnd a section σa : Ua → π −1 (Ua ) to π over the entirety of each Ua (a ∈ ) then (σai ◦ hi ) : V → E is a section to π over the dense open V in ZG (T ). Since U ± is directly spanned by the Ua for non-divisible a ∈ ± , by suitably multiplying all of these sections using the group law on E we get a section to π over the dense open U − · V · U + in G, as desired. It remains to split the central extension 1 → Z → Ea = π −1 (Ua ) → Ua → 1 obtained by restricting the given central extension to Ua for each a ∈ . The centrality of Z in E implies that the G-action on G via conjugation lifts to an action of G = E/Z on E that is trivial on Z. In particular, Ea is a T -equivariant central extension of Ua , and by Proposition 2.3.11 each Ua is a vector group admitting a linear structure with respect to which the weights of T are positive rational multiples of a. The following lemma therefore ﬁnishes the proof of Proposition 5.1.3. 2 Lemma 5.1.6 Let k be a ﬁeld, C a commutative afﬁne k-group scheme of ﬁnite type, and V a k-vector group equipped with an action by a k-split k-torus S such that the weights of S on Lie(V ) are all nontrivial and any two such linearly dependent weights are Q× >0 -multiples of each other in X(S)Q . Every

5.1 Main theorem and central extensions

167

S-equivariant central extension E of V by C (giving C the trivial S-action) admits a unique S-equivariant splitting over k. In the intended applications, C is a group scheme that cannot be assumed to be smooth. Proof. We may assume V = 0. By using uniqueness of the linear structure on V if char(k) = 0 and using Theorem B.4.3 if char(k) > 0, we may and do choose a linear structure on V with respect to which the S-action is linear, so the weights occurring in V match the ones in Lie(V ). For any k-rational character χ ∈ X(S) let Vχ denote the χ -weight space for V with its S-action and let Ga (χ ) denote Ga equipped with an S-action through χ . We have V Vχi for some nontrivial and pairwise distinct χi ∈ X(S). For a nontrivial χ ∈ X(S), the S-action on Ga (χ ) is transitive on the complement of the origin. Thus, Ga (χ ) has no nonzero S-equivariant quotient with a trivial S-action, so the group HomS (V , C) of S-equivariant k-homomorphisms vanishes. An S-equivariant central extension E of V by C over k therefore admits at most one S-equivariant splitting over k. To construct a splitting, by Galois descent we may and do now assume k = ks . In particular, k is inﬁnite. As a ﬁrst step, we show that E is commutative as a k-group. Since E is an S-equivariant central extension, the commutator map E × E → E factors uniquely through an S-equivariant biadditive alternating map V × V → C. Thus, by biadditivity it sufﬁces to show that for any pair of (possibly equal) weights χ and χ for S acting on V , the induced S-equivariant map f : Vχ × Vχ → C of k-schemes is the constant map to 0 ∈ C(k). (Note that C is not necessarily smooth.) The ﬁber f −1 (0) is a closed subscheme of Vχ × Vχ , so to prove f = 0 it sufﬁces (by k-smoothness of the weight spaces, and the condition k = ks ) to check that f vanishes on k-points. The S-equivariance implies that for any (v, v ) ∈ Vχ (k) × Vχ (k), the restriction of f to the Zariski closure of the Sorbit S.(v, v ) is equal to the constant value f (v, v ) ∈ C(k). Hence, to show f (v, v ) = 0 it sufﬁces to ﬁnd a k-point in the Zariski closure of S.(v, v ) at which f vanishes. If χ and χ are independent characters then there is a 1-parameter subgroup a : GL1 → S over k such that the composition χ ◦ a is trivial and χ ◦ a ∈ Z is equal to some n 1. Thus, for t ∈ k × we have a(t).(v, v ) = (t n v, v ). It follows from the inﬁnitude of k that (0, v ) lies in the Zariski closure of S.(v, v ), so f (v, v ) = f (0, v ) = 0. If χ and χ are dependent characters then m n (by hypothesis) they are Q× >0 -multiples of each other in X(S)Q , so χ = χ for some m, n 1. Thus, a variation on the argument for independent characters

168

Ubiquity of the standard construction

gives that the Zariski closure of S.(v, v ) contains (0, 0), so f (v, v ) = f (0, 0) = 0. This completes the proof that E is commutative. Since k is inﬁnite, there exists s0 ∈ S(k) such that χ (s0 ) ∈ k × − {1} for all weights χ of S occurring on V . Consider the k-homomorphism E → E deﬁned by x → s0 .x − x. This kills the central k-subgroup scheme C, so it factors through an S-equivariant k-homomorphism h : V → E. The S-equivariant h

composite mapping V → E E/C = V is the map v → s0 .v − v which is multiplication by χ (s0 ) − 1 ∈ k × on Vχ for each weight χ of S on V . This is an automorphism of V , so h(V ) is an S-stable k-subgroup of E mapping isomorphically onto E/C = V . In other words, h(V ) provides an S-equivariant splitting of the given central extension structure on E. 2 As an application of Proposition 5.1.3 (and Example 5.1.4), we have: Proposition 5.1.7 Let k be a ﬁeld, k a nonzero ﬁnite reduced k-algebra, and G a semisimple k -group whose ﬁbers are connected, absolutely simple, and simply connected. Let Z ⊆ R k /k (ZG ) = ZRk /k (G ) be a central k-subgroup scheme, and G := R k /k (G )/Z the smooth connected afﬁne quotient over k. (1) For any separable ﬁeld extension F/k, every F-automorphism f of GF uniquely lifts to an endomorphism f of R k /k (G )F , and f is an automorphism. Moreover, f restricts to the identity on a smooth connected F-subgroup of R k /k (G )F if f restricts to the identity on its image in GF . (2) If H is a smooth connected k-group equipped with a left action on G then this uniquely lifts to a left action on R k /k (G ). This lifted action restricts to the identity on a smooth connected k-subgroup of R k /k (G ) if H acts trivially on its image in G. An especially useful class of smooth connected subgroups in (1) is the maximal F-tori. Proof. We ﬁrst reduce (2) to a special case of (1), as follows. Let G = R k /k (G ), and consider a smooth connected k-group H equipped with a left action on G. The action map H ×G →H ×G deﬁned by (h, g) → (h, h.g) is an H -group automorphism, so it pulls back over the generic point η : Spec F → H to an F-group automorphism of GF . Since F = k(H ) is separable over k, by (1) this automorphism uniquely lifts to an F-group automorphism αF of GF .

5.1 Main theorem and central extensions

169

For a suitable dense open subscheme U ⊆ H , the lifted F-group automorphism αF of GF extends to a U -group automorphism of U × G . We claim that the resulting dominant composite map pr 2

α :U ×G U ×G →G uniquely extends to an action of H on G , which would settle (2) conditional on (1). The uniqueness is clear, so we need to verify existence, for which we may assume k = ks . For each u ∈ U (k), restricting α to the u-slice deﬁnes a k-group automorphism θu of G . Deﬁne a map Uu × G → G by (u u, g) → α(u , θu (g)). The associated Uu-map Uu × G → Uu × G has F-ﬁber equal to an F-group endomorphism of GF lifting the generic ﬁber of the action of H on G. Thus, the uniqueness aspect of (1) implies that these maps Uu × G → G agree on overlaps (as that can be checked on GF ) and so glue to a k-map H × G → G . The associated H -map H × G → H × G is an H -group map (since it is an F-group map over the generic point of H ). Specializing at 1 ∈ H gives an endomorphism of G covering the identity of G, so it must be the identity (by (1)), and likewise for any h, h ∈ H (k) the endomorphism of G induced by hh is the composition of the endomorphisms induced by h and h separately (as this holds with G in place of G ). It follows that H acts on G in an associative manner, so its action is through automorphisms, thereby ﬁnishing the proof of (2). Now we prove (1). By Galois descent and uniqueness we may assume F = Fs , and since F/k is separable we can extend scalars to F to reduce to the case F = k. To prove the uniqueness of the lifted homomorphism it sufﬁces to observe that a map of pointed k-schemes R k /k (G ) → R k /k (ZG ) is trivial since it corresponds to a map of pointed k -schemes R k /k (G )k → ZG in which the source has smooth connected ﬁbers over Spec k and the target has ﬁbers over Spec k that are ﬁnite. The same argument handles triviality of the action on smooth connected k-subgroups. It remains to show that for the central quotient map π : R k /k (G ) R k /k (G )/Z = G, any k-automorphism f of G lifts through π to a k-automorphism f of R k /k (G ). Consider the pullback diagram 1

1

/ Z

/ Z

/ E

/ R k /k (G )

/ G

/ R k /k (G )

π

/ 1

f ◦π

(5.1.3) / 1

170

Ubiquity of the standard construction

The top line is a central extension of R k /k (G ) by the k-group Z. We claim that Z contains no nontrivial smooth connected k-subgroups. It sufﬁces to prove this claim for the ambient central k-subgroup R k /k (ZG ), and so by the universal property of Weil restriction it sufﬁces to prove the analogous fact for each ﬁber of ZG over Spec k . These ﬁbers are ﬁnite, so the claim is clear. Since Z contains no nontrivial smooth connected k-subgroups, any central extension of G = R k /k (G ) by Z over k (such as the top row of (5.1.3)) is uniquely split over k (Example 5.1.4). The unique splitting of the top row of (5.1.3) provides a canonical copy of G as a k-subgroup of E, and its map down to the middle term along the bottom row of (5.1.3) is an endomorphism f of G that carries Z into itself and induces the automorphism f modulo Z. It remains only to prove that f is an automorphism. But we can apply the same lifting result to f −1 , and this provides an endomorphism f preserving Z such that the composites f ◦ f and f ◦ f preserve Z and reduce to the identity on G /Z. Hence, uniqueness forces f to be an inverse to f. 2

5.2 Properties of standardness and standard presentations In order to prove standardness for a pseudo-reductive group, it is convenient to record how standard presentations (when they exist!) behave with respect to operations on groups. We begin with passage to the derived group. Proposition 5.2.1 A non-commutative pseudo-reductive k-group G is standard if and only if D (G) is standard. If either admits a standard presentation resting on the pair (G , k /k) then so does the other. Proof. Choose a maximal k-torus T in G, and let S = T ∩ D (G) be the corresponding maximal k-torus in D (G). We have G = ZG (T ) · D (G), by Proposition 1.2.6. By Proposition 4.1.4, we may and do restrict our attention to standard presentations that are compatible with T and S. First assume that G is standard, so it arises from a 4-tuple (G , k /k, T , C) compatible with T . In particular, j : R k /k (G ) → G has image D (G) and carries R k /k (T ) into C = ZG (T ). Hence, j(R k /k (T )) ⊆ D (G) ∩ C ⊆ ZD (G) (S). Thus, we can form the standard pseudo-reductive k-subgroup (R k /k (G ) ZD (G) (S))/R k /k (ZG (T ))

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of the standard pseudo-reductive k-group (R k /k (G ) ZG (T ))/R k /k (ZG (T )) = G. This subgroup of G is clearly contained in D (G), yet it contains j(R k /k (G )) = D (G). Thus, we have constructed a standard presentation of D (G), and it rests on the same pair (G , k /k). Conversely, suppose that D (G) is standard, and choose a 4-tuple (G , k /k, T , C ) giving rise to a standard presentation of D (G) that is compatible with S. Hence, C = ZD (G) (S) and there is a factorization diagram φ

ψ

R k /k (ZG (T )) → C → R k /k (T /ZG ) and an isomorphism δ : (R k /k (G ) C1 )/R k /k (ZG (S )) D (G) with the maximal k-torus T in R k /k (T ) mapping into the maximal k-torus S in D (G). Since R k /k (G ) is carried onto D 2 (G) = D (G), it carries T onto S (even by an isogeny). The kernel ker j = ker φ is central (as for any standard presentation). Observe that j(R k /k (T )) ⊆ ZD (G) (S) ⊆ ZG (T ) =: C, so to make a standard presentation of G (compatible with T , and resting on (G , k /k)) we are led to try to lift the natural conjugation action of C on the central quotient D (G) of R k /k (G ) to an action on R k /k (G ) that restricts to the identity on the maximal k-torus T . Such a lift is provided by Proposition 5.1.7, and it is even unique. Using this action and the universal property of R k /k (T /ZG ) from Theorem 1.3.9, we get a factorization diagram R k /k (T ) → C → R k /k (T /ZG ) and hence a standard pseudo-reductive k-group (R k /k (G ) C)/R k /k (T ). There is an evident natural k-homomorphism γ : (R k /k (G ) C)/R k /k (T ) → G

(5.2.1)

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Ubiquity of the standard construction

that restricts to the chosen standard presentation δ of D (G). In particular, γ is surjective since G = ZG (T ) · D (G). To prove that γ is an isomorphism, consider a point (g, z) of R k /k (G ) C (valued in some k-algebra) such that it maps to 1 in G. We have to show that this point lies in the central k-subgroup R k /k (T ). Since g maps into D (G), it follows that z is a point of C ∩ D (G) ⊆ ZD (G) (S) = C . Hence, the triviality of ker δ does the job. 2 A consequence of Proposition 5.2.1 is a description of all isomorphisms between standard pseudo-reductive k-groups. This will enable us to reduce the veriﬁcation of standardness to a problem over separably closed ﬁelds. Proposition 5.2.2 Let (G , k /k, T , C1 ) and (G , k /k, T , C2 ) be two 4-tuples that give standard presentations of a non-commutative pseudo-reductive kgroup G, and choose both to be compatible with a ﬁxed maximal k-torus T in G. Under the resulting identiﬁcations of C1 and C2 with C := ZG (T ), the composite isomorphism (R k /k (G ) C)/R k /k (T ) G (R k /k (G ) C)/R k /k (T ) is induced by a unique pair consisting of a k-algebra isomorphism α : k k and an isomorphism of pairs (G , T ) (G , T ) over Spec α respecting the given factorization diagram (5.2.1) and its counterpart for (G , k /k, T , C). In particular, a standard presentation of G compatible with a speciﬁed maximal k-torus of G is unique up to unique isomorphism. Proof. The uniqueness reduces the problem to the case k = ks by Galois descent. The images of R k /k (G ) and R k /k (G ) in G coincide with D (G), and give rise to standard presentations of D (G) compatible with the maximal k-torus S = T ∩ D (G) (Proposition 5.2.1). Thus, we can replace G with D (G) to reduce to the case where G is perfect. Letting Z1 = ker φ1 and Z2 = ker φ2 be the central kernels arising from the two standard presentations of G, the method of proof of Proposition 5.1.7(1) works without change to show that the k-isomorphism f : R k /k (G )/Z1 G R k /k (G )/Z2 uniquely lifts to a k-isomorphism f : R k /k (G ) R k /k (G ) carrying Z1 into Z2 (via an isomorphism, by working with f −1 and its unique lift f −1 ). The bijective correspondence between sets of maximal k -tori of G , maximal k-tori of G, and maximal k -tori of G implies that f carries the maximal k-torus of

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R k /k (T ) over to the maximal k-torus of R k /k (T ). Passing to centralizers of these maximal k-tori, the Cartan k-subgroup R k /k (T ) is carried over to the Cartan k-subgroup R k /k (T ). It remains to prove that any k-group isomorphism f : R k /k (G ) R k /k (G ) arises from a unique k-algebra isomorphism α : k k and a group isomorphism φ : G G over Spec α, and that if f carries R k /k (C ) over to R k /k (C ) for a Cartan k -subgroup C ⊆ G and a Cartan k -subgroup C ⊆ G then φ(C ) = C (so φ carries the maximal k -torus of C over to the maximal k -torus of C ). The compatibility with Cartans is automatic, due to the behavior of Weil restriction with respect to the formation of Cartan subgroups (Proposition A.5.15(3)), so the problem is to prove the existence and uniqueness of (α, φ). First consider the case when k and k are both ﬁelds. The simply connected datum associated to R k /k (G ) has the form (G , k /k, π ), and similarly for R k /k (G ), so functoriality of the simply connected datum with respect to isomorphisms implies that k = k (as purely inseparable extensions of k) and that there is an isomorphism f : G G over k = k which lifts the isomorphism induced by fk on maximal adjoint quotients over k . But then f coincides with the map induced by fk on maximal reductive quotients over k , so f = R k /k ( f ) by Proposition 1.2.2. To reduce the general case to this settled case of ﬁelds, let {ki } and {kj } be the respective factor ﬁelds of k and k , and let Gi and Gj respectively denote the ki -ﬁber of G and the kj -ﬁber of G . We claim that f arises from a collection ) for a bijection i → j(i) between of k-isomorphisms R ki /k (Gi ) R k /k (Gj(i) j(i) the index sets, so applying the settled ﬁeld case to these isomorphisms would then complete the proof in general. To get this structure for f , it sufﬁces to prove an intrinsic characterization of the collection of nontrivial smooth connected k-subgroups Gi := R ki /k (Gi ) in R k /k (G ): they are the smooth connected normal k-subgroups that are pseudo-simple over k (see Deﬁnition 3.1.1). This consists of two parts, namely that the Gi are themselves pseudo-simple, and that there are no others. To prove the pseudo-simplicity of the Gi , ﬁrst note that each Gi is perfect, by Corollary A.7.11. By pseudo-reductivity of Gi and Lemma 3.1.2, it sufﬁces to prove that (Gi )ss is simple. Since ki /k is purely inseparable, the natural quok tient map qi : (Gi )ki Gi has smooth connected unipotent kernel (Proposition

A.5.11), so (Gi )red = Gi ⊗ki k is simple. k Finally, consider a nontrivial smooth connected normal k-subgroup N in R k /k (G ). We seek some Gi inside N . Since N ⊆ Gi , its smooth connected normal image in some Gi0 is nontrivial. This image is full by pseudo-simplicity

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Ubiquity of the standard construction

of Gi0 , so (N , Gi0 ) also maps onto Gi0 since Gi0 is perfect. But (N , Gi0 ) ⊆ Gi0 , so (N , Gi0 ) = Gi0 . Normality of N implies that (N , Gi0 ) ⊆ N , so Gi0 ⊆ N . 2 Corollary 5.2.3 Let G be a non-commutative pseudo-reductive k-group. If Gks is standard over ks , then so is G over k. In Corollary 5.3.9 we will generalize this to allow any separable extension of ﬁelds in place of ks /k. An interesting case is the extension kv /k for a global function ﬁeld k and a place v of k. Proof. We may and do assume that G is non-commutative, and choose a maximal k-torus T of G. By Proposition 4.1.4, we can consider the unique standard presentation of Gks compatible with Tks . Let (G , k /k) arise from the simply connected datum associated to G. For ks := ks ⊗k k , the standard presentation of Gks compatible with Tks must rest on the pair (Gk , ks /ks ) (due to the coms patibility of the simply connected datum with respect to separable extension of the ground ﬁeld). Uniqueness of the standard presentation compatible with Tks allows us to apply Galois descent to the data in this presentation to make a k-homomorphism j : R k /k (G ) → G that descends the corresponding map in the standard presentation for Gks over ks . By compatibility of the presentation with Tks over ks , this map carries R k /k (T ) into C := ZG (T ). Moreover, the map Cks → R k /k (T /ZG )ks is Galois-equivariant and so descends, thereby deﬁning a standard k-group (R k /k (G ) C)/R k /k (T /ZG ) equipped with an evident k-homomorphism γ to G. But over ks this recovers the standard presentation for Gks compatible with Tks , so γks is an isomorphism and hence γ is an isomorphism. 2 Now we can establish good behavior of standardness relative to torus centralizers and normal subgroups. We ﬁrst take up the case of torus centralizers. Let G be a standard pseudo-reductive group over a ﬁeld k. For any k-torus S in G, the centralizer ZG (S) is a pseudo-reductive k-group, by Proposition 1.2.4. Choose a maximal k-torus T in G containing S. Suppose that G is noncommutative, and let (G , k /k, T , C) be the standard presentation compatible with T . In particular, C = ZG (T ) ⊆ ZG (S). The action of C on R k /k (G ) centralizes R k /k (T ), so it also centralizes R k /k (S ) for any k -torus S in T . Since R k /k (ZG (S )) is the centralizer of R k /k (S ) in R k /k (G ) (Proposition A.5.15), it follows that C normalizes R k /k (ZG (S )) for any k-torus S in T . Thus, the semidirect product R k /k (ZG (S )) C makes sense.

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175

Proposition 5.2.4 For a standard pseudo-reductive k-group G and a k-torus S in G, the centralizer ZG (S) is a standard pseudo-reductive k-group. More speciﬁcally, if G is non-commutative then with notation as above, there is a canonical k -torus S ⊆ T such that (R k /k (ZG (S )) C)/R k /k (T ) = ZG (S)

(5.2.2)

inside (R k /k (G ) C)/R k /k (T ) = G. The formation of S respects separable extension on k. The k -torus S is not uniquely determined, but our construction of it will be canonical. We have little control over ZG (S ) beyond that it has connected reductive ﬁbers, so this result illustrates the usefulness of the generality of the standard construction in the non-commutative case (not requiring G to have absolutely simple and simply connected ﬁbers). Proof. If G is commutative then ZG (S) = G is clearly standard, so we may and do assume that G is non-commutative. For any k -torus S in T , the ﬁbers of ZG (S ) over Spec(k ) are connected reductive but generally are not semisimple, let alone absolutely simple and simply connected.

Step 1. Let G = G /ZG , and consider the map of k -groups Gk G which corresponds to the natural k-map f : G → R k /k (G ) (i.e., the map f in Remark 1.4.8) via the universal property of Weil restriction. This k -group map is surjective, so since the formation of torus centralizers is compatible with the formation of images under maps between smooth connected afﬁne groups over a ﬁeld (Proposition A.2.8), by working on ﬁbers over the factor ﬁelds of k it follows that the image M of ZG (S)k in G is the centralizer of the image of Sk in G . In particular, M is a reductive k -group with connected ﬁbers since G is a semisimple k -group with connected ﬁbers. Let S be the maximal central torus in M (formed ﬁberwise over Spec(k )). The image of Sk in G is central in M and so is contained in S . Since the image of Tk in G contains the maximal torus T = T /ZG , it is equal to T . Hence, Ck = ZG (T )k has image T in G as well. In particular, T is a maximal torus of M . Under a central isogeny H1 → H2 between connected semisimple groups over a ﬁeld, consideration of maximal tori shows that every (not necessarily maximal) torus T2 in H2 is the image of a unique torus T1 in H1 (so T1 is the identity component of the underlying reduced scheme of the preimage of T2 in H1 ). Applying this to the central isogeny G G ﬁberwise over Spec(k ), there is a unique k -torus S in G mapping onto S in G . We shall prove that

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Ubiquity of the standard construction

S satisﬁes (5.2.2). Observe that, by construction, the formation of the k -torus S in G commutes with separable extension on k. In particular, to prove that (5.2.2) holds we may and do now assume that k is separably closed. Step 2. First we check that the inclusion “⊆” holds in (5.2.2). To see that j : R k /k (G ) → G carries R k /k (ZG (S )) into ZG (S) it sufﬁces to check on k-points since k is separably closed. That is, we must prove that j carries the subgroup ZG (k ) (S ) ⊆ G (k ) = R k /k (G )(k) into (ZG (S))(k) = ZG(k) (S). In other words, if a point g ∈ G (k ) centralizes S then we want g := j(g ) ∈ G(k) to centralize S. Since S (k ) = R k /k (S )(k) contains the image of S (k ) = R k /k (S )(k) under the natural map R k /k (T ) → R k /k (T ), the point g = f (g) ∈ G (k ) centralizes S (k ). Thus, if {ki } is the set of factor ﬁelds of k and G i and S i denote the ki -ﬁbers of G and S respectively, then when g is viewed as a k-point of (G i ⊗ki k) it centralizes (S i ⊗ki k). Since k is separably closed, the ﬁnite extensions ki /k are all purely inseparable. By construction of f , the natural map f

Gk −→ R k /k (G )k k

(G i ⊗ki k)

is the projection to the maximal adjoint semisimple quotient. Also, as we have noted already (using the deﬁnition of S ), the image of Sk in the ith factor of this quotient is contained in S i ⊗ki k. Hence, for the point g ∈ G(k) and the k-torus S in G, the image of g as a k-point of the maximal adjoint semisimple quotient of Gk centralizes the image of Sk in this quotient. Since the maximal semisimple quotient G ss maps via a central isogeny onto the maximal adjoint k semisimple quotient of Gk , it follows that the image of g in G ss centralizes the k image of Sk in G ss . By the following lemma, g ∈ ZG (S)(k) as desired (thereby k completing the proof of the containment “⊆” in (5.2.2)). Lemma 5.2.5 Let H be a pseudo-reductive group over a ﬁeld F, and T an F-torus in H . A point h ∈ H (F) centralizes T if and only if its image in HFss (F) centralizes the image of TF in HFss . Proof. The “only if” direction is trivial, so assume that the image of h centralizes the image of TF in HFss . We ﬁrst check that h centralizes TF viewed as a torus in the maximal reductive quotient HFred . If Z denotes the maximal central torus in HFred then it is equivalent to check that the image of h in HFred centralizes the torus Z · TF in HFred . But any torus in HFred that contains Z is uniquely an almost direct product of Z and a torus in D (HFred ). Writing Z · TF as an almost direct product Z · S with a torus S in D (HFred ), we need to prove that the image of

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177

h centralizes S. Since D (HFred ) → HFss is an isogeny, it is equivalent for h to centralize the image of S in HFss , and this image of S is equal to the image of TF that is assumed to be centralized by the image of h. Since we have proved that h centralizes TF viewed as a torus in HFred , it remains to show that a point h ∈ H (F) that centralizes TF in HFred also centralizes T in H . Consider the natural map T → H deﬁned by t → (t, h) = tht −1 h−1 . The closure X of its image has F-ﬁber that is reduced and irreducible and contained in Ru (HF ), so by pseudo-reductivity of H over F we can apply Lemma 1.2.1 to deduce that X = {1}. In other words, h centralizes T in H . 2 Step 3. It remains to prove that the inclusion “⊆” in (5.2.2) is an equality, and for this it sufﬁces to check the reverse inclusion on k-points. Since k = ks we have ZG (S)(k) ⊆ G(k) = (G (k ) C(k))/T (k ), so a point g ∈ ZG (S)(k) is represented by a pair (g , c) ∈ G (k ) C(k). The problem is to show that g ∈ ZG (k ) (S ). Recall from Step 1 that under the quotient map Gk G , ZG (S)k maps onto the connected reductive k -subgroup M in which S is (by deﬁnition) the maximal central torus, and that Ck maps onto the maximal k -torus T of M . In particular, f ((1, c) mod T ) centralizes S , so g := f ( j(g )) ∈ G (k ) centralizes the k -torus S in G . Thus, under the isogeny G → G between semisimple k -groups, the point g ∈ G (k ) maps into ZG (S )(k ). Since S is

(by deﬁnition) the unique k -torus in G mapping onto S , and the map S → S is an isogeny, it follows that g ∈ ZG (S )(k ). This proves the equality (5.2.2). 2

The next result records the preservation of standardness under passage to normal subgroups: Proposition 5.2.6 Any smooth connected normal k-subgroup N of a standard pseudo-reductive k-group G is a standard pseudo-reductive k-group. Proof. Since Ru,k (N ) is normal in the pseudo-reductive k-group G, it is trivial. Thus, N is pseudo-reductive over k. To prove N is standard, by Corollary 5.2.3 we may assume that k is separably closed. The case of commutative G is trivial, so we may also assume that G is non-commutative. Let S be a maximal k-torus of N , and let T be a maximal k-torus of G containing S. Let C = ZG (T ), so by Theorem 4.1.4 (also see Remark 4.1.5) we have an associated standard presentation (R k /k (G ) C)/R k /k (T ) = G.

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Let {ki }i∈I be the set of factor ﬁelds of k . For each i the ﬁnite extension ki /k is purely inseparable and the quotient map Gki G i carries Nki onto a smooth

connected normal ki -subgroup of G i . The connected semisimple ki -group G i is simple (even absolutely simple), so Nki has either trivial or full image in G i .

Suppose for some i0 that Nki has trivial image in G i0 . Since 0

N (k) ⊆ G(k) = (G (k ) C(k))/T (k ) = (( Gi (ki )) C(k))/( Ti (ki )),

any n ∈ N (k) is represented by a point ((gi ), c) whose image in G i0 (ki0 ) is

trivial. But C(k) is carried into T i0 (ki0 ), so gi0 is also carried into this subgroup.

For each i, the map Gi → G i is a central isogeny between connected semisim ple ki -groups, so Ti is the preimage of T i . In particular, gi0 ∈ Ti0 (ki0 ). We can therefore modify our choice of representative for n ∈ N (k) so that gi0 = 1 and all other gi remain unchanged. Let J be the set of i for which Nki maps onto G i , deﬁne kJ := j∈J kj to be the corresponding factor ring of k , and deﬁne GJ := j∈J Gj and TJ = j∈J Tj to be the respective kJ -ﬁbers of the k -groups G and T . The preceding argument

shows that N is contained in the standard pseudo-reductive k-subgroup GJ := (R kJ /k (GJ ) C)/R kJ /k (TJ ) ⊆ (R k /k (G ) C)/R k /k (T ) = G. Observe that J labels the simple factors of the maximal semisimple quotient of (GJ )k . If J is empty then N is contained in C, in which case N is commutative and therefore standard. By induction on the number of simple factors of G ss , k we may reduce to the case where J is non-empty and is equal to the set of all i, which is to say that Nki maps onto G i for all i. Consider the preimage H of N in R k /k (G ) C. This is smooth and connected since it ﬁts into the following exact sequence: 1 → R k /k (T ) → H → N → 1, and it is normal in R k /k (G ) C. Provided that H contains R k /k (G ) we will have that N = (R k /k (G ) CN )/R k /k (T ) with CN denoting the image of H in C (so CN contains the image of R k /k (T ) in C, and it is pseudo-reductive since it is a smooth connected k-subgroup of

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179

the commutative pseudo-reductive k-group C). Such an expression for N is a standard presentation, so we would be done. It remains to prove that a smooth connected normal k-subgroup H in R k /k (G ) C such that Hki maps onto G i for all i must contain R k /k (G ). Equivalently, we claim that H contains R ki /k (Gi ) for each i. For i ∈ I , consider the smooth connected commutator k-group Hi = (H , R ki /k (Gi )). This is contained in both H and R ki /k (Gi ) due to the normality of both the subgroups,

it is a normal k-subgroup of R ki /k (Gi ), and (Hi )ki maps onto D (G i ) = G i . In particular, Hi = 1 for all i. But the k-group R ki /k (Gi ) has no nontrivial smooth connected normal proper k-subgroups (as we showed at the end of the proof of Proposition 5.2.2). This proves that R ki /k (Gi ) = Hi ⊆ H for all i, as required. 2

5.3 A standardness criterion The main result in this section is a criterion for standardness of absolutely pseudo-simple groups. Before discussing it, we reduce the veriﬁcation of standardness to the (absolutely) pseudo-simple case provided that the ground ﬁeld is separably closed: Proposition 5.3.1 Let G be a non-commutative pseudo-reductive group over k = ks , and {Gi } the collection of pseudo-simple normal k-subgroups of G. If each Gi is standard then so is G. Proof. By Proposition 5.2.1 we may replace G with D (G), so G is perfect. Let T be a maximal k-torus of G, and let Ti := T ∩Gi be the corresponding maximal k-torus of Gi (CorollaryA.2.7). By Proposition 4.1.4, we may construct standard presentations (R ki /k (Gi ) Ci )/R ki /k (ZGi (Ti )) Gi compatibly with the Ti , where Ci = ZGi (Ti ). The perfectness of Gi implies that the map ji : R ki /k (Gi ) → Gi is surjective, so Gi = R ki /k (Gi )/Zi for the central k-subgroup Zi = ker φi arising from the standard presentation of Gi , and ji carries the maximal k-torus in R ki /k (Ti ) onto Ti . Moreover, since Gi is absolutely pseudo-simple each ki must be a ﬁeld. Let k = ki , G = Gi , and T = Ti , so T is a maximal k -torus of G . Let T be the maximal k-torus Ti in R k /k (T ). The Gi pairwise commute

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Ubiquity of the standard construction

and generate the perfect G (Proposition 3.1.8), so the product map j : R k /k (G ) =

ji

R ki /k (Gi ) −→

Gi G

is surjective. Since G is pseudo-reductive, and any pseudo-reductive central quotient of R k /k (G ) is automatically standard (see Remark 1.4.6), it sufﬁces to prove that ker j is central. To prove centrality of this normal subgroup scheme, it sufﬁces to prove that conjugation by T is trivial on its Lie algebra (as then Corollary A.8.11 will imply centrality of (ker j)0 , so Lemma 5.3.2 below will yield centrality of ker j in the perfect Rk /k (G )). In other words, we just have to show that the induced map of Lie algebras is injective on the root spaces with respect to T . Each such root space occurs within the Lie algebra of a single factor R ki /k (Gi ), so it sufﬁces to check that each quotient map R ki /k (Gi ) → Gi is injective on the root spaces for Ti . But this is a central quotient, so even the maps between root groups are isomorphisms (due to the structure of the open Bruhat cell, and the fact that central subgroup schemes are contained in Cartan subgroups). 2 The following lemma was used to bypass connectedness problems in the preceding argument, and it will be used later on for a similar reason. Lemma 5.3.2 Let G be a smooth connected afﬁne group over a ﬁeld k and assume that G/ZG is perfect. If N ⊆ G is a normal k-subgroup scheme then N is central in G if and only if N 0 is central in G. Proof. We may assume that k is algebraically closed and that N 0 is central. The ﬁnite étale component group = N /N 0 is normal in the connected G/N 0 , so it is central. Thus, N = γ ∈ Nγ where N1 = N 0 is central in G and Nγ denotes the coset of N 0 corresponding to γ ∈ . The centrality of in G/N 0 says that G-conjugation on N carries each Nγ back into itself. This action is N 0 -invariant since N 0 is central in G by hypothesis, so since each Nγ is an N 0 -torsor we see that the G-conjugation on each Nγ is given by gnγ g −1 = fγ (g mod N 0 ) · nγ for a k-homomorphism fγ : G/N 0 → N 0 . By construction it is clear that fγ pulled back to G factors through G/ZG . But G/ZG is a smooth connected group equal to its own derived group and N 0 is a commutative group scheme, so all such homomorphisms fγ are trivial. 2 For absolutely pseudo-simple groups we will give a useful standardness criterion. This rests on an alternative interpretation, for absolutely pseudo-simple groups, of the k-algebra k appearing in the universal maximal absolutely simple quotient:

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181

Proposition 5.3.3 Let G be a perfect smooth connected afﬁne group over a ﬁeld k. The maximal semisimple quotient of Gk and the maximal adjoint semisimple quotient of Gk have the same ﬁeld of deﬁnition over k. That is, if π : Gk G ss /ZGss denotes the maximal adjoint semisimple k k quotient and the purely inseparable ﬁnite extension K/k is the ﬁeld of deﬁnition of ker π ⊆ Gk over k then the (unipotent) radical R(Gk ) ⊆ Gk has ﬁeld of deﬁnition over k equal to K. Example 4.2.6 shows that the perfectness hypothesis on G cannot be dropped in this proposition. The proof of Proposition 5.3.3 will use faithfully ﬂat descent with base rings that may not be reduced (such as k ⊗k k for an algebraic extension k /k that may not be separable), so we ﬁrst require a lemma for certain groups over a general ring: Lemma 5.3.4 Let k be a ﬁeld, G be a smooth k-group of ﬁnite type. For any k-algebra A and any A-homomorphism f : GA → Z to a commutative A-group Z, f |D (G)A = 1. Proof. We may assume k = ks . Since G is smooth, the commutator subgroup C of G(k) is Zariski-dense in G := D (G) (because k = ks ). Thus, C is a collection of sections of G → Spec k that is schematically dense (in the sense of [EGA, IV3 , 11.10.1]). By [EGA, IV3 , 11.10.6], for any extension ﬁeld K/k, a K-morphism from GK to a separated K-scheme is therefore uniquely determined by its restriction to C ⊆ G (k) ⊆ GK (K). The locally closed ker f ⊆ GA therefore has underlying topological space that contains GA since we can work on ﬁbers over points x ∈ Spec A and use that the ﬁber Zx is separated (as are any group schemes over a ﬁeld). Thus, GA ∩ ker f is a closed subscheme of GA containing C, so GA ∩ ker f = GA (by schematic density). This says f |GA = 1. 2 Proof of Proposition 5.3.3 We may assume that k is separably closed (so K is too). Observe that G red = G ss because G is perfect, so R(Gk ) is unipotent. k k First we address the minimality property for K as a ﬁeld of deﬁnition over k for the geometric radical, assuming that a K-descent of the subgroup R(Gk ) ⊆ Gk exists. Suppose that R(Gk ) descends to a K -subgroup of GK , with K a subﬁeld of K containing k. The quotient G ss of Gk descends to a quok tient Q of GK , so likewise the maximal adjoint semisimple quotient G ss /ZGss of k k Gk descends to a quotient Q/ZQ of GK . But K is the minimal ﬁeld of deﬁnition over k for the maximal adjoint semisimple quotient of Gk , so K must equal K. It remains to descend the unipotent subgroup R(Gk ) ⊆ Gk to a K-subgroup of GK . We will do this via faithfully ﬂat descent. Let : GK G be the

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Ubiquity of the standard construction

G be the simply connected central cover over K-descent of π , and let G K. The maximal semisimple (even maximal reductive) quotient G ss of Gk is k

a central cover of G ⊗K k and so is uniquely isomorphic over G ⊗K k to ⊗K k)/M for a unique k-subgroup scheme M in the ﬁnite multiplicative (G has constant Cartier dual center ZG . The ﬁnite multiplicative center ZG of G k over K since K is separably closed, so the subgroup M ⊆ ZG has the form k μk for a unique K-subgroup μ ⊆ ZG . Thus, the quotient map : GK G over K admits a factorization through the central cover G/μ → G after we extend scalars to k. Such a factorization is unique because there are no nontrivial homomorphisms from the smooth connected group Gk into the ﬁnite commutative group (ZG /μ)k . To descend R(Gk ) to a K-subgroup of GK it sufﬁces to descend the unique factorization Gk → Gkss = (G/μ) k → Gk

(5.3.1)

→G GK → G/μ

(5.3.2)

over k to a factorization ?

over K using the canonical quotient map G/μ → G over K. Consider the central extension E of G by ZG /μ over K given by → G → 1. 1 → ZG /μ → G/μ Pulling this back along : GK G gives a central extension ∗ (E) of GK by ZG /μ over K. This pullback central extension has at most one splitting over K, and likewise after any ground ﬁeld extension (e.g., to k). By (5.3.1), the central extension ∗ (E)k splits, so to give a splitting of ∗ (E) over K is exactly to descend (5.3.1) to a diagram (5.3.2) over K. More generally, if E is a central extension of GK by a commutative ﬁnite type K-group scheme Q and the extension Ek is (necessarily uniquely) split then we claim that the extension E is split over K. To prove this via faithfully ﬂat descent it sufﬁces to show the two pullback splittings over k ⊗K k coincide. Since we are splitting a central extension, any two splittings over a K-algebra A (such as k ⊗K k) are related via multiplication against a homomorphism from GA to QA . By Lemma 5.3.4 all such homomorphisms are trivial (since G = D (G)), so Proposition 5.3.3 is proved. 2 Let k be an arbitrary ﬁeld and G a perfect smooth connected afﬁne k-group. (We are most interested in the case where G is also pseudo-reductive over k

5.3 A standardness criterion

183

with G ss a simple k-group, which is to say that G is an absolutely pseudo-simple k k-group.) Let K be the ﬁeld of deﬁnition over k for the kernel of the projection from Gk onto its maximal adjoint semisimple quotient. By Proposition 5.3.3, R(Gk ) descends to a K-subgroup R(GK ) in GK . Let G = GK /R(GK ) (the K-descent of G ss ), so there is a canonical central

k

K-isogeny αG : G → G onto a connected semisimple K-group of adjoint type. θ → → G uniquely factors as G Hence, the simply connected central cover G αG

G → G with θ a central K-isogeny. In fact, by the preceding considerations /μ G over K. the central isogeny θ arises from a unique G -isomorphism G /μ of KAssociated to the canonical quotient map πG : GK G G /μ) of k-groups groups there is a canonical homomorphism iG : G → R K/k (G that coincides with the canonical map in (1.6.1). Since G is perfect, the map iG factors through the derived group of its target. Proposition 1.3.4 identiﬁes this derived group. The resulting factorization is so important that it deserves a notation: Deﬁnition 5.3.5 For an absolutely pseudo-simple k-group G and the ﬁeld of deﬁnition K/k for its geometric radical, )/R K/k (μ) ξG : G → R K/k (G

(5.3.3)

is the unique k-homomorphism through which iG factors. Our deﬁnition of ξG makes sense more generally when G is pseudo-reductive and perfect, but it is the “wrong” notion except when G is absolutely pseudosimple over k. Indeed, the absolutely pseudo-simple case is the only one for k /k, f ) of G has k a ﬁeld that is morewhich the simply connected datum (G, over purely inseparable over k. Beyond this case, K/k as deﬁned above does not encode useful information about the maximal étale subalgebra of k or the individual factor ﬁelds of k . For any absolutely pseudo-simple k-group G we have ξG (G) = iG (G), so it follows from Proposition 4.2.5 that ξG (G) is always pseudo-reductive. Also, since separable extension of the ground ﬁeld preserves absolute pseudosimplicity (Lemma 3.1.2) and is compatible with the formation of the ﬁeld of deﬁnition K/k (Proposition 1.1.9(2)), it follows that the formation of ξG commutes with separable extension of the ground ﬁeld. Remark 5.3.6 Let us assume now that G is a standard pseudo-reductive k-group which is absolutely pseudo-simple. We now prove two important properties of ξG for such G: it is surjective and ker ξG is a central k-subgroup scheme in G.

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Ubiquity of the standard construction

It follows from Proposition 4.2.4 that G admits a standard presentation and K as above, , and (G , K/k, T , C) with G T a maximal K-torus of G C = ZG (T ), with T the maximal k-torus of G contained in the image of R K/k ( T ). The composite map j ξG ) → )/R k /k (μ) R k /k (G G → R k /k (G

is the canonical surjection, as we may check after composition with the inclusion /μ), so ξG is surjective when G is standard over k. into R k /k (G ) → G is surjective since G = D (G), so ker ξG is a The map j : R k /k (G quotient of R k /k (μ) and hence is a central k-subgroup scheme of the quotient ). G of R k /k (G The aim of the rest of this section is to show (in Theorem 5.3.8) that the surjectivity of ξG and centrality of ker ξG are not only necessary for standardness of G when G is absolutely pseudo-simple over k, as we just saw, but are also sufﬁcient. Let k be an arbitrary ﬁeld and G an absolutely pseudo-simple k-group. In particular, by Lemma 3.1.2 the k-group G is a non-commutative pseudo-reductive k-group, and by Proposition 5.3.3 the ﬁeld of deﬁnition K/k of the kernel of the projection from Gk onto its maximal adjoint semisimple quotient coincides with the ﬁeld of deﬁnition over k for the radical R(Gk ) ⊆ Gk . Consider ξG as in (5.3.3). By functoriality, ker ξG is the maximal k-subgroup scheme Z ⊆ G such that ZK ⊆ R(GK ). Thus, ker ξG is unipotent; a priori it may be disconnected. Note that ker ξG cannot contain a nontrivial smooth connected k-subgroup U , for otherwise the unipotent k-group scheme ker ξG would contain the nontrivial smooth connected normal k-subgroup of G generated by the G(ks )-conjugates of Uks ; this normal k-subgroup would have to be unipotent, contradicting the pseudo-reductivity of G over k. In particular, (ker ξG )(ks ) is ﬁnite. As the following example shows, it can happen, even when the absolutely pseudo-simple k-group G is a standard pseudo-reductive group, that dim(ker ξG ) > 0. (By Theorem 5.3.8, an essential feature of all such examples is that GKss has fundamental group with order divisible by char(k).) Here is an example: Example 5.3.7 Let k be an imperfect ﬁeld with characteristic p > 0 and let k /k be a nontrivial ﬁnite purely inseparable extension. Let k ⊆ k be the n intermediate ﬁeld k ∩ k 1/p with some n > 0 (so k = k). We will be most interested in the case that k is a proper subﬁeld of k (so n is not too large, relative to k /k), but at the outset we do not impose this condition.

5.3 A standardness criterion

185

Let G be a connected semisimple k -group that is absolutely simple and simply connected. Let μ ⊆ ZG be a k -subgroup with p-part that is killed by pn but not by any strictly smaller power of p. (As an example, we can take G = SLpn and μ = ZG = μpn .) By the classiﬁcation of simply connected groups, the p-part of μk must be μpn except if p = 2 and G is of type D2m , in s which case another possibility is μ2 × μ2 over ks . Choose a maximal k -torus T ⊆ G and deﬁne T to be the maximal k-torus in R k /k (T ). Let μ = T ∩ R k /k (μ ), so μk is naturally identiﬁed with μ . The quotient G := R k /k (G )/R k/k (μk ) is equal to its own derived group since R k /k (G ) is perfect (as G is simply connected, so Corollary A.7.11 applies). Moreover, G red = (G /μ ) ⊗k k and k G is the standard pseudo-reductive k-group associated to the factorization R k /k (T ) → C := R k /k (T )/R k/k (μk ) → R k /k (T /ZG ) provided that the smooth connected commutative k-group C is pseudoreductive over k (a property to be veriﬁed below). The map ξG makes sense, and the common kernel ker ξG = ker iG is identiﬁed with Q := R k /k (μ )/R k/k (μk ). By consideration of k-points we will compute dim Q and show that it is > 0 whenever k = k (i.e., when n is not too large relative to k /k). To check that C is pseudo-reductive over k, note that the quotient R k /k (T )/R k /k (μ ) of C is pseudo-reductive over k because it is a k-subgroup of R k /k (T /μ ), so if U ⊆ C is a smooth connected unipotent k-subgroup then U ⊆ ker(C R k /k (T )/R k /k (μ )) = Q. Thus, to prove U = 1 it sufﬁces to prove Q(ks ) is ﬁnite. By decomposing μ into its primary parts and applying Proposition A.5.13 to the (étale) prime-to-p parts, we can pass to the case where μ has p-power order. Hence, μk = μpn s except possibly when p = 2 and G is of type D2m , in which case μk may also s be μ2 × μ2 . We shall prove Q(ks ) = {1}. We may assume k is separably closed, and then we can assume μ = μpn even in the D2m -case when p = 2. In the exact sequence 1 → R k/k (μpn )/μpn → R k /k (μpn )/μpn → Q → 1,

(5.3.4)

186

Ubiquity of the standard construction

the middle term R k /k (μpn )/μpn = (R k /k (GL1 )/GL1 )[pn ] has k-points (k × /k × )[pn ] = k× /k × due to the deﬁnition of k. In other words, in (5.3.4) the map on k-points between the left and middle terms is an isomorphism. This forces Q(k) = {1} because the kernel term R k/k (μpn )/μpn = R k/k (GL1 )/GL1 in (5.3.4) is k-smooth (with dimension [k : k] − 1) and k is separably closed. This concludes the veriﬁcation that G is a standard pseudo-reductive k-group. A calculation with k-points shows that (R k /k (GL1 )/GL1 )[pn ] has dimension n [k : k] − [kk p : k] because the image of the pn -power map on k ⊗k k is n p kk ⊗k k, so dim Q = ([k : k] − [kk n

n

pn

: k]) − ([k : k] − 1). n

n

(5.3.5)

The surjection k⊗k pn k p kk p implies [kk p : k] [k p : k p ] = [k : k] n n n with equality if and only if k and k p are linearly disjoint over k p = k ∩ k p . Since [k : k] > 1, we therefore have n

dim Q [k : k] + 1 − ([k : k] + [k : k]) [k : k] + 1 − ([k : k] + 1) = 0, n

with equality between outer terms if and only if k and k p are linearly disjoint n n over k ∩ k p and [k : k] = 1 (i.e., k p ⊆ k). Thus, dim Q > 0 provided that n k p ⊂ k, which is to say that k is a proper subﬁeld of k . In the special case 2n k = k(a1/p ) with a ∈ k − k p the formula (5.3.5) implies dim Q = (pn − 1)2 . Thus, we can make the common kernel Q of ξG and iG have arbitrarily large dimension (with Q(ks ) = 1). Theorem 5.3.8 Let k be a ﬁeld and let G be an absolutely pseudo-simple k-group. Let ξG be as in (5.3.3). If ξG is surjective then ker ξG is connected, and in general the pseudo-reductive k-group G is standard (over k) if and only if ξG is surjective and ker ξG is central in G. If G is standard then ξG is an isomorphism provided that G ss has fundamental k group with order not divisible by char(k). Proof. This theorem is trivial if char(k) = 0 because in such cases K = k and ξG is an isomorphism whenever G is a connected semisimple k-group. We may and do now assume char(k) = p > 0. In Remark 5.3.6 we proved that if the pseudo-reductive group G is standard then ξG is surjective and ker ξG is central. Thus, in the remainder of the proof we may assume that ξG is surjective. That

5.3 A standardness criterion

187

is, we have an exact sequence ξG )/R K/k (μ) → 1, 1 → Z → G → R K/k (G

(5.3.6)

G = G ss ). where Z = ker ξG and μ := ker(G K We will now use Proposition 5.1.3 to prove the connectedness of ker ξG when ξG is surjective. Since Z is normal in the k-smooth G, so is its identity component Z 0 . Thus, it makes sense to push forward (5.3.6) along Z → Z/Z 0 , yielding the exact sequence )/R K/k (μ) → 1 1 → Z/Z 0 → G/Z 0 → R K/k (G

(5.3.7)

that is a central extension since Z/Z 0 is a ﬁnite étale normal subgroup of the connected k-group G/Z 0 . The π -pullback of (5.3.7) is therefore a central extension ﬁtting into the top row of the diagram 1

1

/ Z/Z 0

/ G

/ Z/Z 0

/ G/Z 0

/ R K/k (G ) p h pppp π p p p xppp / R K/k (G )/R K/k (μ)

/ 1 (5.3.8) / 1

in which the squares commute and the arrow labeled h has not yet been constructed. Since Z/Z 0 is ﬁnite, Proposition 5.1.3 and Example 5.1.4 imply that the top row in (5.3.8) is split over k. Equivalently, we can deﬁne h to make a commutative diagram. An elementary diagram chase shows that h is surjective, so h is a quotient map between smooth afﬁne k-groups. Also, the commutativity of the above diagram implies that h−1 (Z/Z 0 ) = R K/k (μ), so h induces a quotient map R K/k (μ) → Z/Z 0 . We will show that this map is the trivial map, so Z = Z 0 (i.e., ker ξG is connected), as desired. To verify such triviality, ﬁrst observe that the k-group Z = ker ξG is unipotent (since ZK is a K-subgroup of the unipotent K-group R(GK )). Thus, its ﬁnite étale quotient Z/Z 0 is unipotent and hence is a p-group. Consider the primary decomposition μ = M × M , with M the p-part and M the prime-to-p part, so R K/k (μ) = R K/k (M ) × R K/k (M ).

188

Ubiquity of the standard construction

The factor R K/k (M ) must therefore have trivial image in Z/Z 0 for torsionorder reasons. To prove that R K/k (M ) also has trivial image in Z/Z 0 , it sufﬁces to show that R K/k (M ) is connected. By extending scalars to L = k and renaming K ⊗k L as A, it sufﬁces to prove that R A/L (μpn ) is connected for any algebraically closed ﬁeld L of characteristic p > 0, any ﬁnite local L-algebra A, and any n 1. Since A has residue ﬁeld L, we have a canonical decomposition R A/L (GL1 ) = GL1 × U via the unipotent radical of 1-units of A. Using this decomposition, it sufﬁces to show that the pn -torsion subgroup scheme U [pn ] is connected. Let m denote the afﬁne space n n over L associated to the maximal ideal m of A. Since (1 + x) p = 1 + x p in characteristic p, the underlying scheme of U [pn ] is identiﬁed with the kernel of n the endomorphism t → t p of the smooth connected additive L-group m. But n n n c · x p = (c1/p x) p for any c ∈ L and x ∈ A, so this kernel is a vector subspace of m at the level of L-points. Hence, it is connected. It remains to show that if the (connected) group scheme Z = ker ξG is central in G then G is standard, and that moreover Z = 1 (i.e., ξG is an isomorphism) when μ has order not divisible by p. By unipotence of Z and its normality in G, the pseudo-reductivity of G over k implies that Z has no nontrivial smooth connected k-subgroup (by Lemma 1.2.1). Thus, if we now form a pullback diagram of central extensions similar to (5.3.8) in which Z/Z 0 has been replaced with Z and G/Z 0 with G, we may again apply Proposition 5.1.3 to get a ksplitting after pullback, which is to say a commutative diagram of k-groups

1

/ Z

) R K/k (G l ll j llll π l l lll l l ll / R K/k (G / G vl )/R K/k (μ) ξG

/ 1

/Z ) = such that the composition of j and the natural map f : G → R K/k (G G R K/k (G ) is the canonical map. The image of j is normal since the target of ξG is a central quotient of G. Thus, this image is a nontrivial smooth connected normal k-subgroup of G, whence j is surjective since G is absolutely pseudosimple. But ker j is central since ker π is central, so the pseudo-reductive G is ). It then follows from Remark 1.4.6 that G must a central quotient of R K/k (G be standard. Since j is a surjective homomorphism between smooth afﬁne k-groups, it is a quotient map. Hence, it restricts to a quotient map R K/k (μ) → Z between k-group schemes (i.e., it is faithfully ﬂat). In particular, if p does not divide the

5.3 A standardness criterion

189

order of μ then R K/k (μ) is ﬁnite étale of order prime to p (Proposition A.5.13), so its quotient Z inherits these properties in such cases. But Z is a unipotent k-group, so Z = 1 when μ has order prime to p. 2 Corollary 5.3.9 Let F/k be a separable extension of ﬁelds, and G a smooth connected afﬁne k-group. Then G is a standard pseudo-reductive k-group if and only if GF is a standard pseudo-reductive F-group. Proof. By Proposition 1.1.9(1), pseudo-reductivity for G over k is equivalent to that of GF over F, so we may assume that G, or equivalently GF , is pseudoreductive. Standardness is insensitive to passage to a separable closure of the ground ﬁeld (Corollary 5.2.3), so we may assume k and F are separably closed. We can then replace G with D (G), by Proposition 5.2.1, so G is perfect. We may assume G = 1, so by Proposition 3.1.8 (over separably closed ﬁelds) and Proposition 5.3.1 we can even assume that G and GF are pseudo-simple. Thus, G is standard if and only if ξG is surjective with a central kernel (Theorem 5.3.8), and similarly for GF . Separability of F/k implies that (ξG )F = ξGF by Proposition 1.1.9(2), so we are done. 2 Proposition 5.3.10 Let G be an absolutely pseudo-simple group over a ﬁeld k. If the root system of Gks is reduced then ker ξG is central, and in such cases G is standard if and only if ξG is surjective. Proof. We may and do assume k = ks . Let T be a maximal k-torus in G, so G is generated by the G(k)-conjugates of T (since G = D (G)). To prove centrality of ker ξG , by Lemma 5.3.2 and Corollary A.8.11 it is enough to show that ker ξG has no nontrivial T -weights on its Lie algebra. Let K/k be the ﬁeld of deﬁnition over k for the kernel of the projection from Gk onto its maximal adjoint semisimple quotient. By Proposition 5.3.3 this is also the ﬁeld of deﬁnition over k for R(Gk ) ⊆ Gk . By deﬁnition of ξG , it has the same kernel as G → R K/k (G ). The reducedness hypothesis on (G, T ) implies that (G, T ) = (G , TK ) (by Theorem 2.3.10). For each a ∈ (G, T ), consider the associated T -stable root group Ua ⊆ G from Proposition 2.3.11, so Ua is a k-vector group admitting a unique linear structure with respect to which T acts linearly, and Lie(Ua ) is the a-root space for T on Lie(G). Since a ∈ (G , TK ), the restriction of ξG to Ua is a T equivariant map between a-root groups. The 1-dimensional aK -root group of G has a unique linear structure, and the map induced by ξG between a-root groups respects the linear structures since GL1 -compatibility follows from the nontriviality of a. Thus, ker(ξG |Ua ) is also a vector group, so it is a smooth connected k-subgroup of G whose geometric ﬁber lies in the unipotent radical.

190

Ubiquity of the standard construction

By pseudo-reductivity of G it follows that ker(ξG |Ua ) is trivial, so passing to the Lie algebra gives that Lie(ξG ) is injective on the a-root space in Lie(G) for all a ∈ (G, T ). This gives the centrality of ker ξG . By Theorem 5.3.8, the equivalence of standardness and surjectivity of ξG follows. 2 Remark 5.3.11 By Theorem 2.3.10, the reducedness hypothesis in Proposition 5.3.10 always holds except for possibly when k is imperfect of characteristic 2 with G ss of simply connected type Cn for some n 1 (i.e., Sp2n ), as this type is k the only irreducible and reduced root datum for which some roots are divisible in the character lattice.

6 Classiﬁcation results

Let G be an absolutely pseudo-simple group over a ﬁeld k. We have seen in Proposition 5.3.3 that the ﬁeld of deﬁnition K/k of the kernel of the projection of Gk onto its maximal adjoint semisimple quotient is also the ﬁeld of deﬁnition G ss = GK /R(GK )), with G the over k for R(Gk ) ⊂ Gk . Let μ = ker(G K ss simply connected central cover of Gk , so we have a natural map ξG : G → )/R K/k (μ) as in (5.3.3). Observe that if μ is étale (e.g., G ss is of type A1 R K/k (G k and char(k) = 2) then ξG is identiﬁed with the natural map iG : G → R K/k (G ) from (1.3.1). In §5.3 it was shown that the pseudo-reductive G is standard if and only if ξG is surjective with central kernel, that ker ξG is connected if ξG is surjective, and that ker ξG is central if (Gks , Tks ) is reduced (e.g., char(k) = 2). The deﬁnition of ξG makes sense even if we replace K with a ﬁnite extension (i.e., any ﬁnite extension of k over which the subgroup R(Gk ) ⊂ Gk is deﬁned), so for the structure of ker ξG it is irrelevant if we choose K/k to be the (minimal) ﬁeld of deﬁnition of Ru (Gk ) = R(Gk ) ⊂ Gk over k or merely to be some ﬁnite extension of k over which this k-subgroup can be deﬁned. In contrast, the property of ξG being surjective or not depends very much on working with the minimal K/k. It appears to be quite difﬁcult to directly analyze the image of ξG without using a more precise understanding of the structure of G. Our strategy is to ﬁrst handle the case where G ss is of type A1 k by means of concrete calculations with SL2 and sl2 , and to then deduce results in the general case from this case via consideration of torus centralizers. For later purposes in characteristic 2 we also include a separate analysis of types A2 and G2 away from characteristic 3 (but allowing characteristic 2). The study of type A1 in characteristic 2 leads to completely new problems and examples, as we will see in Chapters 9–10.

191

192

Classiﬁcation results

6.1 The A1 -case away from characteristic 2 The goal of this section is to prove the following special case of Theorem 5.1.1. Theorem 6.1.1 Let k be a ﬁeld with char(k) = 2 and let G be a pseudoreductive k-group such that G ss is isomorphic to SL2 or PGL2 . The pseudok reductive k-group G is standard. If G is absolutely pseudo-simple over k and K is the ﬁeld of deﬁnition over k for the k-subgroup Ru (Gk ) = R(Gk ) ⊆ Gk then the natural map ξG = iG : G → R K/k (GKss ) is an isomorphism. To prove Theorem 6.1.1, by Corollary 5.2.3 we may assume k = ks . Since D (G)ss → G ss is a central isogeny (Proposition A.4.8), by Propositions 1.2.6 k k and 5.2.1 we can also assume G is perfect. Thus, by Lemma 3.1.2, G is absolutely pseudo-simple over k. To reduce the PGL2 -case to the SL2 -case, we apply the following lemma that will be useful in some later situations as well. Lemma 6.1.2 Let G be an absolutely pseudo-simple group over a ﬁeld k, with → G ss be the simply K/k the ﬁeld of deﬁnition of its geometric radical. Let G K connected central cover, and μ the kernel of this central isogeny. Consider the cartesian diagram G

ξ

/ R K/k (G )/R K/k (μ0 ) g

f

G

ξG

/ R K/k (G )/R K/k (μ)

(6.1.1)

in which g and ξG are the natural maps. The map g is an étale central isogeny, the k-group G is absolutely pseudosimple (in particular, connected) with K/k the ﬁeld of deﬁnition of its geometric /μ0 , and ξ is identiﬁed with ξG . Moreover, ξG is surjective if radical, GKss = G and only if the map ξG is surjective, and ker ξG is central if and only if ker ξG is central. Proof. The connected-étale sequence 1 → μ0 → μ → μét → 1 is canonically split: if char(k) = 0 then μ0 = 1, and if char(k) = p > 0 then the (functorial) decomposition of the torsion group μ into a direct product of its p-part μ0

6.1 The A1 -case away from characteristic 2

193

and its prime-to-p part gives the splitting. Thus, the inclusion R K/k (μ0 ) → R K/k (μ) has cokernel R K/k (μét ) that is ﬁnite étale (Proposition A.5.13), so the homomorphism g is an étale central isogeny. The map f is therefore also an étale central isogeny, so G 0 is an absolutely pseudo-simple group. It is obvious that ξG is surjective if and only if ξ is surjective and that ker ξ is central if and only if ker ξG is central. We now show that G is connected. It sufﬁces to show that the kernel ker g is contained in the identity component of G . Fix a maximal k-torus T of G; we will prove that f −1 (T ) is connected. Since TK is a maximal K-torus of /μ, its preimage is a maximal K-torus in G . The map GKss = G T in G ξG carries T isomorphically onto the maximal k-torus in R K/k (T )/ R K/k (μ) (recall that ker ξG is unipotent), so in view of the short exact sequence 1 → ker g → R K/k ( T )/ R K/k (μ0 ) → R K/k ( T )/ R K/k (μ) → 1 we just have to show that ker g is contained in the maximal k-torus of R K/k ( T )/ R K/k (μ0 ) (since the quotient map g carries maximal tori onto maximal tori). But ker g is ﬁnite étale, and for n not divisible by char(k), the n-torsion in R K/k ( T )/ R K/k (μ0 ) is identiﬁed with the n-torsion in R K/k ( T ), which in turn clearly lies in the maximal k-torus of R K/k ( T ). This completes the proof that G is connected. The ﬁeld of deﬁnition over k for the geometric radical of G must contain the corresponding ﬁeld of deﬁnition K/k for G. This containment is an equality since fK−1 (R(GK ))0 is a K-descent of R(Gk ) (due to fK -pullback preserving smoothness, since f is étale). It remains to show that ξ is identiﬁed with ξG . Applying extension of scalars by K/k to (6.1.1), and using the canonical map q : R K/k (H )K H for any K-group H , we get the following commutative diagram

GK fK

GK

h

/ G /μ0 π

/ G /μ

in which the bottom and right maps are the canonical ones. We have to show that the top map is the maximal semisimple quotient of GK . By connectedness considerations, the map h is surjective since π ◦ h is surjective with π an étale isogeny. The identity component (ker h)0 must equal fK−1 (R(GK ))0 = R(GK ),

194

Classiﬁcation results

so (ker h)0 is smooth and unipotent. Hence, h factors as /μ0 GK → GKss → G with the ﬁrst map equal to the canonical quotient map and the second an étale isogeny. But the ﬁnal term is the quotient of a simply connected semisimple group by an inﬁnitesimal central subgroup, so it admits no nontrivial connected /μ0 , or ﬁnite étale isogenous covers (as a k-group). In other words, GKss = G more precisely h is the maximal semisimple quotient of GK . 2 Returning to the proof of Theorem 6.1.1, now with GKss SL2 , we will prove that ξG is surjective. Once this is shown, standardness will follow from the hypothesis char(k) = 2 (which ensures that G has a reduced root system, so Proposition 5.3.10 applies). Since SL2 is simply connected, standardness will imply (by the ﬁnal claim in Theorem 5.3.8) that ξG is an isomorphism. We will now choose a convenient K-isomorphism GKss SL2 . This requires some preparations. Fix a maximal k-torus T ⊆ G, so the K-torus TK in GK maps isomorphically onto a maximal K-torus in GKred = GKss SL2 . We abuse notation by identifying TK with its image in GKss since no confusion seems likely (i.e., the context will make clear whether we are viewing TK as a K-torus in GK or in GKss ). Since char(k) = 2, the root system of (G, T ) is reduced and (GKss , TK ) = (G, T ) inside X(T ) = X(TK ). This root system has the form {±a0 }, and Proposition 2.1.8 gives that the multiplication map Ua0 × ZG (T ) × U−a0 → G

(6.1.2)

is an open immersion of k-schemes. By Remark 2.3.6, the image of (U±a0 )K ⊆ GK in the GK -quotient GKss SL2 is the ±a0 -root group for (GKss , TK ). Since TK maps isomorphically onto a maximal K-torus in SL2 , we can choose a K-isomorphism GKss SL2 which carries TK into the diagonal K-torus DK of SL2 . Applying conjugation by a representative of the nontrivial class in W (SL2 , D)(k) if necessary, we can arrange that (Ua0 )K maps onto the strictly upper triangular unipotent matrix group U + , and (U−a0 )K maps onto the strictly lower triangular unipotent matrix group U − . In particular, ξG (U±a0 ) ⊆ R K/k (U ± ) and these images under ξG are nontrivial. The isomorphism TK DK of K-tori uniquely descends to a k-isomorphism T D since k is separably closed. In particular, D(k) = ξG (T (k)). Note that applying the K-automorphism of SL2 arising from the action of an element of the diagonal torus in PGL2 = Aut(SL2 ) will generally change our chosen K-isomorphism GKss SL2 but will preserve the properties that

6.1 The A1 -case away from characteristic 2

195

we have arranged in the preceding considerations. It now sufﬁces to prove the following result. ξG

Lemma 6.1.3 With notation as above (so char(k) = 2), the map G → R K/k (GKss ) R K/k (SL2 ) is surjective. Proof. We will prove that the image is R L/k (SL2 ) for some intermediate ﬁeld L ⊆ K over k after applying a suitable K-automorphism of SL2 arising from the diagonal torus in PGL2 . Granting this for a moment, we ﬁrst claim that the L-map π : GL → SL2 corresponding to the k-map G → R L/k (SL2 ) induced by ξG must be a descent of the canonical quotient K-map : GK GKss = SL2 . Rather generally, consider a tower K/L/k of ﬁnite extensions of ﬁelds and afﬁne (or just quasi-projective) schemes X and Y of ﬁnite type over k and L respectively such that there is a commutative diagram f

/ R K/k (YK ) X G GG O GG GG R L/k ( jY ,K/L ) GG h # R L/k (Y )

(6.1.3)

where jY ,K/L : Y → R K/L (YK ) is as in (A.5.3). We claim that the associated maps XK → YK and XL → Y obtained from the universal property of Weil restriction are related via the ﬁrst being the scalar extension of the second. To verify this, we simply use the mechanism by which these associated maps are constructed: they are respectively the top and bottom composites in the following diagram:

XK

fK

/ R K/k (YK )K O

qY ,K/k

/ YK

R L/k ( jY ,K/L )K

XK

(hL )K

/ (R L/k (Y )L )K

(qY ,L/k )K

/ YK

It therefore sufﬁces to prove that the diagram commutes. The commutativity of the left part is a tautology, and by Proposition A.5.7 the right part evaluated on

196

Classiﬁcation results

A-points for any K-algebra A comes out to be the diagram Y (K ⊗k A) O

/ Y (A)

Y (L ⊗k A)

/ Y (A)

obtained by applying the functor Y to the evident commutative diagram of L-algebras. Returning to our situation of interest, the L-subgroup ker π ⊆ GL must then be a descent of the K-subgroup ker = R(GK ) ⊆ GK . Minimality of K/k as a ﬁeld of deﬁnition of the radical of Gk therefore forces the inclusion L ⊆ K over k to be an equality. Hence, granting the existence of L/k as above, ξG has image R K/k (SL2 ), which is to say that ξG is surjective, as desired. Now we turn to the task of constructing L/k within K as claimed. Let G ⊆ R K/k (SL2 ) denote the image of ξG . The arguments following (6.1.2) show that G is generated by the smooth connected k-subgroup ξG (ZG (T )) ⊆ R K/k (TK ) and some nontrivial smooth connected k-subgroups of R K/k (U + ) and R K/k (U − ), with D ⊆ ξG (ZG (T )). Recall that D(k) = ξG (T (k)), so D(k) ⊆ ξG (ZG (T )(k)). We conclude that G is the Zariski-closure in R K/k (SL2 ) of the subgroup of R K/k (SL2 )(k) = SL2 (K) generated by the subgroup ξG (ZG (T )(k)) ⊆ D(K) that contains D(k) and the subgroups

1 G (k) ∩ U (K) = 0 +

V+ 1 − , G (k) ∩ U (K) = 1 V−

0 1

for some nonzero additive subgroups V ± ⊆ K. Fix the K-isomorphisms t : GL1 DK and u± : Ga U ± via t(x) =

x 0

0 1 , u+ (x) = 1/x 0

x 1 , u− (x) = 1 x

0 . 1

Since t(x)u± (y)t(x)−1 = u± (x±2 y) and D(k) is the image of k × ⊆ K × under the K-isomorphism t : GL1 DK , the additive subgroups V ± ⊆ K are stable under scaling by squares in k × . Pick a nonzero y ∈ V − , so y ∈ K × . Using the natural left action of PGL2 on SL2 over K, changing the isomorphism

6.1 The A1 -case away from characteristic 2 GKss

SL2 by the action of the diagonal element

y 0

197

0 1

∈ PGL2 (K) is

harmless and brings us to the case where 1 ∈ V − . We shall now prove that for any nonzero additive subgroups V ± ⊆ K such that 1 ∈ V − and both V + and V − are stable under scaling by (k × )2 , the Zariskiclosure in R K/k (SL2 ) of the subgroup generated by D(k),

1 0

1 V+ , 1 V−

0 1

(6.1.4)

is R L /k (SL2 ) where L ⊆ K is the subﬁeld k(V + , V − ) generated over k by V + and V − . In our preceding setup this will give R L /k (SL2 ) ⊆ G , and we will then have to address the contribution from the subgroup ξG (ZG (T )(k)) ⊆ D(K) = t(K × ) that contains t(k × ) = D(k). Since the subset (k × )2 ⊆ k is Zariski-dense in the closed k-subgroup Ga ⊆ R K/k (Ga ) over k, we may replace V + and V − with their k-linear spans within K without affecting the Zariski closure in R K/k (SL2 ) of the subgroup of (R K/k (SL2 ))(k) = SL2 (K) generated by the groups in (6.1.4). That is, we may assume that the additive subgroups V ± ⊆ K are nonzero k-linear subspaces (with 1 ∈ V − ). For L = k(V + , V − ), the Zariski-closure H ⊆ R K/k (SL2 ) of the subgroup of SL2 (K) generated by the groups in (6.1.4) is a smooth connected k-subgroup of the smooth connected k-subgroup R L /k (SL2 ). Hence, to prove H = R L /k (SL2 ) it is sufﬁcient to compare dimensions, and for that purpose it is enough to compare Lie algebras. More speciﬁcally, within the Lie algebra sl2 (K) viewed over k, it sufﬁces to show that the Lie subalgebra g over k generated by the k-subalgebras

x 0

0 −x

x ∈ k , 0 0

0 V+ , 0 V−

0 0

is equal to sl2 (L ). For any x ∈ K, deﬁne dt(x) =

x 0

0 0 , du+ (x) = 0 −x

x 0 , du− (x) = 0 x

0 , 0

(6.1.5)

so du− (1) ∈ g since 1 ∈ V − . Also, by hypothesis dt(k) ⊆ g. But we have more in g on the diagonal: g contains [du+ (x), du− (1)] = dt(x) for all x ∈ V + .

198

Classiﬁcation results

Hence, for any x, x ∈ V + , g contains [dt(x), du+ (x )] = du+ (2xx ). Since char(k) = 2, it follows that g contains the image under du+ of the k-linear span A+ of all (non-empty) products among elements in V + . For a nonzero x0 ∈ V + ⊆ K, say with minimal polynomial f (X ) = X n + cn−1 X n−1 + · · · + c0 ∈ k[X ] over k, we have c0 ∈ k × and c0 = − j1 cj x j ∈ A+ , so 1 ∈ A+ . Hence, A+ is a k-subalgebra of K, which is to say it is an intermediate ﬁeld. We may replace V + with A+ for the purposes of carrying out the determination of g, so we can assume that V + is a subﬁeld of K over k. In particular, 1 ∈ V + , so by running through the analogous calculations with focus on the lower triangular part we may replace V − with the subﬁeld it generates in K over k. Since g contains dt(V + ) and dt(V − ), the computation of [dt(x), du+ (x )] shows that we can replace V + with the k-linear span of the set V − · V + of products, which is to say with L . The same goes with the lower left corner, so g contains du± (L ). These generate sl2 (L ). Our arguments show that G = ξG (G) is the Zariski closure in R K/k (SL2 ) of the group generated by SL2 (L ) = R L /k (SL2 )(k) and the subgroup Z := ξG (ZG (T )(k)) ⊆ t(K × ) containing t(k × ). For any z ∈ K × and x ∈ L , t(z)u± (x)t(z)−1 = u± (z ±2 x). Hence, contains u± (z ±2 L ) for any z ∈ K × such that t(z) ∈ Z. Since Z corresponds under t to a multiplicative subgroup of K × , it follows that contains u± (L) where L ⊆ K is the subﬁeld generated by L and squares of elements z ∈ K × for which t(z) ∈ Z. Since the subﬁeld L ⊆ K is separably closed with char(L) = 2, it is stable under the formation of square roots within K. Hence, Z ⊆ t(L× ). But u+ (L) and u− (L) generate SL2 (L), whose diagonal is the group t(L× ) that contains Z. Thus, consideration of k-points gives G = R L/k (SL2 ), as required. 2

6.2 Types A2 and G2 away from characteristic 3 Consider an absolutely pseudo-simple group G over a ﬁeld k with char(k) = 3. The main case of interest will be char(k) = 2. Let K/k be the purely inseparable ﬁnite extension that is the ﬁeld of deﬁnition over k for the (necessarily unipotent) radical R(Gk ) as a subgroup of Gk . This radical descends to a K-subgroup R(GK ) ⊆ GK and by Proposition 5.3.3 the extension K/k is also the ﬁeld of deﬁnition over k for the maximal adjoint semisimple quotient of Gk . Deﬁne G be the simply connected central cover, G = GKss = GK /R(GK ). Let G and let μ be the kernel of this covering map.

6.2 Types A2 and G2 away from characteristic 3

199

Since G = D (G), the canonical map iG : G → R K/k (G ) factors through )/R K/k (μ) whose target is the a canonical k-group map ξG : G → R K/k (G derived subgroup of R K/k (G /μ) = R K/k (G ) (Proposition 1.3.4). The formation of ξG commutes with separable algebraic extension on k, and the reason for our interest in ξG is that the absolutely pseudo-simple k-group G is standard if and only if ξG is surjective and ker ξG is central in G (Theorem 5.3.8), and that centrality is automatic when G ss is not of simply connected type Cn for k some n 1 (Proposition 5.3.10). Assume that G ss is of type A2 or G2 . Our aim is to prove that any such G k is standard. Since char(k) = 3 and G ss is isomorphic to SL3 , PGL3 , or G2 , k → G := G ss has étale kernel the simply connected central covering map G K

μ. Hence, ξG = iG : G → R K/k (G )

(6.2.1)

since R K/k commutes with the formation of quotients by smooth (e.g., étale) subgroups. To prove the standardness of such G, it sufﬁces to prove the following analogue of Theorem 6.1.1 that is the ﬁrst main goal of this section: Proposition 6.2.1 Let k be a ﬁeld with char(k) = 3, and G an absolutely pseudo-simple k-group such that G ss is of type A2 or G2 . The associated natural k map (6.2.1) is an isomorphism, and hence G is standard. To prove Proposition 6.2.1, we may and do assume k = ks . Thus, by Theorem 5.3.8 and Lemma 6.1.2, we may also assume μ is inﬁnitesimal. In our present setting, μ is inﬁnitesimal precisely when μ = 1, so G is isomorphic to either SL3 or G2 . Now we ﬁx notation related to roots. Choose a maximal k-torus T in G, so the quotient map π : GK GKred = GKss = G carries TK isomorphically onto a maximal K-torus T of G . Choose a positive system of roots + in the rank-2 root system (G , T ) ⊆ X(T ) = X(T ) and let = {a, b} be the basis of + , with a long and b short in the G2 -case. G in G , and The image of the coroot b∨ normalizes the root group U±a a the conjugation action on it by relative to the unique linear structure on U±a ∨ the point b∨ (s) ∈ G (K) for s ∈ K × is multiplication by s±a,b ∈ K × (i.e., s∓1 for simply connected type A2 and s∓3 for type G2 ). Note in particular that the exponent of s is not divisible by char(K) = char(k).

200

Classiﬁcation results

Since G is simple but not of type Cn (n 1), the elements of the reduced and irreducible root system (G , T ) are not divisible by 2 in X(T ). Thus, by Theorem 2.3.10, (G, T ) = (G , T ). The proof of Proposition 6.2.1 will proceed by showing that (6.2.1) is surjective, in which case standardness follows from Proposition 5.3.10 and the isomorphism property for ξG follows from the ﬁnal claim in Theorem 5.3.8 (as char(k) = 3). The proof of surjectivity requires some input from subgroups of G that we can understand in concrete terms and which have a nontrivial image under (6.2.1). As in our work in §6.1 away from characteristic 2, to do this we begin by analyzing root groups for (G , T ). For each c ∈ (G, T ), consider the root group Uc ⊆ G as in Proposition 2.3.11. Since the K-subgroup (Uc )K ⊆ GK has Lie algebra K ⊗k Lie(Uc ) that is equal to the cK -root space in Lie(GK ), this Lie algebra maps onto the cK -root space in Lie(G ) under Lie(π ). In particular, the map Uc → R K/k (G ) induced by ξG is nonzero on Lie algebras. Hence, we deduce (from T -equivariance of ξG ) the nontriviality of the k-subgroup V c := ξG (Uc ) ⊆ R K/k (G ) that is contained in R K/k (Uc ), where Uc is the cK -root group of G . By T -equivariance, the map ξG also carries ZG (T ) into R K/k (ZG (T )) = R K/k (T ). Using the basis , choose a pinning of the pair (G , T ) (see Deﬁnition of (G , T ) are thereby A.4.12). In particular, for c ∈ , the root groups U±c over K. (We will equipped with a speciﬁed identiﬁcation u±c : Ga U±c make the choice of pinning more precise below.) Hence, V ±c = ξG (U±c ) is identiﬁed with a smooth connected nontrivial k-subgroup of R K/k (Ga ), and it is stable under the natural GL1 -action, so it is a vector subgroup. In other words, it corresponds to a nonzero k-subspace V±c ⊆ K. We wish to get to the situation (K) = K and V in which the k-subspaces V−a ⊆ U−a −b ⊆ U−b (K) = K both contain 1 (with the equalities Uc (K) = K obtained via our choice of pinning of (G , T )). Choose c ∈ {−a, −b}. For s, s ∈ K × and coroots λ and λ , conjugation by λ(s)λ (s ) on G acts on the cK -root group Uc Ga of G by multiplication by sc,λ s c,λ ∈ K × . Thus, by taking λ = a∨ and λ = b∨ , it sufﬁces to ﬁnd s, s ∈ K × such that the elements ∨

sa,a s

a,b∨

∨

, sb,a s

b,b∨

∈ K×

coincide respectively with any given pair of elements of K × (such as some nonzero elements v−a ∈ V−a and v−b ∈ V−b with respect to the initial choice of pinning). Since K is separably closed, such s, s ∈ K × can be found since the determinant of the Cartan matrix of (G , T ) is not divisible by char(K) (as char(k) = 3). Applying conjugation by a∨ (s)b∨ (s ) for such s, s , adapted to the pair (v−a , v−b ), gives a new pinning such that both V−a and V−b contain 1.

6.2 Types A2 and G2 away from characteristic 3

201

Fix c ∈ {−a, −b}. The pinning on the simply connected pair (G , T ) over K provides a canonical identiﬁcation of SL2 with the K-subgroup Gc ⊆ G and this identiﬁcation carrying U over to the standard generated by Uc and U−c c upper-triangular unipotent K-subgroup of SL2 . We likewise get a corresponding copy of sl2 (K) as a Lie K-subalgebra of Lie(G ). In terms of the resulting identiﬁcation of sl2 (K) as a Lie k-subalgebra of Lie(R K/k (G )) = Lie(G ), the Lie k-subalgebra g := Lie(ξG (G)) must contain

0 0

x 0 , 0 x

0 0

for all x ∈ Vc and x ∈ V−c . But 0 0 so

x 0 , 0 x

0 xx = 0 0

0 , −xx

da∨ (Va · V−a ), db∨ (Vb · V−b ) ⊆ g,

(6.2.2)

where Vc · V−c denotes the k-subspace of K generated by the products xx for x ∈ Vc and x ∈ V−c . We claim that inside of K, Va ⊆ Va · V−a ⊆ V−b .

(6.2.3)

The ﬁrst inclusion holds because 1 ∈ V−a . Since 1 ∈ V−b , to prove the second inclusion it sufﬁces to prove Va · V−a · V−b ⊆ V−b (where the triple product is deﬁned analogously to the products Vc · V−c ). The adjoint representation of any smooth afﬁne k-group induces the adjoint representation of its Lie algebra, so for any x ∈ Va · V−a and y ∈ V−b it follows from (6.2.2) that the Lie k-subalgebra g in Lie(G ) contains [da∨ (x), du−b (y)]. This equals du−b (−b, a∨ xy) because ∨ a∨ (s)u−b (t)a∨ (s)−1 = u−b (s−b,a t) for all s ∈ K × and t ∈ K. Passing to tangent space at the identity yields that −b, a∨ · xy ∈ V−b for all x ∈ Va · V−a and y ∈ V−b . But −b, a∨ has nonzero image in the ﬁeld K, so (6.2.3) is proved. Lemma 6.2.2 The k-subspaces V±a , V±b ⊆ K coincide with a common subﬁeld L containing k. Proof. By Corollary 3.4.3(2), dimk Vc = dimk V−c for all roots c. Thus, the inclusion Va ⊆ V−b implies dimk V−a = dimk Va dimk V−b = dimk Vb .

202

Classiﬁcation results

But we can interchange the roles of a and b in our proof of (6.2.3), so also Vb ⊆ V−a . Hence, all four k-dimensions coincide, so in fact the containments Va ⊆ V−b and Vb ⊆ V−a are equalities, so 1 ∈ Va and 1 ∈ Vb . Also, the containments in (6.2.3) are forced to be equalities, and likewise for the analogous containments with a and b interchanged. This shows that V±a and V±b are all equal to a common k-subspace V ⊆ K that contains 1 and is stable under multiplication against itself. Hence, it is a subﬁeld L ⊆ K containing k, as desired. 2 Our ﬁxed pinning on (G , T ) via the basis {a, b} of speciﬁes a Kisomorphism of G with either SL3 or G2 . Thus, by a reﬁnement of the Isomorphism Theorem (see Theorem A.4.13 for this reﬁnement), there is a unique descent of (G , T ) to a pair (G , T ) over k such that the pinning is deﬁned over k. In particular, T deﬁnes the same k-structure on T = TK that T does, so the subsets T (k), T (k) ⊆ T (K) coincide. By Lemma 6.2.2, V ±a and V ±b generate R L/k (GL ), as we may check using k-rational points and the open Bruhat cell of (GL , TL ). Hence, we have proved the containment R L/k (GL ) ⊆ ξG (G) inside of R K/k (G ) = R K/k (GK ). We can do better: for every c ∈ (G, T ), necessarily V c ⊆ R L/k (GL ) inside of ξG (G). To prove this, observe that W (G , T )(k) = W (G , T )(K) = W (G, T )(K) (the ﬁnal equality by Proposition 3.2.7). Thus, we can ﬁnd n ∈ NG (T )(k) that conjugates an arbitrary but ﬁxed c ∈ (G, T ) to one of a or b. This moves V c to V a or V b , each of which lies in R L/k (GL ), so we get the desired containment for c. We have done enough analysis of root groups to ﬁnally prove the following result, which completes the proof of Proposition 6.2.1. Lemma 6.2.3 The map ξG is surjective. Proof. Lemma 3.1.5 implies that the root groups Uc for c ∈ (G, T ) generate G, and hence the V c = ξG (Uc ) generate ξG (G). But for each c ∈ (G, T ), V c is contained in R L/k (GL ), and therefore ξG (G) ⊆ R L/k (GL ). As we observed above, R L/k (GL ) ⊆ ξG (G). We conclude that ξG (G) = R L/k (GL ), so the map ξG : G → R K/k (G ) = R K/k (GK ) factors through R L/k (GL ). Hence, as in the proof of Lemma 6.1.3 (using (6.1.3)) it follows from the functorial deﬁnition of Weil restriction that the natural quotient map π : GK G GK descends to an L-group map GL GL . But the K-subgroup ker π = R(GK ) in GK has (minimal) ﬁeld of deﬁnition over k equal to K (by deﬁnition of K/k), so L = K; i.e., ξG (G) = R K/k (G ). 2

6.3 General cases away from characteristics 2 and 3

203

Our study of types A2 and G2 will be most useful in characteristic 2, and will be used to prove a general standardness result away from characteristic 3 in Proposition 6.3.6 when the Dynkin diagram of G ss has no edges of multiplicity k 2. We end this section with an application in a special case when there is an edge with multiplicity 2, type F4 : Proposition 6.2.4 Let G be an absolutely pseudo-simple group over a separably closed ﬁeld k, and assume that G ss is of type F4 . Let T be a maximal k-torus k of G, and = (G, T ). For each c ∈ , let Gc be the k-subgroup generated by the ±c-root groups. Then ξGc is an isomorphism. Proof. The case char(k) = 2 is settled by Theorem 6.1.1, so now assume char(k) = 2. Choose a positive system of roots + such that c lies in the set of simple positive roots. The vertex corresponding to c is adjacent to that corresponding to c ∈ of the same length. Let S be the codimension-2 k-torus (ker c ∩ ker c )0red in T , and consider H = D (ZG (S)). This is pseudo-reductive and H red is identiﬁed with D (ZGss (Sk )) by Proposition A.4.8. This is simply k k connected of type A2 since G ss is the (simply connected) group of type F4 . In k particular, H is absolutely pseudo-simple. By Theorem 6.2.1, the pseudo-reductive k-group H is standard. Since H ss k is simply connected, the ﬁnal part of Theorem 5.3.8 implies that ξH is an isomorphism. That is, H R K/k (HKss ) for the ﬁeld of deﬁnition K/k of R(Hk ), with HKss SL3 . This latter isomorphism can be arranged to carry the maximal K-torus (T ∩ H )K to the diagonal K-torus D. Hence, for each a ∈ (H , T ∩ H ) = (SL3 , D) we have that Ha is carried isomorphically onto R K/k (SL2 ) by ξH . Choose a = c, and note that Hc = Gc by construction of H , so Gc R K/k (SL2 ) and hence ξGc is an isomorphism. 2

6.3 General cases away from characteristics 2 and 3 We shall now prove Theorem 1.5.1. This result is obtained by combining Corollary 6.3.5 and Proposition 6.3.6. It will be convenient to ﬁrst carry out some general considerations with torus centralizers, reﬁning Proposition 3.4.1. We initially allow the ground ﬁeld to have any characteristic so that some of our conclusions can be applied later in our work in characteristics 2 and 3. Let G be an absolutely pseudo-simple group over a separably closed ﬁeld k, T a maximal k-torus in G, and K/k the ﬁeld of deﬁnition of the k-subgroup R(Gk ) ⊆ Gk . For each a ∈ (G, T ), let Ga ⊆ G be the smooth connected k-subgroup generated by the root groups Ua and U−a , so by Proposition 3.4.1

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the k-group Ga is absolutely pseudo-simple of rank 1 and it has the following alternative description: if a is not divisible in (G, T ) (as always happens except possibly when k is imperfect of characteristic 2) then Ga = D (ZG (Ta )) where Ta = (ker a)0red , and if a is divisible in (G, T ) (so k is imperfect of characteristic 2) then Ga = D (ZZG (Ta ) (μ)0 ) where μ = S[2] for the rank-1 maximal k-torus S = T ∩ Ga/2 in Ga/2 . Proposition 6.3.1 For each a ∈ (G, T ), let Ka /k be the ﬁeld of deﬁnition of R((Ga )k ) ⊆ (Ga )k . (1) The ﬁelds Ka are all contained in K over k, and they generate K over k. (2) Let πa : (Ga )Ka Ga be the maximal semisimple quotient over Ka . The natural map π : (Ga )K → G := GKss uniquely factors through (πa )K via a K-subgroup inclusion of (Ga )K into G ; equivalently, R((Ga )k ) = (Ga )k ∩ R(Gk ) for all a. The image of (Ga )K in G is the subgroup Ga generated by the ±aK -root groups of (G , T ), where T is the torus TK of GK viewed as a maximal K-torus of G and a = 2a or a according as a is multipliable or not. (3) Letting iG : G → R K/k (GK ) → R K/k (G ) be the natural composite map, the diagram

Ga

G

ξGa

/ D (R Ka /k (Ga ))

/ R Ka /k (Ga )

/ R K/k (G ) a

iG

(6.3.1)

/ R K/k (G )

of k-homomorphisms (with all unlabelled maps equal to the natural inclusions) commutes. Note that in (1) there is no ambiguity concerning how each Ka embeds into K over k since they are all purely inseparable over k. Proof. By Proposition A.4.8 applied initially to ZG (Ta ) in G and then to D (ZG (Ta )) in ZG (Ta ), it follows that the geometric radical of D (ZG (Ta )) is deﬁned over K. Hence, if a is not divisible in (G, T ) (in which case Ga = D (ZG (Ta ))) then Ka ⊆ K over k. To handle the divisible case, we ﬁrst use Proposition A.8.14(2) to see that Ha := ZZG (Ta ) (μ)0 (with μ the 2-torsion in T ∩ Ga/2 ) is a pseudo-reductive k-group whose geometric unipotent radical 0 Ru ((Ha )k ) equals Ru (ZG (Ta )k ) ∩ Ha , and hence, in particular, it has ﬁeld of deﬁnition over k that is contained in the ﬁeld of deﬁnition over k for the

6.3 General cases away from characteristics 2 and 3

205

geometric unipotent radical of ZG (Ta ). But this latter ﬁeld is contained in K by Proposition A.4.8 (applied over k), and so a further application of Proposition A.4.8 to the derived subgroup Ga of Ha gives that Ka ⊆ K in the divisible case as well. This proves the ﬁrst assertion in (1), so the formulation of (2) now makes sense. Before we can prove the second assertion in (1), we need to prove (2). We treat the formulation in terms of geometric radicals, or equivalently (by perfectness of G and Ga ) geometric unipotent radicals. By applying Proposition A.4.8 twice over k (for passage to torus centralizers and to derived subgroups thereof) we can handle the case when a is not divisible in (G, T ), and the same method works when a is divisible by using Proposition A.8.14(2) over k to handle μ-centralizers. Thus, (2) is proved apart from identifying the image of (Ga )K in G . By deﬁnition Ga is generated by the root groups U±a in G, so in view of Theorem 2.3.10 and Remark 2.3.6, the quotient map GK G carries (U±a )K onto the ±aK -root groups in G . Thus, the image of (Ga )K in G is generated by the ±aK -root groups for (G , T ). This coincides with the image of the inclusion (Ga )K → G . Returning to the second claim in (1), we let L ⊆ K be a subﬁeld that contains Ka for all a ∈ (G, T ) (so k ⊆ L) and we aim to prove that L = K. By Proposition 5.3.3, K/k is equal to the ﬁeld of deﬁnition of the kernel of the projection from Gk onto its maximal adjoint semisimple quotient. Hence, it sufﬁces to construct an L-descent of this latter quotient of Gk . Let R be the smooth connected normal L-subgroup of GL generated by the G(k)-conjugates of the radical R((Ga )L ) as we vary a ∈ (G, T ). (This radical over L makes sense since L contains Ka for all a, and R is normal in GL since G(k) is Zariski-dense in GL due to the fact that k = ks .) We shall use R to descend the maximal adjoint semisimple quotient of Gk to a quotient of GL . Since R((Ga )k ) = (Ga )k ∩ R(Gk ) by (2) we see that Rk is contained in R(Gk ). Let us show that it sufﬁces to prove that R(Gk )/Rk is central in (GL /R)k . If this centrality holds then the central quotient (GL /R)/Z(GL /R) is semisimple and so over k it dominates the maximal adjoint semisimple quotient of Gk . That is, the quotient of (GL /R)/Z(GL /R) modulo its ﬁnite scheme-theoretic center (over L) is an adjoint semisimple quotient of GL that descends the maximal adjoint semisimple quotient of Gk , as desired. To prove the centrality of a smooth connected normal k-subgroup H of (GL /R)k (such as the radical) it sufﬁces to prove that H is centralized by the maximal k-torus image of Tk , since the conjugates of a maximal torus of (GL /R)k generate (GL /R)k (as GL /R is perfect, since G is). By Corollary A.8.11, a torus action on a smooth connected group is trivial if and only if the induced action on the Lie algebra is trivial. Thus, it remains to prove that Tk acts trivially on

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the Lie algebra of R(Gk )/Rk . Equivalently, for every nontrivial weight a for T acting on Lie(G) we claim that Lie(Gk )a ∩ Lie(R(Gk )) ⊆ Lie(Rk ). Since the a-root space of Lie(G) is contained in Lie(Ga ) (as Ga contains the a-root group Ua of G), we conclude that (Ga )k ∩ R(Gk ) = R((Ga )k ) = R((Ga )L ) ⊗L k ⊆ R ⊗L k = Rk and our claim follows from this since the formation of Lie subalgebras of subgroup schemes is compatible with (scheme-theoretic) intersections. Finally, we prove (3). The composite map across the top in (6.3.1) corresponds (under the universal property of Weil restriction) to the natural quotient map (Ga )K Ga , and iG likewise corresponds to the natural quotient map GK G = GKss . It therefore sufﬁces to prove that the diagram of K-groups (Ga )K

/ G a

GK

/ G

commutes, and this follows from the deﬁnition of Ga .

2

Let G be absolutely pseudo-simple over a ﬁeld k. By Theorem 5.3.8, the pseudo-reductive k-group G is standard over k if and only if ξG is surjective and ker ξG is central in G. Moreover, by Proposition 5.3.10, if char(k) = 2 then ker ξG is always central in G and hence G is standard if and only if ξG is surjective. We therefore wish to prove the surjectivity of ξG when char(k) = 2, subject to an extra hypothesis in characteristic 3. A key ingredient is the following proposition: Proposition 6.3.2 Let k be a separably closed ﬁeld, and G an absolutely pseudo-simple k-group whose root system is reduced. Let K/k be the ﬁeld of deﬁnition over k for the subgroup R(Gk ) = Ru (Gk ) in Gk . Fix a maximal k-torus T ⊆ G and for each a ∈ := (G, T ) let Ka ⊆ K be the ﬁeld of deﬁnition over k for the kernel of the projection from (Ga )k onto its maximal semisimple quotient, where Ga is generated by the root groups Ua and U−a . (1) If all root lengths in coincide then Ka = K for all a ∈ . (2) Assume there are two distinct root lengths and ξGa is surjective for all a. Then Ka = K for all a except possibly when k is imperfect with char(k) = p ∈ {2, 3} and G ss has an edge of multiplicity p in its Dynkin diagram. k In such exceptional cases, Kb = K for all short roots b and the ﬁelds

6.3 General cases away from characteristics 2 and 3

207

Ka for long a are all equal to a common subﬁeld K> ⊆ K satisfying kK p ⊆ K> ⊆ K. Before we prove Proposition 6.3.2, we make some remarks. Since all choices for T are G(k)-conjugate, in case (2) the subﬁeld K> ⊆ K is independent of T , so it is an intrinsic invariant of G. In particular, if we consider an arbitrary ﬁeld k and an absolutely pseudo-simple k-group G such that Gks satisﬁes the requirements for case (2) then (with K/k denoting the ﬁeld of deﬁnition of the geometric radical) we can deﬁne the subﬁeld K> ⊆ K over k by Galois descent of the subﬁeld (Ks )> ⊆ Ks . Also, Theorem 6.1.1 ensures that the surjectivity hypothesis in (2) holds when char(k) = 2 (in fact ξGa is always an isomorphism in such cases) and it will turn out that in characteristic 2 it is satisﬁed whenever [k : k 2 ] = 2 (by Proposition 9.1.10). Proof. Let G = GKss and let T be TK viewed naturally as a maximal K-torus of G . Since is assumed to be reduced, by Theorem 2.3.10 we have R(G, T ) = R(G , T ) as root data, compatibly with the identiﬁcation of character lattices and cocharacter lattices for T and T . The reducedness of the root system also implies that D (ZG (Ta )) coincides with the k-subgroup Ga considered in Proposition 6.3.1. By Corollary 3.4.3(2), if a pair of roots a, b ∈ have the same length then some n ∈ NG (T )(k) moves a to b and thus conjugates Ga into Gb . Hence, Ka = Kb inside of K in such cases. It follows that the subﬁelds Ka /k are all equal when there is only one root length, whereas in the case of two root lengths we get a pair of subﬁelds K< and K> in K over k (with K< = Ka for short a and K> = Ka for long a). Since the ﬁelds Ka generate K over k (Proposition 6.3.1(1)), it follows that the ﬁelds Ka are all equal to K in case all roots are of the same length and that K> and K< generate K over k in case there are roots of unequal lengths. This proves (1). Of course, if k is perfect then we always have K< = K> = K since K = k in such cases. To prove (2) we can now assume that there are two distinct root lengths, that ξGa is surjective for all a, and that k is imperfect. It sufﬁces to prove that K> ⊆ K< , and that the reverse inclusion holds except possibly when char(k) = p ∈ {2, 3} and there is an edge of multiplicity p in the Dynkin p diagram of , in which case kK< ⊆ K> ⊆ K< . The key point is to analyze some root groups for (G, T ). Choose any a ∈ . By Proposition 6.3.1, whose notation we now freely use, the ﬁeld of deﬁnition Ka /k for R((Ga )k ) ⊆ (Ga )k is contained in K and (Ga )ss K is identiﬁed with the K-subgroup Ga ⊆ G of rank 1 generated by the ±aK -root groups of G .

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Classiﬁcation results

Recall from Corollary 2.1.9 that for any surjective homomorphism between pseudo-reductive k-groups equipped with a compatible choice of maximal k-tori, Cartan k-subgroups are carried onto Cartan k-subgroups and root groups are carried onto root groups when the root systems are reduced. Thus, since the map ξGa in (6.3.1) is surjective by hypothesis, it carries the Cartan ksubgroup ZGa (T ∩ Ga ) onto the centralizer of R Ka /k (Ta ) in the k-subgroup ss D (R Ka /k (Ga )) of R K/k (Ga ) for the Ka -descent Ga = (Ga )ss Ka of Ga = (Ga )K in which (T ∩ Ga )Ka is identiﬁed with the Ka -descent Ta of the maximal K-torus T ∩ Ga of Ga . We claim that ξGa (ZGa (T ∩ Ga )) is the image of R Ka /k (aK∨a ) : R Ka /k (GL1 ) → R Ka /k (TKa ) → R K/k (TK ) = R K/k (T ). To prove this, we will separately treat the cases where Ga is simply connected or adjoint (i.e., isomorphic to SL2 or PGL2 ). First consider the simply connected case, so R Ka /k (Ga ) is its own derived group (since Ga SL2 ). Thus, ξGa (ZGa (T ∩Ga )) = R Ka /k (Ta ) inside of R K/k (T ∩Ga ), so we must prove that R Ka /k (Ta ) is the image of R Ka /k (aK∨a ) inside of R K/k (T ). Since we are in the simply connected case, the 1-parameter subgroup aK∨ : GL1 → T ∩Ga ⊆ T = TK has trivial kernel and so is a direct factor. Thus, likewise aK∨a : GL1 → TKa is a direct factor, so by left exactness of Weil restriction, the image of the map R Ka /k (aK∨a ) is obtained by applying R Ka /k to the image of aK∨a . This image is a∨ (GL1 )Ka ⊆ TKa , and since a∨ (GL1 ) = T ∩ Ga we get that a∨ (GL1 )Ka is the subtorus (T ∩ Ga )Ka which (by deﬁnition) goes over to Ta under the quotient map (Ga )Ka Ga . Hence, we get the desired result when Ga is simply connected. Suppose instead that Ga is adjoint. In this case Ga PGL2 = SL2 /μ2 , so if qa : Ga → Ga denotes the simply connected central cover and μ μ2 denotes the kernel of the covering map then Proposition 1.3.4 provides an identiﬁcation R Ka /k (Ga )/R Ka /k (μ) = D (R Ka /k (Ga )). ⊆ G denote the qa -preimage of the maximal Ka -torus T = Letting T a a a (T ∩ Ga )Ka ⊆ Ga , we have )/R K /k (μ). ξGa (ZGa (T ∩ Ga )) = R Ka /k (T a a

6.3 General cases away from characteristics 2 and 3

209

In this adjoint case the coroot a∨ : GL1 → T ∩ Ga is an isogeny between 1-dimensional tori and has kernel of order 2, so over Ka there is a unique factorization / T a BB BB BB BB qa ∨ aK a Ta

GL1

in which the horizontal map is an isomorphism for degree reasons. Thus, once again ξGa (ZGa (T ∩ Ga )) is the image of R Ka /k (aK∨a ). Now consider the action of ZGa (T ∩ Ga ) on the b-root group of (G, T ) for any b ∈ . This b-root group coincides with that of (Gb , T ∩ Gb ), which the surjective homomorphism ξGb : Gb D (R Kb /k (Gb )) carries onto R Kb /k (Ub ) where Ub is the b-root group in Gb that is a Kb -descent of the bK -root group Ub of (G , T ). (The intervention of the derived group may shrink Cartan subgroups, but preserves root groups.) Applying iG and using (6.3.1), we conclude that under the natural action of R K/k (GL1 ) on R K/k (Ub ) induced by aK∨ : GL1 → T ∨ and the action of R K/k (T ) on RK/k (G )(i.e., t.x = t b,a x), the k-subgroup R Ka /k (GL1 ) preserves R Kb /k (Ub ). Upon ﬁxing a Kb -group isomorphism Ub Ga we recover on Ub (K) = Ub (K) the natural K × -action arising from the natural 1-dimensional K-vector space structure on Ub (K), so the restriction to R Ka /k (GL1 )(k) = Ka× of the b, a∨ -power map on K × preserves the subgroup R Kb /k (Ub )(k) = Ub (Kb ) inside of Ub (K) = Ub (K). But this subgroup is a Kb -line, and the only K × -scalings on a 1-dimensional K-vector space which preserve a Kb -line are scalings by Kb× . Hence, the b, a∨ -power map on Ka× takes values in Kb× . Likewise, the a, b∨ -power map on K × carries Kb× into Ka× . Since there are distinct root lengths, there is a nonorthogonal pair {a, b} of simple positive roots (with respect to a ﬁxed ordering on ) with a long and b short. Then b, a∨ = −1, so K> ⊆ K< and hence K< = K. The edge joining the vertices corresponding to a and b in the Dynkin diagram has multiplicity 2 or 3, which is to say a, b∨ is equal to −2 or −3 respectively. Hence, K 2 ⊆ K> or K 3 ⊆ K> respectively. Since k is separably closed we therefore get K> = K except possibly when p = char(k) ∈ {2, 3} and the Dynkin diagram has an edge with multiplicity p, in which case kK p ⊆ K> ⊆ K. 2 Deﬁnition 6.3.3 Let k be a ﬁeld. A pseudo-reductive k-group G is exceptional if it is absolutely pseudo-simple and the following properties hold:

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Classiﬁcation results

(i) the root system of Gks is reduced, (ii) k is imperfect with char(k) = p ∈ {2, 3}, and the Dynkin diagram of has an edge of multiplicity p, (iii) for every a ∈ the map ξ(Gks )a is surjective, (iv) K> = K, where K/k is the ﬁeld of deﬁnition of the geometric radical of G. Note that when properties (i) and (iii) hold, Proposition 6.3.2 (and the remarks immediately following it) provides the existence of the ﬁeld K> used in (iv). There is some redundancy in this deﬁnition: properties (i) and (iii) always hold when p = 3 (by Theorem 2.3.10 and Theorem 6.1.1). Later in Proposition 9.1.10 we will prove that (iii) is redundant when p = 2 if [k : k 2 ] = 2 since we assumed reducedness of in (i). Proposition 6.3.2 implies that (iv) holds, i.e., K> = K, only if (ii) holds. The deﬁnition is formulated so that we can use it now for both values of p without logical dependence on results not yet proved when p = 2. Under hypotheses that always hold away from characteristic 2 and that will later be proved to hold in characteristic 2 when [k : k 2 ] = 2, the exceptional pseudo-reductive groups account for all absolutely pseudo-simple k-groups that are not standard and have a reduced root system: Theorem 6.3.4 Let G be an absolutely pseudo-simple group over a ﬁeld k, and let T be a maximal k-torus in G. Assume that (Gks , Tks ) is reduced and that ξ(Gks )a is surjective for all a ∈ (Gks , Tks ), where (Gks )a ⊆ Gks is the pseudo-simple ks -subgroup of rank 1 generated by the ±a-root groups. The pseudo-reductive k-group G is standard if and only if G is not exceptional. Proof. We may assume that k is separably closed. By reducedness of the root system and Proposition 5.3.10, standardness of G is equivalent to the surjectivity of ξG . Assuming that G is standard, we will prove that the inclusion Ka ⊆ K is an equality for all a ∈ (G, T ) (and hence G is not exceptional). Since G is )/Z pseudo-simple, standardness implies (by Remark 5.3.6) that G R K/k (G for a connected semisimple K-group G that is (absolutely) simple and simply connected, with Z a k-subgroup of R K/k (ZG T )/Z is ) such that C := R K/k ( . Note pseudo-reductive for some (equivalently, any) maximal K-torus T of G that the pseudo-reductivity of C is also a consequence of the pseudo-reductivity )/Z. Indeed, R K/k ( of the quotient G = R K/k (G T ) is a Cartan k-subgroup of R K/k (G ) (Proposition A.5.15(3)), so its image C is a Cartan k-subgroup of G (Proposition A.2.8), and Cartan k-subgroups of pseudo-reductive groups are

6.3 General cases away from characteristics 2 and 3

211

always pseudo-reductive (Proposition 1.2.4). The idea for proving Ka = K for all a in the standard case is to reduce to the case of rank 1. By Corollary 4.1.6, the choice of T corresponds canonically to a choice , of maximal k-torus T in G and we naturally have (G, T ) = (G T ). → G carries Beware that the canonical map G T onto T with nontrivial

kernel if G is not simply connected, so X(T ) is just a ﬁnite-index subgroup of X( T ). The precise relationship is that the maximal k-torus T in R K/k ( T ) maps isogenously onto T in the quotient G of R K/k (G ), since this quotient , ), T ) map is j as in Remark 5.3.6. We also have (G T ) = (R K/k (G via the equality X( T ) = X(T ) arising from the natural identiﬁcation of TK with T . , For each root a in the common root system of (G, T ), (G T ), and ), T ), deﬁne Ta , (R K/k (G Ta , and Ta to be the codimension-1 tori contained in the kernel of a in the respective groups. Since is reduced, the root groups U±a in G are the only ones with trivial Ta -action, and similarly for the root groups in G ) in R (relative to ±a ±a U Ta ) and the root groups R K/k (U K/k (G ) (relative ) → G carries R K/k (U ±a ) onto U±a to Ta ). Thus, the quotient map j : R K/k (G is simply connected, so G SL2 . Hence, (Remark 2.3.6) for all a. But G a (K), so R (K) is generated by its subgroups U ±a G K/k (G )a = R K/k (Ga ), and a this group is mapped onto Ga by j. This says that the pseudo-reductive subgroup Ga is standard with the same associated ﬁeld K/k in a standard presentation. Thus, the link between the parameters in standard presentations and the simply connected datum (Proposition 4.2.4) implies that K/k is the ﬁeld of deﬁnition for the maximal adjoint semisimple quotient of (Ga )k . By perfectness of Ga , this ﬁeld coincides with the ﬁeld of deﬁnition Ka /k of the geometric radical of Ga (Proposition 5.3.3). Thus we have proved that Ka = K for all a when G is standard. Now we assume G is not exceptional (so Ka = K for all a by Proposition 6.3.2) and prove that G is standard. Since we assume that all maps ξGa are surjective, it follows from the commutativity of (6.3.1) that for every a ∈ (G, T ) the image group ξG (Ga ), inside of R K/k (G ), is equal to R K/k (Ga ). The full image of ξG is therefore a smooth connected k-subgroup of R K/k (G ) that contains the k-subgroups R K/k (Ga ) as we vary through all a ∈ (G, T ) = (G , T ). Thus, it is enough to show that these k-subgroups of R K/k (G ) gen )/R K/k (μ) ⊆ R K/k (G ) of ξG , with μ denoting the erate the target R K/k (G → G = G ss . kernel of the simply connected central covering map π : G K −1 . For each a ∈ The preimage T = π (T ) is a maximal K-torus of G , in (G , T ) there is a unique root a ∈ (G T ) such that the root group U a relative to Ua . G T lies in π −1 (Ua ), and π restricts to an isomorphism U a

212

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) R K/k (Ua ). Thus, likewise R K/k (π ) restricts to an isomorphism R K/k (U a , The map a → a is a bijection between the root systems (G , T ) and (G T ), so it sufﬁces to observe that since R K/k (G ) is a perfect pseudo-reductive group ) (Corollary A.7.11), by Lemma 3.1.5 it is generated by the root groups R K/k (U a , for a ∈ (G T ). 2 Corollary 6.3.5 Let k be a ﬁeld with char(k) = 2, and G a pseudo-reductive k-group. If k is imperfect with char(k) = 3, assume G ss has no factor of type k G2 . Then G is standard over k. Proof. By Corollary 5.2.3 and Proposition 5.3.1 we may assume that k = ks and that G is (absolutely) pseudo-simple over k. Theorem 2.3.10 implies reducedness of the root system = (G, T ) (for any maximal k-torus T in G), and by Theorem 6.1.1 all maps ξGa (a ∈ ) are isomorphisms. Thus, by Theorem 6.3.4, a necessary condition for G to be non-standard is that k is imperfect and the Dynkin diagram of has an edge with multiplicity p = char(k) ∈ {2, 3}. But we assume char(k) = 2, and that if k is imperfect with characteristic 3 then the irreducible Dynkin diagram of is not G2 (the unique one with an edge of multiplicity 3). 2 We end with a standardness result that is only of interest over imperfect ﬁelds of characteristic 2, where it gives the best analogue of Corollary 6.3.5 (and, together with Corollary 6.3.5, ﬁnishes the proof of Theorem 5.1.1(1)). Proposition 6.3.6 Let k be a ﬁeld with char(k) = 3, and G a pseudo-reductive k-group such that the Dynkin diagram of G ss has no isolated vertices and no k edges with multiplicity 2. The pseudo-reductive k-group G is standard. Proof. We may assume that k is separably closed (Corollary 5.2.3), and we may also assume that G is not commutative and hence is non-solvable. By Proposition 5.3.1 we can assume that G is pseudo-simple (so D (G) = G and G ss is simple). Finally, by Proposition 6.2.1 we can assume that the simple k group G ss is not of type A2 or G2 , and by hypothesis it is not of type C1 = A1 k nor of type C2 = B2 . Hence, it has rank at least 3. Lemma 6.1.2 further reduces our problem to the case where the covering of G := G red = G ss is group μ for the simply connected central cover G K K inﬁnitesimal (hence of degree not divisible by 3, since char(k) = 3). Let T be a maximal k-torus in G, so T = TK may be viewed as a maximal K-torus of is a maximal K-torus of G G . The scheme-theoretic preimage T of T in G with T /μ = T in G /μ = G . Let = (G , T ) ⊆ X(T ) = X(T ), so has rank at least 3.

6.3 General cases away from characteristics 2 and 3

213

The list of options for the Dynkin diagram ruled out from consideration implies that no irreducible component is of type Cn (n 1), so the irreducible root system (G, T ) = of rank 3 is reduced and all roots are of the same length. In particular, the Weyl group W (G, T )(k) = W () (see Proposition 3.2.7) acts transitively on and for all a ∈ the ﬁeld of deﬁnition over k for the geometric radical of Ga is K (Proposition 6.3.2(1)). Moreover, reducedness of the root system implies that to prove standardness it is enough to show that ξG is surjective (Proposition 5.3.10). We ﬁx a positive system of roots + ⊆ , and choose a simple positive root a, as well as another simple positive root b that is adjacent to a in the Dynkin diagram associated to (, + ). Deﬁne the k-subtori Ta = (ker a)0red , Ta,b = ((ker a) ∩ (ker b))0red of T , and let Ga and Ga,b respectively denote the pseudo-simple derived groups of the centralizers ZG (Ta ) and ZG (Ta,b ). The root systems of (Ga , Ta ) and (Ga,b , Ta,b ) are of type A1 and A2 respectively (as a and b have the same length). Let Ka,b /k be the ﬁnite extension that is the ﬁeld of deﬁnition over k for the (unipotent) radical R((Ga,b )k ). Since (G, T ) is reduced, Ga can also be described as the k-subgroup of G generated by the root groups Ua and U−a (Proposition 3.4.1(2)). We can therefore determine Ga in exactly the same way using (Ga,b , T ∩ Ga,b ) in place of (G, T ), so Ka = Ka,b . Hence, Ka,b = K. The absolutely pseudo-simple group Ga,b is of type A2 so it is standard with ξGa,b an isomorphism by Proposition 6.2.1. Since char(k) = 3, the maps iGa,b and ξGa,b coincide (see Deﬁnition 5.3.5). The isomorphism property of iGa,b implies that the root groups in Ga,b have the form R K/k (Ga ). These root groups have dimension [K : k], and the ±a-root groups of (G, T ) coincide with those of both (Ga , T ∩ Ga ) and (Ga,b , T ∩ Ga,b ). Since these root groups of (Ga , T ∩ Ga ) are its only ones, the root groups of (Ga , T ∩ Ga ) have dimension [K : k]. Let ss Ga := (Ga )ss K = (Ga )K /R((Ga )K ), Ga,b := (Ga,b )K = (Ga,b )K /R((Ga,b )K )

denote the maximal semisimple quotients (over their minimal ﬁelds of deﬁnition over k) of (Ga )K and (Ga,b )K . Using Proposition A.4.8, we view Ga and Ga,b as subgroups of G with Ga ⊆ Ga,b . The map iGa : Ga → R K/k (Ga ) carries the ±a-root groups of Ga into those of R K/k (Ga ) (apply Remark 2.3.6 to (Ga )K

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Classiﬁcation results

Ga ), yet ker iGa = 1 due to the commutative diagram of natural maps / Ga,b

Ga iGa

R K/k (Ga )

iGa,b =ξGa,b

) / R K/k (Ga,b

But we have seen that the root groups of (Ga , T ∩ Ga ) have dimension [K : k], as do those of the target R K/k (Ga ), so the image of iGa contains the root groups of R K/k (Ga ). Consider the commutative diagram

iGa

Ga

/ G

R K/k (Ga )

iG

/ R K/k (G )

We just proved that the image of the map along the left contains R K/k (Ua ) for the root group Ua of (G , T ). Any a ∈ is simple positive for some + , → G restricts to an isomorphism between root groups and the natural map G (relative to T and T respectively), so it follows from the factorization ξG )/R K/k (μ) → R K/k (G /μ) = R K/k (G ) iG : G → R K/k (G

that the image of ξG contains the images of the Weil restrictions of all root , groups of (G T ). Thus, to deduce the surjectivity of ξG it sufﬁces to prove ) is generated by the Weil restrictions of the root groups of (G , that R K/k (G T ). As in the proof of Theorem 6.3.4, this follows from Corollary A.7.11 and Lemma 3.1.5. 2 Remark 6.3.7 Note that Corollary 6.3.5 and Proposition 6.3.6 imply Theorem 1.5.1.

PA RT III General classiﬁcation and applications

7 The exotic constructions

7.1 Calculations in characteristics 2 and 3 Over imperfect ﬁelds k of characteristics 2 and 3 there are pseudo-reductive groups that are not standard. Interesting classes of such groups (to be called exotic) were discovered by Tits, resting on special properties of the group G2 in characteristic 3 and the group F4 in characteristic 2, as well as groups of type Bn and Cn in characteristic 2 with n 2. A basic common feature of these types is that the unique pair of nonorthogonal positive simple roots {a, b} with distinct root lengths, say with a long and b short, satisﬁes a, b∨ = −p. In other words, the Dynkin diagram has an edge with multiplicity p. The deﬁnition of exotic pseudo-reductive k-groups requires a lot of preparation (essentially all of §§7.1–7.2), due to the intervention of subtleties related to the ﬁeld of deﬁnition over k of a Levi ks -subgroup. (See Example 7.2.2.) The case of G2 provides everything we need for a complete result in characteristic 3. Tits’ method for constructing exotic examples in characteristics 2 and 3 uses general axiomatic arguments with root systems that he sketched in [Ti2, §5] over a separably closed ﬁeld (and applied to the root systems of a split G2 in characteristic 3 and a split F4 in characteristic 2, as well as variants for types B and C in rank 2 in characteristic 2). We will use an alternative approach that is well suited to working with arbitrary k-forms of type G2 (resp. types F4 and Bn and Cn with n 2) for any imperfect k of characteristic 3 (resp. 2). This section largely focuses on the calculations with (forms of) these types that underlie the constructions to be given in §7.2. Let k be an arbitrary ﬁeld of characteristic p ∈ {2, 3}, and let G be a connected semisimple k-group that is absolutely simple and simply connected with Dynkin diagram having an edge with multiplicity p. We do not assume that G is k-split. Fix a maximal k-torus T in G. Let g = Lie(G) and t = Lie(T ), and let k be a separable algebraic extension of k such that T := Tk is k -split. Let 217

218

The exotic constructions

G = Gk . For each root c ∈ := (G , T ), let Uc be the associated root group in G , which is to say the unique T -stable smooth connected unipotent k -subgroup of G whose Lie algebra is the 1-dimensional c-root space gc in g = Lie(G ) = gk . There are two lengths for the roots in , and we call a root group Uc short (resp. long) when c is a short root (resp. long root). We also write < (resp. > ) to denote the set of short (resp. long) roots in . If + is a positive system of + + + roots then we deﬁne + < = ∩ < and > = ∩ > . We write for + the set of simple positive roots in , and < (resp. > ) to denote the set of short roots (resp. long roots) in . Since G is assumed to be simply connected, the coroots c∨ for c ∈ are a basis of the cocharacter group X∗ (T ). In particular, in such cases k -subgroups c∨ : GL1 → T for c ∈ have tangent lines Lie(c∨ (GL1 )) that are a k -basis of t := Lie(T ) = tk . (Recall that for any c ∈ , c∨ (GL1 ) is the unique 1 and lies in the SL generated dimensional k -torus that normalizes Uc and U−c 2 by Uc and U−c . This is essentially how the coroots c∨ are deﬁned.) Likewise, the roots c ∈ are a basis of the character group X(T /ZG ) of the k -split maximal k -torus T /ZG in the adjoint quotient G /ZG . In the speciﬁed characteristics something remarkable happens: Lemma 7.1.1 Let + be a positive system of roots in , and the basis of simple positive roots. (1) The k -lines Lie(c∨ (GL1 )) ⊆ tk for c ∈ < directly span a subspace that coincides with the span of the k -lines Lie(c∨ (GL1 )) for all c ∈ < . In particular, this subspace is independent of the ordering on . (2) For all a ∈ > and b ∈ < , a, b∨ ∈ pZ. (3) Under the isomorphism T /ZG c∈ GL1 deﬁned by t → (c(t)), the subgroup c∈< μp has preimage annihilated by precisely those roots in that are long. In particular, this k-subgroup is independent of the ordering on . By Galois descent, the k -subspace in (1) and the k -subgroup in (2) respectively descend to a canonical k-subspace in t and a k-subgroup in T [p] (independent of k /k); we respectively denote them as t< and μ< . Proof. We may assume k = k. Write < = {bj },

> = {ai }.

By [Bou2, VI, §1.2, Prop. 7] and the irreducibility of , up to scaling there is a unique W ()-invariant inner product (·, ·) on the vector space X(T )R . Fix a

7.1 Calculations in characteristics 2 and 3

219

choice of such an inner product, and let a0 ∈ > and b0 ∈ < be the unique long and short roots in that are non-orthogonal under this inner product (equivalently, they correspond to adjacent vertices in the Dynkin diagram). ∨ By inspection, in each case we have a0 , b∨ 0 = −p and b0 , a0 = −1, so (a0 , a0 ) = p(b0 , b0 ). It follows that (a, a) = p(b, b) for any a ∈ > and b ∈ < . We use the inner product to identify the dual spaces X∗ (T )R and X(T )R , and this carries c∨ to 2c/(c, c) for all c ∈ . In other words, it identiﬁes the inverse root system (X∗ (T )R , ∨ ) (terminology from [Bou2, VI, §1.1]) with the dual root system (X(T )R , {2c/(c, c)}c∈ ) (terminology from [Hum1, 4.2]). The positive system of coroots {c∨ }c∈+ has basis ∨ = {c∨ }c∈ , so the sets of simple positive short and long coroots are respectively given by ∨ ∨ ∨ < = (> ) := {ai },

∨ ∨ ∨ > = (< ) := {bj }.

Thus, if a ∈ > with unique expression a = mi , nj ∈ Z then a∨ =

mi ai +

nj bj for integers

2bj 2a 2ai mi · nj · mi ai∨ + (nj /p)b∨ = + = i . (a, a) (a, a) (a, a) i

j

i

j

But these coefﬁcients must be integral, so p|nj for all j; i.e., each long root has its < -coefﬁcients in pZ. The same argument applied to short roots gives that if b ∈ < with unique expression b = mi ai + nj bj then b∨ = ∨ ∨ ∨ pmi ai∨ + nj b∨ j . Hence, the element b ∈ > has < -coefﬁcients in pZ, ∨ ∨ so Lie(b (GL1 )) in characteristic p has no contribution from ∨ < = (> ) . This proves (1). Pushing the calculation further, we now prove (2). By introducing a sign if necessary, we can assume a, b ∈ + . We saw above that the expansion of a in has all < -coefﬁcients in pZ, so by linearity in a it sufﬁces to treat a ∈ > . Likewise, by considering the expansion of b∨ in ∨ we reduce to the case b ∈ < . In this case the pairing vanishes except when the corresponding vertices in the Dynkin diagram are adjacent, which is to say a = a0 and b = b0 , so a, b∨ = −p. Finally, to prove (3), let μ< ⊆ T [p] correspond to c∈< μp . By construction this k-subgroup of T [p] is killed by all elements of > . But we saw above that every a ∈ > has < -coefﬁcients in pZ, so a(μ< ) is generated by the images of μ< under > ; these latter images all vanish, so a(μ< ) = 1. It remains to show that if c ∈ < then c(μ< ) is nontrivial. Since NG (T )(k) acts transitively on the set of positive systems of roots in , and for some such choice the

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The exotic constructions

root c is a simple positive root, it sufﬁces to check for any g ∈ NG (T )(k) that the expansion of μ< in T [p] relative to the μp -decomposition indexed by the basis g() has vanishing components along (g())> (as then by consideration of order of the group scheme it must be c∈(g())> μp ). But this vanishing is a special case of the fact that μ< is killed by all long roots in . 2 With the maximal k-torus T in G ﬁxed, we let g< ⊂ g denote the k-descent of the k -span of the root spaces in g for roots in < and we likewise let g> ⊂ g denote the k-descent of the k -span of the root spaces in g for the roots in > . Both g< and g> are independent of k /k, and neither is a Lie subalgebra of g (since any pair of opposite root groups U±c generate an SL2 in the usual manner with diagonal c∨ (GL1 ), as G is simply connected, and the Lie bracket between opposite root spaces in sl2 is non-vanishing and valued in the diagonal subalgebra). There is a remarkable nontrivial k-isogeny π : G → G onto a simply connected k-form of a connected semisimple and absolutely simple group whose root system is isomorphic to the dual of the root system of G. For the convenience of the reader and to set some notation to be used later, we recall how this is constructed. The key point is to give a nontrivial factorization over k of the Frobenius isogeny FG/k : G → G (p) . This rests on the following lemma. Lemma 7.1.2 Every proper G-submodule of g under the adjoint action of G is contained in t ⊕ g< , and the subspace n = t< ⊕ g< ⊂ g is a G-stable p-Lie subalgebra of g. It is the unique irreducible G-submodule of g except for types Bn and Cn (n 2), in which case z := Lie(ZG ) = Lie(μ2 ) is the only other one. In type B (with rank at least 2) the center is contained in the coroot group c∨ (GL1 ) for any short root c, so z ⊆ t< in such cases. Also, for types F4 and G2 the subspace n is the unique nonzero proper G-submodule of g. Proof. We may assume that k = ks , and we ﬁrst address the uniqueness aspect. Since a G-stable subspace in g is a Lie ideal (as ad g is obtained by differentiating the adjoint representation of G; see Proposition A.7.5), we are aiming to ﬁnd nonzero proper G-stable Lie ideals h ⊂ g. The Weyl group W (G, T )(k) maps isomorphically onto the Weyl group of the irreducible and reduced root system, so NG (T )(k) acts transitively on all roots of a given length. Hence, by consideration of T -weight spaces it follows that a G-stable subspace of g must be a direct sum among g< , g> , and a subspace of t. Each pair of opposite root groups for (G, T ) generates an SL2 since G is simply connected, so each pair of root spaces in g for opposite roots generates an sl2 . Since sl2 is generated as

7.1 Calculations in characteristics 2 and 3

221

a Lie algebra by the root spaces for a pair of opposite roots, we conclude that if a Lie ideal h contains a root space then it contains the sl2 arising from such a root. In particular, if h = g then it cannot contain both g< and g> . Choose a positive system of roots and consider the unique pair {a, b} of nonorthogonal simple positive roots, with a long and b short. For the codimension-2 subtorus S = (ker a ∩ ker b)0red , the connected semisimple group D (ZG (S)) is of rank 2 with root datum of type B2 when p = 2 and of type G2 when p = 3 (as {a, b} is a basis of the root system and {a∨ , b∨ } is a basis of the coroot system). The Chevalley commutation relations among the root groups in these two cases [Hum2, §33.4(b), §33.5(b)] (where α is short and β is long!) give that the root spaces ga and gb have nonvanishing pairing under the Lie bracket, valued in the root space ga+b for the short root a + b. Thus, if a G-stable subspace of g contains g> then it contains g< . We conclude that a proper G-stable subspace h must be contained in g< ⊕ t. If h ⊆ t = Lie(T ) then by G(k)-conjugacy of all maximal k-tori we have h ⊆ T Lie(T ) = Lie( T T ). But G is generated by its maximal k-tori (as they generate a smooth connected normal k-subgroup), so T T = T ZG (T ) = ZG . In other words, in such cases h ⊆ z. This vanishes for F4 and G2 , and it is 1-dimensional for types B and C. Thus, all other h (if any exist) contain t< ⊕g< . It remains to prove that n := t< ⊕ g< is a G-stable p-Lie subalgebra. Since n is spanned by the Lie algebras of smooth k-subgroups of G, by functoriality of the p-operation and the universal Lie-theoretic formula for how this operation interacts with addition we see that n is stable under the p-operation provided that n is a Lie subalgebra of g. This will even be a Lie ideal if it is G-stable, so it remains to check G-stability of n. It sufﬁces to check stability under the actions of T and the root groups Uc for c ∈ . This is trivial for T . Choose c ∈ and c ∈ < , and ﬁx an isomorphism xc : Ga Uc . Consider the basis vectors Xc = dxc (∂u |u=0 ) ∈ Lie(Uc ) and Hc = dc ∨ (∂t |t=1 ) ∈ Lie(c ∨ (GL1 )). For u ∈ Ga , Ad(xc (u))(Hc ) = −uc, c ∨ Xc +Hc . Only long c need to be considered (due to how n is deﬁned), and then for long c and short c we have c, c ∨ ∈ pZ (Lemma 7.1.1). To handle the behavior of g< under the adjoint action of Uc on g for c ∈ , we have to study Uc -conjugation on Lie(Uc ) for c ∈ < . If c = ±c there is nothing to do since Uc and U−c generate an SL2 whose diagonal is c∨ (GL1 ) and hence has Lie algebra contained in n. Thus, we can assume c and c are independent, and hence we can choose + to contain c and c . It therefore

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The exotic constructions

sufﬁces to prove that the positive short root groups relative to a positive system of roots directly span a smooth connected k-subgroup U<+ ⊂ G. Since the collection of all positive root groups directly spans (in any order) a smooth connected (unipotent) k-subgroup of G, we just have to check that the commutator (Uc , Uc ), for c short, is contained in the direct product of the short positive root groups. If p = 3 then this follows from how the Chevalley commutation relations among positive root groups in G2 [Hum2, §33.5, (b)] specialize in characteristic 3 (e.g., certain positive short root groups commute only in this characteristic). If p = 2 then our problem is intrinsic to the central isogeny class of the rank-2 connected semisimple k-subgroup D (ZG (Tc,c )) for Tc,c = (ker c ∩ ker c )0red . If this rank-2 subgroup is of type A1 × A1 or A2 then there is nothing to do. The only other possibility to consider is that the subgroup is of type B2 = C2 , and we can again conclude by using how the Chevalley commutation relations among positive root groups in B2 [Hum2, §33.4, (b)] specialize in characteristic 2 (the two distinct positive short root groups commute, which never happens away from characteristic 2). 2 By Proposition A.7.14, there is a unique k-subgroup scheme N ⊂ ker FG/k whose Lie algebra is n inside of Lie(ker FG/k ) = g, and by Example A.7.16 it is normal in G (due to Lemma 7.1.2). Another characterization of N (due to Lemma 7.1.2) is that it is the unique non-central normal subgroup scheme of G with vanishing Frobenius and Lie algebra which is irreducible under the natural G-action. Concretely, in case k = ks (or more generally, if T is k-split) the k-group N is directly spanned by the kernels of the Frobenius isogenies (p) Uc → Uc for the short roots c and the Frobenius kernels ker Fc∨ (GL1 )/k for the short roots c (or even just short roots contained in a basis of the root system). Deﬁnition 7.1.3 With notation as above, the canonical quotient map π : G → G/N is the very special isogeny associated to the absolutely simple and simply connected k-group G. More generally, if k is a nonzero ﬁnite product of ﬁelds and G is a semisimple k-group whose ﬁbers are connected, absolutely simple, and simply connected with Dynkin diagram having an edge of multiplicity p then we deﬁne the very special isogeny for G by performing this construction on ﬁbers over Spec k. (We will generally write G to denote the quotient G/N , making sure to avoid confusion with the similar notation used to denote adjoint semisimple quotients of connected semisimple groups. This is the very special quotient of G.) The Frobenius isogeny FG/k : G → G (p) over the ﬁeld k has kernel that contains N (or equivalently, the Lie algebra Lie(ker FG/k ) = g contains n), so

7.1 Calculations in characteristics 2 and 3

223

we get a unique factorization π

π

G −→ G −→ G (p)

(7.1.1)

of FG/k through π . The uniqueness in Lemma 7.1.2 implies that up to unique isomorphism this is the only nontrivial factorization of FG/k into isogenies such that the ﬁrst step admits no nontrivial factorization into isogenies. Remark 7.1.4 In general, if G is a connected semisimple absolutely simple group over a ﬁeld k, an isogeny G → G is called special (in the sense of Borel and Tits) if the induced map g → g on Lie algebras does not factor through the G-coinvariants of g. (Equivalently, if we ﬁx a maximal k-torus T , the Lie algebra map is nonzero on the root space in gk for some element of (Gk , Tk ).) For example, every central isogeny is special and the Frobenius isogeny in nonzero characteristic is never special. By construction, the map π in Deﬁnition 7.1.3 is special in the sense of Borel and Tits, thereby explaining our terminology. Borel and Tits classiﬁed the special isogenies between absolutely simple semisimple groups, and noncentral ones only exist in characteristics 2 and 3. It is important that we determine the root datum for G. More precisely, if T is a k-split maximal k-torus of G and T := π(T ) denotes the associated k-split maximal k-torus of G under the very special isogeny then we want to relate := (G, T ) to := (G, T ) and determine where G sits within its central isogeny class (e.g., whether it is simply connected or adjoint). For each c ∈ (resp. c ∈ ) we denote by Uc (resp. U c ) the corresponding root group. Proposition 7.1.5 The root system is dual to that of , and G is simply connected. More precisely: (1) For each long root c ∈ > , the map π carries Uc isomorphically onto the root group U c for a short root c ∈ < satisfying c ◦ π = c in X(T ) and c∨ = π ◦ c∨ in X∗ (T ). (2) For each short root c ∈ < the map π restricts to a purely inseparable isogeny Uc → U c of degree p (i.e., it induces an isomorphism Uc / ker(FUc /k ) U c ) for a long root c ∈ > satisfying c ◦ π = pc in X(T ) and c∨ = (1/p)(π ◦ c∨ ) in X∗ (T ). (3) In the factorization (7.1.1), π is the very special isogeny of G. (4) The bijection → deﬁned by c → c carries the basis of a positive system of roots to the basis of the corresponding positive system of roots.

224

The exotic constructions

For a reduced root system R spanning a ﬁnite-dimensional R-vector space V and a choice of W (R)-invariant inner product (·, ·) on V (which is unique up to scaling factors associated to the irreducible components of R [Bou2, VI, §1.2, Prop. 7]), the dual root system is the subset {2a/(a, a)}a∈R in V . In the setting of the proposition, if we choose the inner product uniquely so that (a, a) = 2 for the long roots then (1) and (2) imply that c → c ◦ π identiﬁes with the dual of . Proof. Since π is bijective on geometric points and carries T onto T , by Lemma 3.2.1 it carries NG (T ) onto NG (T ) and induces an isomorphism W (G, T ) → W (G, T ). (In particular, π carries ZG (T ) onto ZG (T ).) Root groups relative to a split maximal torus are characterized as being the minimal nontrivial smooth connected unipotent subgroups normalized by the torus. Hence, for each c ∈ there is a unique c ∈ such that π(Uc ) = U c . Thus, c → c deﬁnes a bijection → . This bijection is deﬁned in terms of root groups, not in terms of the ﬁnite-index inclusion X(T ) → X(T ) of character lattices, so we are not claiming that this correspondence is well behaved with respect to the additive structure (and it generally is not). However, since π carries Borel subgroups onto Borel subgroups, and the set of Borel subgroups containing a given maximal torus is in natural bijection with the set of positive systems of roots in the corresponding root system (via consideration of the root groups contained in the choice of Borel subgroup), the bijection → carries positive systems of roots to positive systems of roots. The line Lie(Uc ) ⊆ Lie(G) is the c-root space, and Lie(ker π ) = ker(Lie(π)) = t< ⊕ g< by construction of π . Hence, if c is long then π : Uc → U c is an isomorphism. Likewise, if c is short then ker(π |Uc ) is nontrivial yet is inﬁnitesimal of height 1 (as this is true for ker π = N by construction), so it is the unique such k-subgroup of Uc Ga , namely αp . This has order p by inspection (or because Uc is smooth of dimension 1). Since the roots of a given length constitute a single Weyl orbit of roots, it follows that the map c → c either takes all short (resp. long) roots to short (resp. long) roots, or it maps short roots onto long roots and long roots onto short roots. Now we choose c ∈ and explicitly compute c◦π in terms of c and compute π ◦ c∨ in terms of c∨ . Fixing isomorphisms xc : Ga Uc and xc : Ga U c , π : Uc → U c becomes u → au for some a ∈ k × when c is long and u → au p for some a ∈ k × when c is short. The T -action on Uc is through c and the T -action on U c is through c, so by equivariance of π : Uc → U c with respect to π : T → T we get a · c(t)u = c(π(t)) · au for u ∈ Ga and t ∈ T when c is long, and a · (c(t)u)p = c(π(t)) · au p for u ∈ Ga and t ∈ T when c is short.

7.1 Calculations in characteristics 2 and 3

225

Hence, c ◦ π is equal to c when c is long and pc when c is short. In particular, −c = −c. To relate c∨ and π ◦ c∨ , we ﬁrst observe that in the subgroup Gc generated by Uc and U−c , c∨ (GL1 ) is the unique maximal k-torus normalizing U±c . Since π(Gc ) = G c , it follows that π(c∨ (GL1 )) = c∨ (GL1 ). Thus, π ◦ c∨ = qc · c∨ for some qc ∈ Q× . For long c we get 2 = c ◦ c∨ = c ◦ π ◦ c∨ = qc · c ◦ c∨ = 2qc , so π ◦ c∨ = c∨ for long c. Likewise, if c is short then 2 = c ◦ c∨ = (1/p) · (c ◦ π ◦ c∨ ) = (qc /p) · c ◦ c∨ = 2qc /p, so π ◦ c∨ = p · c∨ for short c. It follows from these formulas that is the dual root system to via the isomorphism X(T )R X(T )R induced by π that is equivariant with respect to the actions of the Weyl groups W () = W (G, T )(k) and W () = W (G, T )(k). Since the ratio of the squared root lengths in is p, it even follows from the explicit formula for c ◦ π in terms of c that c → c carries long roots to short roots and carries short roots to long roots. The duality of the root systems implies that the Dynkin diagrams for and are dual to each other, so if G has type G2 (resp. Bn , resp. Cn , resp. F4 ) then G has type G2 (resp. Cn , resp. Bn , resp. F4 ). In the case of types F4 and G2 it is automatic that G is simply connected. If G is simply connected of type Cn (n 2), to detect if its quotient G of type Bn is simply connected or adjoint + we will use the crutch of the unique short root c ∈ < in the Dynkin diagram + for . The key point is that a group of type Bn has center c∨ (μ2 ), so whether or not G is simply connected can be detected from the isomorphism type of the rank-1 group G c generated by U ±c : SL2 in the simply connected case and PGL2 in the adjoint case. If c ∈ > corresponds to c then π carries Gc onto G c and Lie(ker π ) meets the Lie algebra g−c ⊕ Lie(c∨ (GL1 )) ⊕ gc of Gc trivially (by weight space considerations with respect to T ). Hence, Gc → G c is an isomorphism, and Gc SL2 since G is simply connected. Thus, G is simply connected of type Bn when G is simply connected of type Cn . Now consider the factorization (7.1.1) with G simply connected of type Cn (n 2), or more generally any example for which the very special quotient G is known to be simply connected (i.e., all examples so far except possibly G of type Bn , n 3). We have shown that π carries long root groups isomorphically onto the short root groups with degree equal to p#< +#< , and the composite map in the diagram (7.1.1) has kernel equal to the k-group ker FG/k of order pdim G whose own Frobenius morphism vanishes. Thus, the other isogeny π : G →

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G (p) in (7.1.1) has degree p#> +#> = p#< +#< and kernel of height 1 that contains the root spaces for the short roots. The Lie subalgebra generated by root spaces for short roots contains t< in g, so we get a containment g< ⊕ t< ⊆ ker π that must be an equality for dimension reasons. Hence, π is the very special isogeny of G, so the very special quotient of G is G (p) , which is simply connected. Up to isomorphism there is only one connected semisimple absolutely simple group of type Bn that is split and simply connected, and we just realized it as a very special quotient G above. But in that calculation we also computed the very special isogeny of G and observed that its own very special quotient is simply connected. Thus, we can run the preceding calculation with G simply connected of type Bn to deduce that (7.1.1) exhibits π as the very special isogeny of G in the remaining case where G is of type Bn . 2 Remark 7.1.6 The very special isogeny from simply connected type Bn to simply connected type Cn in characteristic 2 is a very classical construction related to the defect of non-degenerate quadratic spaces (V , q) of odd dimension 2n + 1 for n 2: if V ⊥ ⊆ V is the 1-dimensional radical of the associated alternating bilinear form Bq (v, w) = q(v + w) − q(v) − q(w) then Bq induces a non-degenerate alternating form Bq on the 2n-dimensional space V = V /V ⊥ and the very special isogeny is the map Spin(q) → SO(q) → Sp(Bq ). To use the above results to construct some new pseudo-reductive groups over imperfect ﬁelds of characteristics 2 or 3, we need reﬁned information concerning the group-theoretic properties of the root groups relative to the long roots and short roots in (G, T ) when T is k-split. Proposition 7.1.7 Let k be a ﬁeld of characteristic p ∈ {2, 3} and let G be a connected semisimple group that is absolutely simple and simply connected with Dynkin diagram having an edge with multiplicity p. Let T be a maximal k-torus in G. (1) Assume that T is k-split. The long root groups of (G, T ) generate a ksubgroup G> ⊆ G containing T that is semisimple and simply connected, and the root groups of (G> , T ) are precisely the long root groups of (G, T ) with (G> , T ) = > . The short root groups of (G, T ) generate a ksubgroup G< ⊆ G containing T that is of the same type as G > , and its center equals ZG . The root groups of (G< , T ) are precisely the short root groups of (G, T ) with (G< , T ) = < . (2) The pair (> , ) is in the list (D4 , F4 ), (Dn , Bn ) for n 2, (A1n , Cn ) for n 2, and (A2 , G2 ).

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227

(3) In general, the ks -subgroups (Gks )> and (Gks )< of Gks arising from the ks -split torus Tks uniquely descend to respective k-subgroups G> and G< of G containing T . The Lie algebras are given by Lie(G> ) = t ⊕ g> and Lie(G< ) = t ⊕ g< . It is understood that D3 means A3 and D2 means A1 × A1 . We introduce G< because it is a useful device for the study of G> in certain cases, but in later applications we will only use G> . The fact that G> is always simply connected (in contrast with G< for G of certain types in characteristic 2) is useful. Proof. Assertion (3) follows via Galois descent from ks to k once (1) is proved, so for the proof of (3) it sufﬁces to focus on assertion (1). In particular, T is k-split (so G is k-split). The basic idea for proving that G> is no larger than expected (i.e., it contains no short root groups) is to indirectly construct G> by another procedure (resting on a technique of Borel and de Siebenthal). We then analyze G< by showing that G> covers G < via the very special k-isogeny π : G → G onto another k-split form G of the same type as G. Let μ< ⊆ (T /ZG )[p] be the canonical k-subgroup as in Lemma 7.1.1(3). Using the natural left action of G/ZG on G, we get an action of μ< on G. Let H = ZG (μ< )0 denote the identity component of the centralizer of the μ< -action on G. By Proposition A.8.10(2) applied over k, H is a connected reductive group. The k-split T is a maximal k-torus in H , so the connected reductive k-group H is k-split and its Lie algebra is the span of t and the root spaces gc for which c|μ< = 1 (Proposition A.8.10(3)). By Lemma 7.1.1(2), such c are precisely the long roots. We conclude that the root groups of H are the long root groups of G, so H = G> with (H , T ) the root system formed by the long roots in (G, T ). It is a classical fact that when is one of the types Bn (n 2), Cn (n 3), F4 , or G2 then > is respectively of type Dn , A1n , D4 , and A2 (with Dn interpreted as explained above when n = 2, 3). In all cases, the preimage M of μ< under T → T /ZG is a central subgroup scheme in H . Its order is #μ< · #ZG = p#< · #ZG . In case p = 3 the k-group G> is k-isomorphic to either the simply connected SL3 or the adjoint PGL3 , but the central subgroup M has order 3, so G> must be simply connected. In each case it is easy to check that M has order equal to the order of the center of the simply connected group of the same type as G> , so G> is always simply connected (with M equal to its center). Consider the very special isogeny π : G → G that carries T onto T . The map π carries long root groups onto short root groups and carries short root groups onto long root groups. The map π therefore induces a purely inseparable

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The exotic constructions

isogeny from G< onto G > and from G> onto G < . So we conclude that G< is a smooth connected semisimple k-subgroup of G containing T and its root system (G< , T ) equals < . To prove that the center of G< equals the center ZG of G, it sufﬁces to observe that the former certainly contains the latter, and to recall that the short roots in generate the root lattice so the center of G< cannot be bigger. 2

7.2 Basic exotic pseudo-reductive groups Building on the results in §7.1, we are now in position to present a twisted version of Tits’ examples of non-standard pseudo-reductive groups over imperfect ﬁelds with characteristics 2 and 3. Let k be a ﬁeld of characteristic p ∈ {2, 3} and let k /k be a ﬁnite extension with k p ⊆ k. Let G be a connected semisimple k -group that is absolutely simple and simply connected with Dynkin diagram having an edge with multiplicity p, and let π : G → G be the very special k -isogeny. Consider the k-map

f = R k /k (π ) : R k /k (G ) → R k /k (G ).

We wish to consider f -preimages of certain Levi k-subgroups of R k /k (G ). As a preliminary step, it is convenient to parameterize Levi k-subgroups of R k /k (G ). These are related to k-isomorphism classes of k-structures on G in the sense of the following general lemma. Lemma 7.2.1 Let k /k be a ﬁnite and purely inseparable extension of ﬁelds and let G be a connected reductive k -group. There is a natural bijection between the set of Levi k-subgroups of R k /k (G ) and the set of k-isomorphism classes of pairs (G, i) where G is a k-group and i : Gk G is a k -isomorphism. In particular, R k /k (G ) admits a Levi k-subgroup if and only if the k -group G admits a descent to a k-group. Each abstract k-group isomorphism between a pair of Levi k-subgroups of R k /k (G ) is induced by the action of a unique element of Aut k (G ). In particular, if G is k -split then Aut k (G ) acts transitively on the set of k-split Levi k-subgroups of R k /k (G ). This lemma becomes more concrete when G is semisimple and the Dynkin diagram for Gk has no nontrivial automorphisms, since then Aut k (G ) = s (G /ZG )(k ) = R k /k (G /ZG )(k) (by Corollary A.4.7(1)). Proof. By Theorem 1.6.2(2), the unipotent radical of R k /k (G )k has ﬁeld of deﬁnition over k equal to k . The maximal reductive quotient of R k /k (G )k is

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naturally identiﬁed with G , so if j : G → R k /k (G ) is a Levi k-subgroup then the natural map i : Gk → G corresponding (via the universal property of Weil restriction) to j is an isomorphism. Conversely, if G is an abstract k-group equipped with a k -isomorphism i : Gk G then the k-map G → R k /k (G ) corresponding to i has trivial kernel (as it is the composition of G → R k /k (Gk ) with R k /k (i)) and it is a Levi k-subgroup since k /k is purely inseparable. This establishes the desired bijection. It remains to check that any k-group isomorphism α between two Levi k-subgroups G and H of R k /k (G ) has the form R k /k (α ) for a unique α ∈ Aut k (G ). The inclusions of G and H as Levi k-subgroups induce k isomorphisms Gk G and Hk G , so a k-isomorphism α : G H induces a k -automorphism α of G . The induced k-automorphism of R k /k (G ) carries G into H via α, so it sufﬁces to show that the k-automorphism R k /k (α ) of R k /k (G ) uniquely determines α . Since the natural map q : R k /k (G )k G is the maximal reductive quotient and intertwines R k /k (α ) and α , the desired uniqueness follows from Proposition 1.2.2. 2 Example 7.2.2 Let k be an imperfect ﬁeld of characteristic p > 0, and k /k a ﬁnite purely inseparable extension. Let G be a k -form of SLp , GLp , or PGLp . We seek examples of such G that do not admit a descent to a k-group, since then Lemma 7.2.1 implies that the (standard) pseudo-reductive k-group R k /k (G ) has no Levi k-subgroup. Such examples demonstrate the need for the k-split hypothesis in Theorem 3.4.6. For any n > 1, the automorphism groups of SLn , GLn , and PGLn over a ﬁeld are given as follows: Aut(SL2 ) = Aut(PGL2 ) = PGL2 and in all other cases it is the semidirect product (Z/2Z) PGLn in which Z/2Z is generated by transpose-inverse. We now use the classical identiﬁcation of H1 (k, PGLn ) with the subset Br(k)ind|n ⊆ Br(k)[n] consisting of elements with index dividing n (where the index of a Brauer class is d when the central division algebra representative has dimension d 2 ). This carries the effect of transpose-inverse on H1 (k, PGLn ) over to inversion on Br(k)ind|n . Thus, the ﬁbers of the map Br(k)ind|n = H1 (k, PGLn ) → H1 (k, (Z/2Z) PGLn ) are Brauer classes up to inversion (using [Ser3, I, §5.5, Prop. 39(ii)]). Since the restriction map H1 (k, Z/2Z) → H1 (k , Z/2Z) is an isomorphism, by taking n = p we conclude that the existence of the desired examples would follow from the non-surjectivity of the restriction map Br(k)ind|p → Br(k )ind|p . (Note, in contrast, that Br(k) → Br(k ) is surjective since it is the map in degree-2 Galois

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The exotic constructions

cohomology induced by the maximal k-torus inclusion GL1 → R k /k (GL1 ) whose cokernel is unipotent and hence has vanishing H2 [Oes, IV, 2.2].) Consider the case k = k 1/p (assumed to be of ﬁnite degree over k). In this case the isomorphism k k deﬁned by t → t p carries the inclusion k → k over to the p-power map on k. Thus, the resulting isomorphism Br(k ) Br(k) carries the restriction map Br(k) → Br(k ) over to the p-power map on Br(k). In particular, Br(k)[p] is killed by restriction to Br(k ), so any central division algebra over k of rank p2 cannot arise from a k-algebra. For example, if k is a local or global function ﬁeld (so Br(k )ind|n = Br(k )[n] for any n 1, by local and global class ﬁeld theory) and k = k 1/p then local and global class ﬁeld theory for k ensure the existence of a central division algebra over k of rank p2 , so we obtain the desired G over k for which R k /k (G ) has no Levi k-subgroup. The following is another example. Let k be a local function ﬁeld and k be a quadratic extension of k. Let q be a non-degenerate quadratic form over k in 2n + 1 variables and of Witt index n − 1. Then the k -group G := SO(q ) does not admit a descent to a k-group. To prove this, we ﬁrst note that the existence of such a descent is equivalent to the existence of a quadratic form q over k such that q is a nonzero scalar multiple of q by extension of scalars to k . Since k is a local function ﬁeld, if a non-degenerate quadratic form over k in 2n + 1 variables is obtained by extension of scalars from a quadratic form over k then its Witt index is necessarily n. Thus, no such q exists, so G cannot be descended to a k-group.

Returning to the very special k -isogeny π : G → G , assume there is given a k-descent G of G , or equivalently (by Lemma 7.2.1) a Levi k-subgroup G ⊆ R k /k (G ). We want to relate k-structures on G and k-structures on G , or equivalently Levi k-subgroups of R k /k (G ) and Levi k-subgroups of R k /k (G ). This can go in two directions: given a k-structure on G can we make one on G with respect to which it is compatible via π , and given a k-structure on G is there one on G with respect to which it is compatible via π ? The easy (and less useful) direction is going from G down to G . Namely, if G is a k-structure on G then the very special k-isogeny π : G → G provides a unique k-structure G on G with respect to which the very special k -isogeny π descends to the very special k-isogeny π. In particular, G is naturally identiﬁed with a Levi k-subgroup of R k /k (G ). To go in reverse, we need to work with preimages under the k-homomorphism

f = R k /k (π ) : R k /k (G ) → R k /k (G ). It is the preimage f −1 (G) that will be Tits’ construction of exotic pseudoreductive groups, at least when there is a G compatible with G via π . The

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point is that if there is a compatible k-descent pair (G, G) as above then (by functoriality) f carries the Levi k-subgroup G ⊆ R k /k (G ) into the Levi k subgroup G ⊆ R k /k (G ) via the very special k-isogeny π , so if we are given G and seek to ﬁnd G then a natural idea is to look for a Levi k-subgroup of f −1 (G) (assuming f −1 (G) is smooth). But there is a serious difﬁculty with this preimage procedure: f is not sur jective when k = k, so for some choices of k-descent G of G the preimage f −1 (G) may not map onto G (and may not even be smooth). The reason that f is not surjective when k = k is that ker f = R k /k (ker π ) has dimension > 0 in such cases, since it contains a k-subgroup R k /k (μ ) whose ks -ﬁber contains the positive-dimensional R ks /ks (μp ) = R ks /ks (GL1 )[p]. In Proposition 7.3.1 it is proved that failure of f −1 (G) to map onto G is actually equivalent to non-smoothness of f −1 (G). Here is the main result concerning properties of f −1 (G) when there does exist a k-structure G on G compatible via π with the chosen k-structure G on G . Theorem 7.2.3 Let k be a ﬁeld of characteristic p ∈ {2, 3} and let k /k be a ﬁnite extension with k p ⊆ k. Let G be a connected semisimple k-group that is absolutely simple and simply connected with Dynkin diagram having an edge with multiplicity p. Let π : G → G be the very special isogeny, T a maximal k-torus of G, = (Gks , Tks ), and the basis of a positive system of roots in . Let f = R k /k (πk ) and let G = f −1 (G). (1) The k-group G is pseudo-reductive, and dim G = (#< + #< )[k : k] + (#> + #> ). (2) Let T = Tk be the associated maximal k -torus in G := Gk , and n ⊆ g as in Lemma 7.1.2. We have Lie(G ) = Lie(G> ) + nk

(7.2.1)

in Lie(R k /k (G )) = Lie(G ) = gk . If T is k-split then G is the k-subgroup of R k /k (G ) generated by G> and the Weil restrictions R k /k (Uc ) of the short root groups Uc of (G , T ). Also, still assuming T is k-split, the Cartan k-subgroup ZG (T ) is the smooth

232

The exotic constructions connected k-subgroup of R k /k (T ) generated by T and the k-subgroups R k /k (ck∨ (GL1 )) for c varying through short roots in (G, T ) = (G , T ). More precisely, if is a basis of (G, T ) then ZG (T ) =

a∨ (GL1 ) ×

a∈>

R k /k (b∨ k (GL1 )).

b∈<

(3) The natural map q : Gk → G corresponding to the inclusion G → R k /k (G ) is a smooth surjection with connected unipotent kernel. Roughly speaking, G is obtained from the k-group G by thickening the short root groups from k to k and thickening part of a maximal k-torus from k × to k × (using the k-subtorus arising from the coroots associated to short simple positive roots). In particular, if k = k then G = G, so this construction is only interesting when k = k. Proof. Observe that (2) over ks implies that G is smooth and connected with the expected dimension as in (1) (using (7.2.1)), so we may and do assume k = ks . We ﬁrst prove (2), then address the pseudo-reductivity claim in (1). Step 1. To prove (2), let H be the smooth connected k-subgroup of R k /k (G ) generated by G> and the k-subgroups R k /k (Uc ) for short c ∈ (G, T ). Our ﬁrst aim is to show H = G . By functoriality of the natural map X → R k /k (Xk ) for afﬁne k-schemes X , f carries G> into π(G> ) = G < ⊆ G. For each short root c ∈ (G, T ), say with c ∈ (G, T ) the associated long root such that π carries Uc onto U c , we claim that the k-map R k /k (Uc ) → R k /k (U c ) induced by f has image contained in U c . It sufﬁces to check this on k-points, where it becomes the assertion that the map Uc (k ) → U c (k ) induced by π has image contained in U c (k). Under suitable k-isomorphisms Ga Uc and Ga U c , the k-map π : Uc → U c becomes x → x p . Since k p ⊆ k by hypothesis, we get the desired containment. Thus, f (H ) ⊆ G, so we have an inclusion H ⊆ G as subschemes of R k /k (G ). For any long root c ∈ , π carries Uc isomorphically onto U c for some short root c ∈ . Hence, f carries Uc isomorphically onto U c as well. We have just seen that if c ∈ is short and c ∈ is the long root associated to c via π then f carries the smooth connected R k /k (Uc ) into U c Ga by a nonzero map, so f (R k /k (Uc )) = U c for dimension reasons. Since f carries T onto T , we conclude that f carries H onto G. To prove H = G , it remains to show

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that the k-subgroup ker f = R k /k (ker πk ) is contained in H . Step 2. The kernel of f is directly spanned (as a k-group scheme) by the k-groups R k /k (ker FUc /k ) ⊆ H , for c ∈ < , and the k-group R k /k (μk ) where μ is directly spanned by the Frobenius kernels of the k-tori c∨ (GL1 ) for short roots c in a basis of a positive system of roots over k. Hence, we just have to check that R k /k (μk ) ⊆ H . Due to the direct spanning description of μ, by treating the short simple positive roots c separately, this becomes a problem for SL2 over k: if D is the diagonal k-torus and U ± are the two nontrivial D-stable smooth connected unipotent k-subgroups, then we claim that R k /k (Uk+ ) and R k /k (Uk− ) generate a k-subgroup of R k /k (SL2 ) containing the k-group scheme R k /k (D[p]k ). In fact, they generate the entirety of R k /k (SL2 ) because on k-points of the Weil restrictions we get the k-subgroups U + (k ) and U − (k ) in SL2 (k ) that generate SL2 (k ) (a classical fact for any ﬁeld k ). This completes the proof that H = G .

Step 3. To prove the rest of (2), let G = G k and T = T k , with T = π(T ) ⊆ G. Since the centralizer of T in R k /k (G ) is a smooth k-subgroup and k is separably closed, consideration of k-points shows that this centralizer scheme coincides with R k /k (T ). Hence, the smooth k-group ZG (T ) is the schemetheoretic intersection G ∩ R k /k (T ). This is the scheme-theoretic preimage of G ∩ R k /k (T ) = T under the map

fT : R k /k (T ) → R k /k (T ). By decomposing T and T compatibly according to the coroots associated to a basis of a positive system of roots in (as we may do since G and G are simply connected), fT decomposes into a direct product of two kinds of nontrivial maps: the identity on R k /k (GL1 ) for the coroots c∨ with simple positive long c and the p-power map on R k /k (GL1 ) for the coroots c∨ with simple positive short c. The second of these two maps lands in the k-subgroup GL1 ⊆ R k /k (GL1 ) (since k p ⊆ k) and so it maps onto GL1 for dimension reasons. This gives the desired description of ZG (T ) = fT−1 (T ) for part (2).

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To complete the proof of (2), it only remains to determine the Lie algebra of G = f −1 (G). This Lie algebra is the preimage of the k-subspace g := Lie(G) ⊆ g under the k -linear map Lie(π ) : g → g arising from π = πk . To determine Lie(π )−1 (g) we ﬁrst observe that it is T -stable, so it admits a k-linear weightspace decomposition with respect to T . Hence, we can determine Lie(π )−1 (g) by working one weight at a time, as follows. For the trivial weight, under the map

tk = t → t = tk (with t = Lie(T ) and t = Lie(T )) the preimage of t is t + t< (as a Lie-theoretic consequence of our determination that fT−1 (T ) = ZG (T )). For each c ∈ > the map Lie(π ) carries gc isomorphically onto gc for a suitable c ∈ < . Likewise, for each c ∈ < the map Lie(π ) kills the entire weight space gc since π carries Uc onto U c by a map that is given in suitable coordinates by x → x p and hence vanishes on Lie algebras over k. This gives the expected components for Lie(π )−1 (g) in the weight spaces for the nontrivial weights of the T -action on the k-Lie algebra g , namely gc for c ∈ > and gc for c ∈ < . This completes the proof of (2). Step 4. To prove (1), since G is now known to be smooth and connected of the expected dimension it remains to prove that it is pseudo-reductive over k. Observe that G ⊆ G ⊆ R k /k (Gk ), with G a Levi k-subgroup of R k /k (Gk ) (since k /k is purely inseparable). Also, R k /k (Gk ) is a pseudo-reductive kgroup, by Proposition 1.1.10. Hence, the pseudo-reductivity of G follows from the following useful general result. Lemma 7.2.4 Let H be a pseudo-reductive group over any ﬁeld k, and L ⊆ H a Levi k-subgroup. Any smooth connected k-subgroup M ⊆ H containing L is also pseudo-reductive over k, with L a Levi k-subgroup of M and Ru (Mk ) = Mk ∩ Ru (Hk ). Proof. Since L is a Levi k-subgroup of H , the natural map Lk Ru (Hk ) → Hk is an isomorphism. But Lk ⊆ Mk , so likewise Mk = Lk (Mk ∩ Ru (Hk )).

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Hence, Mk ∩ Ru (Hk ) is smooth and connected, so it must be Ru (Mk ). This shows that L is a Levi k-subgroup of M , and that Ru (Mk ) ⊆ Ru (Hk ). Now we can apply Lemma 1.2.1 to X = Ru,k (M ) to deduce that M is pseudo-reductive over k. 2 Step 5. Finally, we will prove (3). Recall that the Levi k-subgroup G ⊆ R k /k (G ) is contained in G . Since f : G → G is surjective and the diagram Gk

q

fk

/ G

Gk

/

G

π

commutes, with π an isogeny, it follows that q is surjective. Thus, the map Gk /(ker q)0 → G with étale kernel (ker q)/(ker q)0 is an étale isogeny between smooth connected k -groups. It is therefore a central isogeny, and hence is an isomorphism since the connected semisimple k -group G is simply connected. In other words, ker q is connected. The kernel of q is unipotent because it is a k -subgroup of the unipotent kernel of the natural k -map R k /k (G )k G . To prove smoothness of q, it is equivalent to prove that Lie(q) is surjective. That is, the Lie subalgebra Lie(G ) ⊆ Lie(G ) over k spans Lie(G ) over k . This follows from (7.2.1) since G> contains a maximal k-torus of G. 2 Corollary 7.2.5 Let k be a ﬁeld of characteristic p ∈ {2, 3}, and let (G, k /k) be as in Theorem 7.2.3. Let G ⊆ R k /k (Gk ) be the associated pseudo-reductive k-subgroup as in Theorem 7.2.3. (1) The scheme-theoretic center ZG is equal to G ∩R k /k ((ZG )k ); in particular, G /ZG is naturally a k-subgroup of R k /k ((G/ZG )k ). (2) The center ZG is trivial if G is of type F4 or G2 . If G is of type Bn with n 2 or type Cn with even n 2 then ZG = R k /k ((ZG )k ) R k /k (μ2 ) inside of R k /k (Gk ). If G is of type Cn with odd n 3 then ZG = ZG μ2 inside of R k /k ((ZG )k ). (3) Assume there exists a k-split maximal k-torus T of G. For each long root c ∈ (G, T ), with associated root group Uc ⊆ G and 1-parameter subgroup c∨ : GL1 → T , we have G ∩ R k /k ((Uc )k ) = Uc , G ∩ R k /k (c∨ (GL1 )k ) = c∨ (GL1 ).

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The exotic constructions

Proof. To establish the description of ZG , we may and do assume k = ks . Letting T vary through the maximal k-tori of G, we get (by Theorem 7.2.3(2)) ZG ⊆ ∩T ZG (T ) ⊆ ∩T R k /k (Tk ) = R k /k (∩T Tk ) = R k /k ((∩T T )k ) = R k /k ((ZG )k ). This implies (1), so ZG = 1 when ZG = 1, settling (2) for types F4 and G2 . For type Bn with n 2 or type Cn with even n 2, the center ZG is a copy of μ2 diagonally embedded into a product of the k-groups c∨ (μ2 ) for certain short simple positive roots c relative to a ﬁxed choice of positive system of roots (the unique such c for type Bn , and n/2 of them for type Cn ). Since R k /k (c∨ (GL1 )k ) ⊆ G for short c, it follows that R k /k ((ZG )k ) ⊆ G in these cases, and this is clearly central in G (as it is even central in R k /k (Gk )). In contrast, for type Cn with odd n 3 the center ZG μ2 has nontrivial projection to the direct factor c∨ (GL1 ) of T for the unique long positive simple root c. Thus, once we establish that G ∩ R k /k (c∨ (GL1 )k ) = c∨ (GL1 ) then we will get the containment ZG ⊆ ZG = μ2 , with equality then being immediate. Now let T be a maximal k-torus of G and assume that T is k-split. Let T = π(T ) be the associated k-split maximal k-torus in G and for c ∈ (G, T )> let c ∈ (G, T ) be the short root such that π : G → G carries Uc isomorphically onto U c , so π −1 (U c ) = Uc × ker π . Hence, G ∩ (R k /k ((Uc )k ) × ker f ) = f −1 (G) ∩ f −1 (R k /k ((U c )k ) = f −1 (G ∩ R k /k ((U c )k )) = f −1 (U c ). But the map f restricts to π on Uc ⊆ G ⊆ R k /k (Gk ), so f −1 (U c ) = Uc ×ker f . Hence, G ∩ R k /k ((Uc )k ) = Uc . The analogous claim concerning c∨ (GL1 ) with a long root c was settled in the determination of fT−1 (T ) = ZG (T ) in the proof of Theorem 7.2.3 since R k /k (c∨ (GL1 )k )∩ker f = 1 (as we can check via Lie algebras and the triviality of étale multiplicative subgroups of ker π ). 2 In the proof of Theorem 7.2.3(1), we observed that the pseudo-reductive k-group G contains G as a Levi k-subgroup (and this Levi property was used

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237

to establish the pseudo-reductivity of G after we proved that G is smooth and connected). In Proposition 8.1.1 we will show that such G are never standard pseudo-reductive k-groups, and so over imperfect ﬁelds of characteristics 2 and 3 we will need to incorporate these k-groups into our classiﬁcation of pseudoreductive k-groups. Observe that these k-groups G (and their Weil restrictions) contain Levi k-subgroups (and not merely Levi ks -subgroups). To circumvent the fact that (in the non-pseudo-split case) pseudo-reductive groups over an arbitrary ﬁeld k can fail to have Levi k-subgroups (Example 7.2.2), we now introduce a class of non-standard pseudo-reductive k-groups in characteristic p ∈ {2, 3}, including the above k-groups G as well as variants, whose analysis only requires working with Levi subgroups over ks . The deﬁnition of this class of k-groups (Deﬁnition 7.2.6) will involve a condition that amounts to the existence of a Levi subgroup over ks . In view of the existence of Levi ks -subgroups in pseudo-reductive groups over ks (Theorem 3.4.6), imposing the requirement of the existence of a Levi subgroup over ks is not unreasonable. However, it is also not especially elegant. In Proposition 7.3.1 we establish an equivalence between this hypothesis and several other properties, some of which look more natural than the existence of a Levi ks -subgroup. The proof of this equivalence is somewhat technical, and its statement is never used in the rest of §7.2. Deﬁnition 7.2.6 Let k be an imperfect ﬁeld with characteristic p ∈ {2, 3}. A basic exotic pseudo-reductive k-group is a k-group G that is k-isomorphic to a k-group scheme of the form f −1 (G) where (i) f = R k /k (π ) for a nontrivial ﬁnite extension k /k satisfying k p ⊆ k and π denoting the very special k -isogeny G → G for a connected semisimple k -group G that is absolutely simple and simply connected with an edge of multiplicity p in its Dynkin diagram, (ii) the k-subgroup G ⊆ R k /k (G ) is a Levi k-subgroup, (iii) f −1 (G)ks contains a Levi ks -subgroup of R k /k (G )ks = R ks /ks (Gk ). s

We make a similar deﬁnition when k is a nonzero ﬁnite product of ﬁelds, treating the ﬁbers over Spec k separately. The Levi ks -subgroup in (iii) allows us (via Lemma 7.2.1) to apply Theorem 7.2.3 and Corollary 7.2.5 over ks to see that a k-group G as in Deﬁnition 7.2.6 really is pseudo-reductive (and in particular, smooth and connected) over k with center ZG = G ∩ R k /k (ZG ) (which is trivial precisely when G is of type F4 or G2 ), so G /ZG ⊆ R k /k (G /ZG ).

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Note that in Deﬁnition 7.2.6 we require k = k. This will be needed to prove several interesting properties of basic exotic pseudo-reductive k-groups, including that they are non-standard (and in particular, not reductive). By using Weil restriction one can give a more general construction of non-standard absolutely pseudo-simple pseudo-reductive k-groups, to be called exotic, as we will see in Deﬁnition 8.2.2. It is a very useful fact that a basic exotic pseudo-reductive k-group G as in Deﬁnition 7.2.6 uniquely determines both the pair (G , k /k) and the Levi k-subgroup G ⊆ R k /k (G ) that give rise to it, up to unique k-isomorphism. This is part of the following proposition. Proposition 7.2.7 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3}. Let k /k be a nontrivial ﬁnite extension with k p ⊆ k, and let G be a connected semisimple k -group that is simply connected and absolutely simple with an edge of multiplicity p in its Dynkin diagram. Denote its very special k -isogeny as π : G → G . Let G ⊆ R k /k (G ) be a Levi k-subgroup such that for f = R k /k (π ) the −1 preimage f (G)ks = fk−1 (G ks ) contains a Levi ks -subgroup of R k /k (G )ks = s R ks /ks (Gk ). Let G = f −1 (G) ⊂ R k /k (G ) be the associated basic exotic s pseudo-reductive k-group. (1) The ﬁeld of deﬁnition over k for the unipotent radical of Gk is equal to k (so G is not reductive), and if T is a maximal k-torus of G then (Gks , Tks ) = (Gk , Tks ) inside of X(Tks ) = X(Tks ). In particular, s (Gks , Tks ) is reduced. (2) The k -descent of the maximal reductive quotient of Gk is uniquely k isomorphic to the canonical k -map Gk → G corresponding to the inclusion G → R k /k (G ), and the image of G under the Weil restric tion R k /k (π ) of the very special k -isogeny π : G → G is the Levi k-subgroup G ⊆ R k /k (G ). (3) Let K/k be any separable extension ﬁeld. An arbitrary k-group H is a basic exotic pseudo-reductive group if and only if the K-group HK is a basic exotic pseudo-reductive group. In Corollary 8.1.3 we prove that such k-groups G are absolutely pseudosimple. Proof. By Proposition 1.1.9(2), the ﬁeld of deﬁnition Ek over k of the k-unipotent radical of a smooth afﬁne k-group is a ﬁnite purely inseparable extension of k whose formation commutes with any separable extension on k. Hence, (1) and (2) together imply (3) via Galois descent and specialization.

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239

To prove (1) and (2) we may assume that k is separably closed. In particular, by the hypotheses, the k-group G = f −1 (G) contains a Levi k-subgroup G of R k /k (G ). The identiﬁcation of root systems in (1) is immediate from (7.2.1). The natural map G → G induced by f must be the very special k-isogeny of G because over k it is identiﬁed with the very special k -isogeny G → G (due to Lemma 7.2.1 and the fact that G and G are Levi k-subgroups of Rk /k (G ) and Rk /k (G ) respectively). By Theorem 7.2.3(3), the natural map q : Gk → G adjoint to the inclusion G → R k /k (G ) is a k -descent of the maximal reductive quotient of Gk . Thus, our problem is to prove that the k -subgroup ker q ⊆ Gk has ﬁeld of deﬁnition over k equal to k (and then the rest is straightforward). It sufﬁces to show that the k -subspace Lie(ker q) = ker(Lie(q)) in Lie(Gk ) = k ⊗k Lie(G ) cannot be deﬁned over a proper subﬁeld of k containing k. The k -linear map k ⊗k Lie(G ) → Lie(G ) is induced by the k-linear inclusion Lie(G ) → Lie(G ) = k ⊗k Lie(G) whose image is the k-subspace described by (7.2.1) in terms of the weight spaces on g = Lie(G) relative to a choice of maximal k-torus T in G. Since a descent of the k -subspace Lie(ker q) to a K-subspace of K ⊗k Lie(G ) for an intermediate ﬁeld k ⊆ K ⊆ k must be equivariant for the action by TK , it sufﬁces to study the descent problem separately on the different weight spaces in k ⊗k Lie(G ). In particular, it sufﬁces to show that the canonical k -linear map k ⊗k gc → gc on gc = k ⊗k gc for a short root c ∈ (G, T ) has kernel whose (minimal) ﬁeld of deﬁnition over k is k . Upon choosing a k-basis of the line gc , this becomes the claim that the kernel of the multiplication map m : k ⊗k k → k has its (minimal) ﬁeld of deﬁnition K ⊆ k over k equal to k when using the k -linear structure on k ⊗k k from the left tensor factor. This is shown in the proof of Proposition A.7.8. 2 By Proposition 7.2.7(2), for G as in Deﬁnition 7.2.6 (so G is k-smooth and connected) the inclusion G → R k /k (G ) induces an isomorphism on maximal reductive quotients over k. Hence, a Levi ks -subgroup of Gks is the same as one of R k /k (G )ks = R ks /ks (Gk ) that lies in Gks (and likewise over k). In particular, s in condition (iii) in Deﬁnition 7.2.6 it is equivalent to require that G is a smooth connected k-group such that Gks admits a Levi ks -subgroup, which is a natural condition in view of Theorem 3.4.6.

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We prefer to think in terms of Levi ks -subgroups of R ks /ks (Gk ) rather than of s Gks because the former have a useful interpretation as explained in Lemma 7.2.1. Also, in Proposition 7.2.7(2) we saw that the Levi k-subgroup G in (ii) is canonically determined by the k-group G . Remark 7.2.8 Let G be a basic exotic pseudo-reductive group over an imperfect ﬁeld k of characteristic p ∈ {2, 3}, with (G , k /k, G) the corresponding triple of data underlying its construction. Assume k = ks , so the k-group G is k-split. The very special k -isogeny G → G = G k descends to the very special k-isogeny G → G using any Levi k-subgroup G in G . Let T be a maximal ktorus in G, so T := Tk is a maximal k -torus in Gk = G . Let Uc and Uc denote the corresponding roots groups in G and G for any c ∈ (G, T ) = (G , T ). Pick a positive system of roots + in (G , T ) = (G, T ) and let be the set of positive simple roots. The corresponding open Bruhat cell (as in Corollary 3.3.16) can be written down explicitly: by Theorem 7.2.3(2) and Corollary 7.2.5, it is the product scheme c∈− <

R k /k (Uc ) ×

Uc × ZG (T ) ×

c∈− >

R k /k (Uc ) ×

c∈+ <

Uc

(7.2.2)

c∈+ >

(embedded in G via multiplication) where ZG (T ) =

b∈<

R k /k (b∨ (GL1 )) ×

a∨ (GL1 ).

a∈>

7.3 Algebraic and arithmetic aspects of basic exotic pseudo-reductive groups In our work with basic exotic pseudo-reductive groups over imperfect ﬁelds of characteristics 2 and 3, there are several specialized results that are not logically relevant in the proof of the classiﬁcation theorems but are nonetheless of interest, so we record them here. Some are very useful in arithmetic applications (such as in [Con2]). Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3}. In the deﬁnition of a basic exotic pseudo-reductive k-group (Deﬁnition 7.2.6), we imposed a hypothesis (hypothesis (iii)) requiring the existence of a Levi ks -subgroup (for an auxiliary k-group). That deﬁnition was entirely sufﬁcient for our needs, but the hypothesis (iii) over ks may appear to be ad hoc. The following proposition establishes an equivalence between that hypothesis in Deﬁnition 7.2.6 and several alternative conditions that may seem more natural.

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241

Proposition 7.3.1 Let k be an imperfect ﬁeld with characteristic p ∈ {2, 3}, k /k a ﬁnite extension satisfying k p ⊆ k, and G a connected semisimple k -group that is absolutely simple and simply connected with an edge of multiplicity p in its Dynkin diagram. Let π : G → G

be the associated very special k -isogeny, and let f = R k /k (π ). Let G be a Levi k-subgroup of R k /k (G ). The following conditions on the k-subgroup scheme H = f −1 (G) ⊆ R k /k (G ) are equivalent: (a) H is k-smooth, (b) Hks has maximal dimension among the fk−1 (L) for Levi ks -subgroups L ⊆ s

R ks /ks (G ks ), (c) the natural map f : H → G is surjective, (d) G is contained in the image of f , (e) Hks contains a Levi ks -subgroup of R k /k (G )ks . When these ﬁve equivalent conditions hold, H = f −1 (G) is connected. Proof. We may and do assume that k is separably closed. The key to the proof is to show that im(Lie( f )) ∩ Lie(G) has dimension independent of G. To show this, we will ﬁrst check:

Lemma 7.3.2 The Lie ideal im(Lie( f )) in Lie(R k /k (G )) is stable under the natural action of R k /k (G /ZG ). Proof. Consider the unique nontrivial factorization in k -isogenies π

π

G → G → G

(p)

of the Frobenius isogeny FG /k of G . By Proposition 7.1.5(3), π is the very special isogeny of G . Moreover, the proof of that result showed that the dimension of ker(Lie(π )) = Lie(ker π ) is equal to the dimension of im(Lie(π )) so the containment im(Lie(π )) ⊆ ker Lie(π ) inside of Lie(G ) is an equality. Viewing these k -vector spaces as k-vector spaces, this says that im(Lie( f )) is the Lie algebra of the kernel of the map R k /k (π ), so the Lie ideal im(Lie( f )) in Lie(R k /k (G )) is stable under the natural action of R k /k (G /ZG ), as desired. 2

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To deduce that the k-dimension of im(Lie( f )) ∩ Lie(G) is independent of the choice of G, it now sufﬁces to show that all choices of G are conjugate to each other under the natural action of R k /k (G /ZG )(k) on Rk /k (G ). Such

conjugacy follows from Lemma 7.2.1 (since G is a k -split form of Bn (n 2), Cn (n 2), F4 , or G2 , as k is separably closed). In general dim H dim Lie(H ) with equality if and only if H is smooth, and dim Lie(H ) = dim Lie( f )−1 (Lie(G)) = dim Lie(ker f ) + dim(im(Lie( f )) ∩ Lie(G)),

where the ﬁrst equality uses that the formation of Lie algebras commutes with ﬁber products. Thus, dim Lie(H ) is independent of G due to the result proved above. Since k is separably closed, G is k -split, so it admits a k-descent G. Hence, by Theorem 7.2.3, there exists a Levi k-subgroup G in the image of f such that H = f −1 (G) is smooth. Thus, for this H we have dim H = dim Lie(H ), so the equivalence of (a) and (b) follows. Since we have just shown that, when varying the Levi k-subgroup G in R k /k (G ), the containment G ⊆ im( f ) does sometimes occur, the maximal value attained by dim H = dim ker f + dim(im( f ) ∩ G) as we vary G is obtained precisely when G ⊆ im( f ). In other words, (b) and (d) are equivalent. Since H = f −1 (G), we see that (c) and (d) are equivalent. The Levi k-subgroup G ⊆ H in condition (e) (with k = ks !) is a k-descent of G such that the map π : G → G descends π , so it identiﬁes the k-group H = f −1 (G) with an instance of the k-group G as in Theorem 7.2.3. Hence, Theorem 7.2.3(1) gives that (e) implies (a), and also that H is connected when (e) holds. To conclude the proof, we need to show that (d) implies (e). We have put ourselves in the case where k is separably closed, so we have to show that if (d) holds with k separably closed then the k-group H = f −1 (G) contains a Levi k-subgroup of R k /k (G ). Since (d) implies (a), H is smooth. Since (d) implies (c), the natural map H → G is surjective and hence the natural map Hk → G is also surjective. Fix a maximal k-torus T in H , so T is also a maximal torus of R k /k (G ). It follows that the image T of T in G is a maximal k-torus and T := Tk is a maximal torus of the maximal reductive quotient G of R k /k (G )k . The identity component H 0 of H is pseudo-reductive over k because H 0 maps onto G with kernel containing no nontrivial k-points.

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243

By Corollary A.7.6 we have (R k /k (G ), T ) = (G , T ) (via the equality X(T ) = X(T )). The surjectivity of the natural map Hk → G implies that the containment of root systems (H 0 , T ) ⊆ (G , T ) is an equality (by Remark 2.3.6 and the reducedness of (G , T )). Using the equality of root systems (H 0 , T ) = (R k /k (G ), T ), we can choose a common positive system of roots and let be the resulting basis of simple positive roots. For each a ∈ , let Ea be a 1-dimensional k-linear subgroup of the a-root group Ua of (H 0 , T ). By Theorem 3.4.6, there is a unique Levi k-subgroup L of H 0 containing T such that L contains Ea for each a ∈ . Explicitly, if we choose ua ∈ Ua (k) − {1} for each a ∈ then there is a unique va ∈ U−a (k) such that na := va ua va normalizes T and induces the reﬂection ra on X(T ), and La is generated by T and the k-subgroups {Ea , na Ea n−1 a }a∈ . In view of the uniqueness aspects of this description, the containment of U±a in the ±a-root group for (R k /k (G ), T ) for each a ∈ implies that L coincides with the unique Levi k-subgroup of R k /k (G ) containing T and Ea for each a ∈ . Thus, H contains a Levi k-subgroup of R k /k (G ). 2 In arithmetic applications, a very useful feature of standard pseudo-reductive groups is that the non-commutative aspect of a standard presentation rests on a simply connected semisimple group. It was proved by Harder (resp. Bruhat– Tits) that connected semisimple simply connected groups over global function ﬁelds (resp. local function ﬁelds) have vanishing degree-1 Galois cohomology. For basic exotic pseudo-reductive groups, there is a related vanishing theorem that plays an equally essential role when proving general theorems for algebraic groups over local or global function ﬁelds of characteristics 2 or 3. The key point is a consequence of Proposition 7.2.7 that allows us, in certain contexts, to replace basic exotic pseudo-reductive groups over local or global function ﬁelds of characteristics 2 and 3 with forms of simply connected semisimple groups over the same ground ﬁeld: Proposition 7.3.3 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3} such that [k : k p ] is ﬁnite. Let G be the basic exotic pseudo-reductive k-group arising from a triple (G , k /k, G) with k = k 1/p . (1) The natural map G (k) → G(k) is an isomorphism of groups and the natural map H1 (k, G ) → H1 (k, G) is bijective. (2) If k is complete with respect to an absolute value then the natural map G (k) → G(k) is a topological isomorphism. (3) If k is a global function ﬁeld then the natural map G (Ak ) → G(Ak ) on adelic points is a topological isomorphism.

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The exotic constructions

Note that if [k : k p ] = p then every basic exotic pseudo-reductive k-group arises as in the above proposition. This applies when k is a local or global function ﬁeld of characteristic p. Before we take up the proof of Proposition 7.3.3, it is convenient to ﬁrst review the description of open Bruhat cells (Corollary 3.3.16) in basic exotic pseudo-reductive groups over imperfect ﬁelds k of characteristic p. Let k be such a ﬁeld, and k /k a nontrivial ﬁnite extension such that k p ⊆ k. Let G be a connected semisimple k -group that is absolutely simple and simply connected of type Bn or Cn with n 2, F4 when p = 2 and of type G2 when p = 3, with associated very special k -isogeny π : G → G . Let f = R k /k (π ) and let G ⊆ R k /k (G ) be a Levi k-subgroup such that f −1 (G)ks contains a Levi ks -subgroup of R k /k (G )ks , so G := f −1 (G) is a basic exotic pseudo-reductive k-group. Assume that G contains a Levi k-subgroup G. This is a property that holds after some ﬁnite separable extension on k (since it holds over ks ). Under this assumption, the map π : G → G induced by f must be the very special k-isogeny of G and more speciﬁcally over k it is identiﬁed with π . Impose the further hypothesis that G is k-split, so we may choose a k-split maximal k-torus T in G. Let T = π(T ) be the corresponding k-split maximal + k-torus in G. Let + be a positive system of roots in = (G, T ), and let be the corresponding positive system of roots in = (G, T ). Let V ⊆ G be the open Bruhat cell of G associated to T and the choice of + . By calculating as in the determination of where G meets thickened root groups and tori in the proof of Corollary 7.2.5, the open subscheme R k /k (Vk ) ⊆ R k /k (Gk ) meets the closed k-subgroup G ⊆ R k /k (Gk ) in the open subscheme V ⊆ G given by (7.2.2) in Remark 7.2.8 (where we assumed k = ks , which is not necessary under the above hypotheses). There is an analogous open subscheme V of G given by the open Bruhat cell Uc × T × U c. c∈

−

+

c∈

The k-map f = R k /k (πk ) carries V into the open Bruhat cell V of G compatibly with the product decompositions (with long roots of going over to short roots of and short roots of going over to long roots of ). It is also compatible with product decompositions of ZG (T ) and T according to the simple positive coroots since the bijection → carries the basis of + to the basis + of (Proposition 7.1.5(4)). The open locus V is the open Bruhat cell in the pseudo-reductive k-group G equipped with the choice of k-split maximal k-torus T ⊆ G ⊆ G and the choice of + .

7.3 Algebraic and arithmetic aspects

245

Proof of Proposition 7.3.3 The hypotheses and the formation of G and (G , k /k, G) are compatible with separable algebraic extension on k. Thus, once it is proved that G (k) → G(k) is bijective in general, the natural Gal(ks /k)equivariant map G (ks ) → G(ks ) is bijective. Hence, H1 (k, G ) → H1 (k, G) is bijective. For part (1) we can therefore focus on the group-theoretic aspects of the assertion. By Galois equivariance, to prove that the map G (k) → G(k) is an isomorphism of abstract groups or (in the case of ﬁelds complete with respect to an absolute value) a topological isomorphism, it is harmless to replace k with a ﬁnite separable extension. Thus, we can assume that the k-group G = f −1 (G) contains a Levi k-subgroup G (and hence G is canonically identiﬁed with Gk ). By making a further ﬁnite separable extension on k we can assume that G is k-split. Fix a k-split maximal k-torus T in G and a positive system of roots + in (G, T ), so we get an associated k-split maximal k-torus T in the quotient G + and an associated positive system of roots ⊆ (G, T ). In terms of this data, construct both the open Bruhat cells V ⊆ G and V ⊆ G as above. We shall now prove that the map V (k) → V (k) is bijective in general and a homeomorphism when k is complete with respect to an absolute value. We may work factor by factor in the product decompositions of V and V and we may treat long roots in (G, T ) separately from short roots. On some factors the map V (k) → V (k) induces the identity on k or k × , and on other factors it is the p-power map from k to k or from k × to k × . These p-power maps are bijective since k = k 1/p , and they are trivially homeomorphisms when k is topologized by an absolute value. In such cases, R k /k (Ga )(k) = k is endowed with its topology as a ﬁnite-dimensional kvector space and this coincides with the valuation topology on k when k is complete. For an arbitrary imperfect ﬁeld k of characteristic p such that k = k 1/p has ﬁnite degree over k we now prove that the map of groups G (k) → G(k) is bijective. It is injective because the kernel is a subgroup of (ker f )(k) ⊆ (ker π )(k ) and π is a purely inseparable isogeny. To prove surjectivity, ﬁrst note that the preceding bijectivity result for the map V (k) → V (k) implies that for the positive Borel k-subgroup B ⊂ G the abstract subgroup B(k) ⊂ G(k) lies in the image of G (k). The map W (G, T ) → W (G, T ) between Weyl groups induced by π is an isomorphism, so each w ∈ W (G, T )(k) = NG (T )(k)/T (k) is represented by the image of some point of NG (T )(k) ⊆ G(k) ⊆ R k /k (Gk )(k). But G = f −1 (G), so f (G (k)) contains a set of representatives for W (G, T )(k). Such representatives together with B(k) generate the group G(k), by the Bruhat decomposition for the k-split G, so we obtain the desired surjectivity on kpoints. When k is complete with respect to an absolute value we have shown

246

The exotic constructions

that the continuous bijective map on k-points restricts to a topological isomorphism between open neighborhoods of the identity points, so it is a topological isomorphism globally. Finally, we assume that k is a global function ﬁeld of characteristic p and we check that the continuous map G (Ak ) → G(Ak ) on adelic points is a topological 1/p isomorphism. For any place v of k, k 1/p ⊗k kv = kv . Thus, the natural map G (kv ) → G(kv ) is a topological isomorphism, so v G (kv ) → v G(kv ) is one as well. There is a ﬁnite non-empty set of places of k such that the k-map G → G extends to a map G → G between smooth afﬁne group schemes over the ring Ok, of -integers in k. Our problem is to prove that for all but ﬁnitely many places v ∈ , the induced map G (Ov ) → G (Ov ) between compact open subgroups of G (kv ) and G(kv ) is bijective. Since we have a topological isomorphism on kv -points for every v, the problem is to prove surjectivity of the map on Ov -points for all but ﬁnitely many v ∈ . By Galois equivariance, it sufﬁces to establish this after ﬁrst making a ﬁnite separable extension on k (and replacing with the set of places over it in the ﬁnite separable extension), so we can assume after perhaps also increasing that G is a Chevalley group over Ok, and that G contains a closed Chevalley Ok, -group G whose quotient via its very special isogeny (obtained over Ok, by base change from Fp , and uniquely characterized as being the very special isogeny between the generic ﬁbers over k) is identiﬁed with G . Moreover, we can assume (after possibly increasing ) that G is a closed subgroup scheme of R Ok , /Ok, (G ⊗Ok, Ok , ) respecting the canonical isomorphism G Gk and inclusion G → R k /k (G ) on generic ﬁbers. In other words, we have spread out the entire situation over k to one over Ok, , including split maximal tori and Weyl groups. We therefore have a “Bruhat decomposition” of G (Ov ) for all v ∈ , using double cosets for an Iwahori subgroup in the role of a Borel subgroup and using the Weyl group as the index set for the decomposition. Since there is no concept of Iwahori subgroup for G (Ov ) arising from G , it will be convenient to replace this decomposition of G (Ov ) with an integral open cell and to suppress the use of an Iwahori subgroup. To be precise, since an Iwahori subgroup of G (Ov ) relative to an Ov -split maximal Ov -torus T v of G ,v is contained in the Ov -points of an open Bruhat cell over Ov (determined by a choice of positive system of roots), the abstract group G (Ov ) is generated by the Ov -points of the open cell and any Ov -point representatives for the elements of the Weyl group W (G ,v , T v )(Ov ).

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The proof of surjectivity of G (kv ) → G (kv ) can therefore easily be carried over to work with Ov -points (using the open cell in place of a Borel subgroup). This modiﬁed argument works integrally since G (Ov ) → G (Ov ) induces a bijection between “open cells” (relative to compatible Ov -split maximal Ov tori and compatible positive systems of roots) since the p-power isomorphism kv kv for any place v of k and the unique place v of k over v induces an isomorphism between the valuation rings. 2 A basic exotic pseudo-reductive k-group G uniquely determines, up to unique k-isomorphism, the triple (G , k /k, G) of data that gives rise to it (Proposition 7.2.7), so it is natural to try to relate properties of G and G . Here is one such relationship, assuming k = k 1/p . Corollary 7.3.4 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3} such that [k : k p ] is ﬁnite. Let G and (G , k /k, G) be as in Deﬁnition 7.2.6 with k = k 1/p , and let f : G → G denote the canonical surjection. If T ⊆ G is a maximal ktorus (resp. maximal k-split k-torus) then there exists a unique maximal k-torus (resp. maximal k-split k-torus) T ⊆ G such that T ⊆ f −1 (T ), and T → T is an isogeny. In particular, there is a natural bijection between the set of maximal k-tori (resp. maximal k-split k-tori) in G and G, and G is k-isotropic (resp. pseudosplit over k) if and only if G is k-isotropic (resp. k-split). Proof. By Galois descent, we may and do assume that k = ks . Since f : G → G is surjective, it carries maximal k-tori onto maximal k-tori. By Proposition 7.3.3(1) we have G (k) G(k) via f since k = k 1/p , so by the conjugacy of maximal k-tori in smooth afﬁne k-groups (Proposition A.2.10) each maximal ktorus T of G is the image of one in G . It remains to show that f −1 (T ) contains a unique maximal k-torus of G . Since ( f −1 (T ))(k) is commutative (as it is isomorphic to a subgroup of T (k)), the desired uniqueness follows. 2 A variant on the construction in Corollary 7.3.4, using maximal tori in G rather than in G, works for any imperfect ﬁeld k of characteristic p ∈ {2, 3} and nontrivial ﬁnite extension k /k with k p ⊆ k; see Proposition 11.1.3. A defect of Corollary 7.3.4 is that it imposes restrictive hypotheses on [k : k p ] and k . However, these hypotheses are always satisﬁed in arithmetic cases (local and global function ﬁelds) and under such hypotheses there are no further constraints involving descent of ﬁelds of deﬁnition of tori as in Proposition 11.1.3. It is for this reason that Corollary 7.3.4 is easier to use than Proposition 11.1.3, at least when its hypotheses are satisﬁed. The following result relates k-forms of G to k-forms of G when k = k 1/p .

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Proposition 7.3.5 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3} such that [k : k p ] is ﬁnite. Let G1 and G2 be basic exotic pseudo-reductive k-groups such that the associated triples (G1 , k1 /k, G 1 ) and (G2 , k2 /k, G 2 ) satisfy k1 = k2 = k 1/p . Let fi : Gi → G i be the canonical surjection. (1) For every k-isomorphism α : G1 G2 there is a unique k-isomorphism α : G 1 G 2 such that f2 ◦ α = α ◦ f1 . The resulting map Isomk (G1 , G2 ) → Isomk (G 1 , G 2 ) is bijective. (2) Assume [k : k p ] = p, and let denote one of the following types: G2 if p = 3, and Bn (n 2), Cn (n 2), or F4 if p = 2. The assignment G → G is a bijection from the set of k-isomorphism classes of basic exotic pseudo-reductive k-groups of type to the set of k-isomorphism classes of absolutely simple simply connected k-groups of type ∨ . Notable cases for which the hypothesis in (2) is satisﬁed are local and global function ﬁelds. Thus, for such k the basic exotic pseudo-reductive k-groups of type Bn (resp. Cn ) for n 2 are classiﬁed up to k-isomorphism by the the classiﬁcation of k-forms of simply connected groups of type Cn (resp. Bn ) [Spr, 17.2]. Proof. To prove (1), by Galois descent we may and do assume k = ks , so each Gi contains a Levi k-subgroup Gi and the map Gi → G i induced by fi is the very special isogeny for Gi . Since G1 and G2 are absolutely simple and simply connected semisimple k-groups of the same type and k = ks , there exists a k-isomorphism G1 G2 over a k-isomorphism G 1 G 2 and hence a k-isomorphism G1 G2 over a k-isomorphism G 1 G 2 . For the proof of (1) it therefore sufﬁces to work with a single G (corresponding to a triple (G , k /k, G)) and study k-automorphisms of G and G; this makes the notation a bit simpler. Let G ⊂ G be a Levi k-subgroup, so f |G : G → G is the very special isogeny for G. We claim that the reductive quotient mapping f : G → G is maximal over k in the sense that any surjective k-homomorphism G → H onto a reductive k-group uniquely factors through f . To prove this, ﬁrst observe by a noetherian argument that a maximal such quotient exists (it is G /N for the minimal normal k-subgroup scheme N such that Ru (Gk ) ⊆ Nk ), and it is clearly unique. If : G H denotes this quotient then dominates f via a unique map φ : H → G. We assume that φ is not an isomorphism and seek a contradiction. By Theorem 7.2.3(3), the maximal reductive quotient of Gk is the natural map q : Gk → G which restricts to an isomorphism Gk G on the Levi k -subgroup Gk . The reductive quotient k over k induces a quotient map π : G → H k , and the

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249

composite map π

φk

G −→ H k −→ G k is the very special isogeny of G (due to the isomorphism Gk G , which carries φk ◦ π over to ( f |G )k ). But the very special isogeny admits no factorization in surjections that are not isomorphisms, due to the irreducibility aspect of Lemma 7.1.2, so since we assume φ is not an isomorphism it follows that k = q. Since ker q is a nontrivial smooth connected unipotent normal k -subgroup of Gk , so ker is a nontrivial smooth connected unipotent normal k-subgroup of G , we have contradicted the pseudo-reductivity of G over k. In view of the uniqueness of the maximal reductive quotient map f over k, it follows that for every k-automorphism α of G there is a unique k-automorphism α of G such that f ◦ α = α ◦ f . The resulting map Aut k (G ) → Aut k (G) is injective by Proposition 1.2.2 (since the maximal reductive quotient G of Gk k

maps onto G k via an isogeny; see Proposition 7.2.7(2) ). To prove surjectivity, consider a Levi k-subgroup G in G (which exists since k = ks ). As we noted earlier, f restricts to a map π : G → G that is the very special isogeny. In the factorization π

G → G → G (p) of FG/k , the second map is the very special isogeny for G (as follows from Proposition 7.1.5(1),(2) applied to G and Lemma 7.1.2 applied to G, together with degree considerations). We conclude that for any k-automorphism α of G, there is an induced kautomorphism α of the very special quotient G (p) , which in turn lies over the (p) k-automorphism α (p) of G via π (p) . The p-power map of k that is used to deﬁne G (p) is identiﬁed with the inclusion of k into k := k 1/p , so α may be viewed as a k -automorphism of Gk that lies over the k -automorphism α k of G k via πk . Thus, the map R k /k (πk ) : R k /k (Gk ) → R k /k (G k ) is compatible with R k /k (α ) and R k /k (α k ). Since R k /k (α k ) restricts to α on the Levi k-subgroup G, we conclude that R k /k (α ) restricts to a k-automorphism of R k /k (π)−1 (G) = G that is carried to α ∈ Aut k (G) via f : G G. Now we deduce (2) from (1). Under the assumption in (2), the condition k1 = k2 = k 1/p required in (1) is always satisﬁed since ki /k is a nontrivial ﬁnite extension satisfying ki ⊆ k 1/p . In particular, since all possibilities for G become

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isomorphic over ks , it follows that all possibilities for G become isomorphic over ks . Moreover, at least one G exists, namely the one associated to a triple (Gk , k /k, G) with k-split G. Thus, upon ﬁxing one such G , the set of all kforms is in natural bijection with H1 (ks /k, Aut ks (Gks )). Moreover, (1) provides a natural Gal(ks /k)-equivariant isomorphism Aut ks (Gks ) → Aut ks (G ks ), and the induced bijection in degree-1 Galois cohomology corresponds exactly to the assignment considered in (2). Thus, that assignment is bijective. 2 An interesting consequence of Proposition 7.3.5 is the existence of basic exotic pseudo-reductive k-groups without a Levi k-subgroup, over any local or global function ﬁeld k of characteristic 2. In Example 7.2.2 we saw standard pseudo-reductive k-groups without Levi k-subgroups over any local or global function ﬁeld, but examples in the basic exotic case are especially interesting because they demonstrate that in the condition in Deﬁnition 7.2.6(iii) on existence of a Levi subgroup it really is necessary to extend scalars to ks . Proposition 7.3.6 Let k be a local or global function ﬁeld of characteristic p = 2, and G a basic exotic pseudo-reductive k-group with G ss of rank n 2. k Assume G is not pseudo-split over k. The k-group G does not contain a Levi k-subgroup if the root system of G ss k is either of type Bn with n 2 or of type Cn with n even. If the root system of G ss is of type Cn with n odd, then G contains a Levi k-subgroup. k In all cases, if k /k denotes the ﬁeld of deﬁnition of Ru (Gk ) then the k -rank of Gkss = Gkred is larger than the k-rank of G . By Proposition 7.3.5, there exists a non-pseudo-split G with G ss of any k type Bn or Cn with n 2 since non-split absolutely simple and semisimple k-groups of each of these types exist (due to the classiﬁcation of such forms). The hypothesis that G is not pseudo-split is necessary, since by Theorem 3.4.6 a pseudo-split pseudo-reductive group over any ﬁeld k admits a Levi k-subgroup. The inequality rank k (G ) < rank k (Gkss ) is interesting because this never occurs for absolutely pseudo-simple standard pseudo-reductive k-groups (due to Proposition 4.1.4(2)). Proof. In all cases, G corresponds to a triple (G , k /k, G) giving rise to G with k = k 1/p equal to the ﬁeld of deﬁnition over k for Ru (Gk ). There is a surjective k-homomorphism f : G G. For n 2, a non-split absolutely simple and semisimple group of type Bn (resp. type Cn ) over a local or global function ﬁeld has rank n − 1 (resp. rank n/2 or (n − 1)/2, depending on the parity of n). The k-group G has type dual to that of G ss with the same absolute rank, so by k

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Corollary 7.3.4 it follows that G has k-rank equal to n/2 or (n − 1)/2 if G ss is k of type Bn and k-rank equal to n − 1 if it is of type Cn . Suppose there is a Levi k-subgroup G in G . If G ss is of type Bn , so G is nonk split of type Bn and hence of k-rank n − 1. But G has k-rank n/2 or (n − 1)/2 (depending on the parity of n 2) which is strictly smaller than n − 1, so no such G can exist. Now let us assume that G ss is of type Cn with n even. Then G is a non-split k absolutely simple simply connected k-group of type Cn , and hence its k-rank is n/2. Let S be a maximal k-split torus of G and let A = D (ZG (S)). In the Tits index of G (see [Ti0, Table II]) there are n/2 nondistinguished vertices, they are all isolated. Thus, there is a central quaternion division algebra D over k such that A is the direct product of n/2 copies of the anisotropic k-group SL1,D . Since the root corresponding to each nondistinguished vertex in the Tits index of G is short, the very special isogeny π : G → G restricts to a quotient map A → A := π(A) = D (ZG (π(S))) whose kernel is the Frobenius kernel of A, so this latter map is identiﬁed with the Frobenius morphism for A. Hence, A is identiﬁed with the direct product of n/2 copies of SL2 . We conclude that the krank of G is equal to dim π(S) + n/2 = n. Corollary 7.3.4 implies then that the k-rank of G is also n, and hence G is pseudo-split, contrary to our hypothesis. Hence, G cannot contain a Levi k-subgroup. To prove the existence of Levi k-subgroups when G ss is of type Cn with odd k n > 1, we carry out a Galois-twisting argument beginning in the pseudo-split case. To do this, let G be an absolutely simple and simply connected k-split kgroup of type Cn with odd n > 1 and let π : G → G be the very special isogeny. Let G be the pseudo-split basic exotic pseudo-reductive k-group corresponding to the triple (G , k /k, G), where k = k 1/2 and G = Gk . Then G ⊂ Rk /k (G ) and the inclusion G → Rk /k (G ) is a Levi k-subgroup of G . It sufﬁces to show that all k-forms of G admit a Levi k-subgroup (since over ks there is only one isomorphism class of basic exotic pseudo-reductive group of each semisimple type, by the end of Proposition 7.2.7(2)). As n is odd, the center Z of G is isomorphic to μ2 and is contained in G, and π maps Z isomorphically onto the center Z of G. Therefore, π induces a homomorphism Aut ks (Gks ) = (G/Z)(ks ) → (G/Z)(ks ) = Aut ks (G ks ). By Proposition 7.3.5(1), we have H1 (ks /k, Aut ks (Gks )) = H1 (ks /k, Aut ks (G ks )) = H1 (ks /k, (G/Z)(ks )). The well-known triviality of degree-1 Galois cohomology of simply connected groups over local and global function ﬁelds implies that in the commutative

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diagram of natural maps H1 (ks /k, (G/Z)(ks )) H1 (ks /k, (G/Z)(ks ))

δ

/ H2 (k, Z)

δ

/ H2 (k, Z)

(in which the right side is the isomorphism Z Z induced by π ), the horizontal connecting maps are injective. The terms on the right side are identiﬁed with H2 (k, μ2 ) = Br(k)[2], and for local and global function ﬁelds this is the set of isomorphism classes of central simple quaternion algebras over k. The explicit construction of all k-forms of groups of types Bn and Cn in terms of quaternion division algebras implies that the connecting maps are bijective. Thus, the k-forms of both G and G are compatibly parametrized by H2 (k, Z). For any continuous 1-cocycle c : Gal(ks /k) → (G/Z)(ks ) the Galois-twists G and c G make sense, with c G a Levi k-subgroup in c G . Thus, it sufﬁces c to show that as c varies, we obtain all k-forms of G ; that is, for each x ∈ H2 (k, Z) we seek a continuous 1-cocycle c = c(x) on Gal(ks /k) with values in (G/Z)(ks ) ⊂ Aut ks (Gks ) such that the image of the cohomology class of c(x) in H2 (k, Z) is equal to x. The existence of such a c(x) is immediate from the above commutative diagram of natural maps. Now we turn to a comparison of the k-rank of G and the k -rank of Gkss when G is not pseudo-split over k and Gkss is either of type Bn or of type Cn . The natural quotient map q : Gk G is a k -descent of the maximal reductive quotient of Gk (Proposition 7.2.7(2)), so G has the same type as G ss . k It follows that if G ss is of type Bn then the k -rank of G is at least n − 1. We k saw above that in such cases the non-pseudo-split G has k-rank n/2 or (n−1)/2, which is strictly smaller than the k -rank of G = Gkss . Suppose instead that G ss k is of type Cn , so the k-rank of G is n − 1. The quotient G = Gkss of Gk therefore has k -rank at least n − 1. But G is of type Cn , so if G were not k -split then its k -rank would be n/2 or (n − 1)/2, which is strictly smaller than n − 1. Hence, G is k -split, so its k -rank is n, which strictly exceeds the k-rank of G . 2 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3} and let G be a basic exotic pseudo-reductive k-group with associated unique triple (up to unique k-isomorphism) (G , k /k, G) as in Deﬁnition 7.2.6 and Proposition 7.2.7, so G red G ; assume this is of type F4 when p = 2 and of type G2 when p = 3, k k so it is adjoint and thus we can try to use Corollary 3.4.8 to construct Levi k-subgroups in G . The case of most interest is k = k 1/p (so [k : k p ] is ﬁnite).

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If [k : k p ] = p, such as local and global function ﬁelds over ﬁnite ﬁelds, then this always holds. Proposition 7.3.7 Let G be a basic exotic pseudo-reductive k-group such that G red is of type F4 or G2 . Let T be a maximal k-torus in G . If there is a ﬁnite k extension K/k of degree not divisible by p such that TK is K-split then G admits a Levi k-subgroup containing T . If k = k 1/p and p = 3 then there exists a Levi k-subgroup of G . We do not know any examples without a Levi k-subgroup when p = 2 for type F4 , nor for p = 3 if k = k 1/3 . Proof. First assume that such a K/k exists. The explicit structure of ZG (T )ks and root groups for (Gks , Tks ) shows that the principal homogeneous space hypothesis in Corollary 3.4.8 is satisﬁed. Thus, the necessary and sufﬁcient condition for the existence of such a Levi k-subgroup is the vanishing of a certain class c ∈ H1 (k, ZG (T )/T ) whose formation commutes with separable extension on k. By hypothesis, c|K therefore vanishes for a separable extension K/k of degree not divisible by p. But the smooth connected commutative kgroup ZG (T )/T is unipotent, so its group of ks -points is p-power torsion (in fact it is p-torsion by inspection since k p ⊆ k). Hence, the splitting of a class c by a ﬁnite separable extension of degree not divisible by p implies that c = 1. Now assume that p = 3 and k = k 1/p . In this case it sufﬁces to prove that some choice of T is split by a separable quadratic extension of k. The hypothesis k = k 1/p is preserved under separable algebraic extension on k, so by Corollary 7.3.4 the problem of constructing a maximal k-torus in G that is split by a separable quadratic extension of k is reduced to the analogous problem for the k-form G of G2 . For any ﬁeld F with characteristic different from 2, the set of isomorphism classes of F-forms of G2 is naturally identiﬁed (functorially in F) with the subset of H3 (F, Z/2Z) consisting of triple cup products from H1 (F, Z/2Z) [Spr, 17.4.6]. In particular, any such form is split by some quadratic extension of F. Thus, Lemma 7.3.8 below completes the proof. 2 Lemma 7.3.8 Let F be a ﬁeld and G a connected reductive F-group that is split by a separable quadratic extension F /F. There exists a maximal F-torus in G that is F -split.

Proof. Let B ⊆ GF be a Borel subgroup deﬁned over F and let B denote its conjugate by the nontrivial element of Gal(F /F). By [Bo2, 14.22(i), 20.7(i)], B ∩ B is a smooth connected F -subgroup of GF and it contains a maximal F -torus of GF .

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Since B is an F -Borel subgroup of the F -split connected reductive F -group GF , its F -unipotent radical Ru,F (B ) descends Ru (B ), so every maximal F F -torus in B maps isomorphically onto the quotient B /Ru,F (B ) that is an F -split torus. Thus, all maximal F -tori in B are F -split. The smooth connected Gal(F /F)-stable F -subgroup B ∩ B in GF descends to a smooth connected F-subgroup H in G, and the maximal F-tori T in H are also maximal in G (as may be checked over F ). Choosing such a T , we conclude that TF is F -split since it is a maximal F -torus in B . 2

8 Preparations for classiﬁcation in characteristics 2 and 3

To complete the proof of Theorem 5.1.1 we need to understand the signiﬁcance of the constructions in Chapter 7, especially the notion of basic exotic pseudoreductive group that was introduced in Deﬁnition 7.2.6 over any imperfect ﬁeld k of characteristic 2 or 3.

8.1 Further properties of basic exotic pseudo-reductive groups This section lays the groundwork for the deﬁnition (in §8.2) of the concept of an exotic pseudo-reductive group over an imperfect ﬁeld k of characteristic 2 or 3, generalizing the concept of a basic exotic pseudo-reductive group. The real work will be to show that it accounts for all exceptional absolutely pseudosimple groups (in the sense of Deﬁnition 6.3.3), at least when G ss is simply k connected. Proposition 8.1.1 Let k be an imperfect ﬁeld of characteristic 2 or 3. A basic exotic pseudo-reductive k-group G is never standard over k. Proof. By Corollary 5.2.3 we may assume k = ks . Let (G , k /k, G) be the unique triple giving rise to G as in Proposition 7.2.7. By Corollary 7.2.5(3) the long root groups of G have dimension 1, and by construction the short root groups have dimension > 1 (in fact, [k : k]). The pseudo-reductive k-group G is absolutely pseudo-simple (Corollary 8.1.4), and an absolutely pseudo-simple standard pseudo-reductive group has all root groups of the same dimension. Hence, G is not standard. 2 We wish to apply Proposition 5.1.3 in the context of basic exotic pseudoreductive groups, as well as their Weil restrictions: 255

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Proposition 8.1.2 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3}, K a nonzero ﬁnite reduced k-algebra, and G a K-group whose ﬁbers are basic exotic pseudo-reductive groups in the sense of Deﬁnition 7.2.6. The k-group R K/k (G ) is pseudo-reductive over k, perfect and not reductive. Moreover, if Z is a commutative k-group scheme of ﬁnite type with no nontrivial smooth connected k-subgroup then any central extension of R K/k (G ) by Z over k is split over k. Proof. By Proposition 1.1.10, R K/k (G ) is pseudo-reductive over k. It is not reductive, by Proposition 7.2.7(1) applied to the (non-reductive) quotient G of Rk/k (G )k (also see Lemma A.5.10). To prove that R K/k (G ) is its own derived group, we can assume k is separably closed and K is a ﬁeld. By Proposition A.7.10, it sufﬁces to prove that G is its own derived group and that Lie(G ) has a vanishing space of coinvariants with respect to the adjoint action by G . The key point is to explicitly describe the open Bruhat cell in G relative to a maximal K-torus T and a positive system of roots + in (G , T ). By Theorem 7.2.3(2) and Corollary 7.2.5, for each root c the opposite root groups Uc and U−c in G generate a copy of SL2 (with diagonal c∨ (GL1 )) if c is long and a copy of R K/k (SL2 ) (with diagonal R K/k (c∨ (GL1 )) if c is short. Moreover, if ⊆ + is the set of simple positive roots then ZG (T ) =

a∈>

a∨ (GL1 ) ×

R K/k (b∨ (GL1 )).

(8.1.1)

b∈<

Thus, the vanishing of the G -coinvariants of Lie(G ) is reduced to the analogous vanishing property for SL2 and for R K/k (SL2 ). Since Lie(R K/k (SL2 )) = sl2 (K) as a Lie algebra over k (Corollary A.7.6), it has vanishing space of coinvariants over k with respect to the adjoint action of R K/k (SL2 )(k) = SL2 (K). By a similar argument, G contains a Zariski-dense subgroup of G (k) generated by copies of SL2 (k) and SL2 (K), so G is its own derived group. This completes the proof that R K/k (G ) is perfect. It remains to split central extensions of R K/k (G ) by commutative k-groups Z of ﬁnite type that contain no nontrivial smooth connected k-subgroups. Since R K/k (G ) is perfect for any k (perhaps not separably closed), such a splitting is provided by Proposition 5.1.3 provided that ZG (T )ks is rationally generated by root groups, which is to say that there exist rational maps hi on ZG (T ) as in Proposition 5.1.3 when k = ks . By (8.1.1), it sufﬁces to check the analogous property for Cartan k-subgroups of SL2 and R K/k (SL2 ). This in turn is settled by Example 5.1.4. 2

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257

Corollary 8.1.3 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3} and let G be a basic exotic pseudo-reductive k-group. Then G is absolutely pseudo-simple over k. Proof. By Proposition 8.1.2 (with K = k), the pseudo-reductive k-group G is perfect. By Theorem 7.2.3(3), G ss is simple. Hence, absolute pseudo-simplicity k follows from Lemma 3.1.2. 2 Corollary 8.1.4 Let (G , K/k) be as in Proposition 8.1.2, {Ki } the set of factor ﬁelds of K, and Gi the ﬁber of G over Ki . Every smooth connected normal k-subgroup of the pseudo-reductive k-group G := R K/k (G ) is a product of some of the k-subgroups R Ki /k (Gi ). In particular, G is absolutely pseudo-simple if and only if K/k is a purely inseparable ﬁeld extension. Proof. By Galois descent we may reduce to the case of separably closed k. We ﬁrst check that each factor Gi = R Ki /k (Gi ) is pseudo-simple. The natural map (Gi )Ki Gi has smooth connected unipotent kernel since Ki /k is purely inseparable (Proposition A.5.11). By Corollary 8.1.3, Gi is absolutely pseudosimple over ki . Since Gi is perfect (by Proposition 8.1.2), pseudo-simplicity of Gi follows from Lemma 3.1.2. Now let N be a smooth connected normal k-subgroup of G. To prove that N has the asserted form, the pseudo-simplicity of the factors Gi reduces us to the case in which N maps onto each factor. Thus, the smooth connected k-subgroup (N , Gi ) is normal and also maps onto each Gi (as Gi is perfect). But (N , Gi ) is contained in the pseudo-simple Gi , so (N , Gi ) = Gi for all i. Since (N , Gi ) is contained in N , we get N = G. 2 We next wish to make precise the sense in which the pseudo-reductive k-group G = R K/k (G ) for a basic exotic pseudo-reductive K-group G over a ﬁnite extension K/k in characteristic p ∈ {2, 3} uniquely determines the pair (G , K/k) up to canonical k-isomorphism (much like the uniqueness of the pair (G , k /k) in the non-commutative standard case via Proposition 4.2.4). We also want to reconstruct the k-group identiﬁcation G R K/k (G ) from the k-group G. This rests on the determination of the universal maximal absolutely simple adjoint quotient of G in the sense deﬁned after Proposition 4.2.1, as we now explain. Proposition 8.1.5 Let k be imperfect of characteristic p ∈ {2, 3}, K/k a nonzero ﬁnite reduced k-algebra, and G a K-group with basic exotic ﬁbers arising from a triple (G , K /K, G). Let G = R K/k (G ) ⊆ R K/k (R K /K (G )) = R K /k (G ).

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(1) The universal maximal absolutely simple adjoint quotient of G is (K /k, π ), where π : GK → R K /k (G )K G → G /ZG is the canonical composite map. In particular, the universal maximal absolutely simple adjoint quotient map π uniquely factors through the simply connected central cover G → G /ZG of this quotient. (2) The k-subalgebra K ⊆ K is the maximal one such that the k-subspace Lie(G) ⊆ Lie(R K /k (G )) = Lie(G ) of the K -module Lie(G ) is a K-submodule. Also, the K-map GK → R K /K (G ) corresponding to G → R K /k (G ) = R K/k (R K /K (G )) has image G . The case K = k (equivalently G = G ) is the most important one, but for the purpose of obtaining a classiﬁcation theorem for all pseudo-reductive groups in characteristics 2 and 3 we need to permit the possibility that K = k and that K may not be étale over k. Proof. If {Ki } and {Ki } are the compatible respective collections of factors ﬁelds of K and K then every intermediate k-algebra F between K and K has the form F = Fi for some intermediate ﬁelds Ki ⊆ Fi ⊆ Ki . This applies to the maximal k-subalgebra as deﬁned for (2), so we may use Galois descent to reduce to the case k = ks and pass to factor ﬁelds of K to reduce to the case where K and K are ﬁelds. We ﬁrst prove (2), and then (1). The k-subspace Lie(G) ⊆ Lie(G ) is the K-subspace Lie(G ), so to prove the maximality claim concerning K in (2) it sufﬁces to treat the case k = K (i.e., G = G ). Since K is separably closed, G contains a Levi K-subgroup L of R K /K (G ). The K-subspace Lie(G ) ⊆ Lie(G ) = K ⊗K Lie(L) is therefore described explicitly in (7.2.1). Inspection of the contribution from long roots therefore gives the maximality claim for K. For the second claim in (2) we use a functoriality argument as follows. For any afﬁne schemes X and Y of ﬁnite type over k and K respectively, a k-map f : X → R K/k (Y ) corresponds to the K-map fK

XK → R K/k (Y )K → Y , where the second map is given on R-valued points for a K-algebra R by the map Y (K ⊗k R) → Y (R) arising from the K-algebra multiplication map K ⊗k R → R. (See Proposition A.5.7.)

8.1 Further properties of basic exotic pseudo-reductive groups

259

Thus, if j : G → R K /K (G ) is the inclusion then α := R K/k ( j) : G → R K /k (G ) is the inclusion and the map GK → R K /K (G ) of interest in (2) is the composition of αK with the natural projection q : R K /k (G )K = R K/k (R K /K (G ))K R K /K (G ). Consider q : GK → G that corresponds to the equality G = R K/k (G ) deﬁning G. The diagram of K-groups GK

αK

/ R K /k (G )K

q

G

j

q

/ R K /K (G )

commutes and the left side is a smooth surjection with connected unipotent kernel (Proposition A.5.11), so the ﬁnal assertion of (2) is proved. Turning to the proof of (1), the natural map GK = R K/k (G )K → G is a smooth surjection with a connected unipotent kernel as observed above. Hence, by extending scalars from K to K we obtain a smooth surjection GK → GK

(8.1.2)

over K with a smooth connected unipotent kernel. By Theorem 7.2.3(3), GK admits a canonical smooth surjection onto G with a smooth connected unipotent kernel. Composing this with (8.1.2), we get a quotient map : GK G over K with a smooth connected unipotent kernel. Thus, is a K -descent of the maximal simple semisimple quotient of Gk . In particular, the maximal simple adjoint quotient of Gk is deﬁned over K (as a quotient) and as such is naturally identiﬁed with the quotient G /ZG . In view of how the universal maximal absolutely simple adjoint quotient was constructed over a separably closed ground ﬁeld in the proof of Proposition 4.2.1, our problem is to show that K /k is the minimal extension over which the maximal adjoint semisimple quotient of Gk is deﬁned (since this quotient is simple).

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The k-group G is perfect (Proposition 8.1.2), so the minimal ﬁeld of deﬁnition over k for the maximal adjoint semisimple quotient of Gk is the same as the minimal ﬁeld of deﬁnition over k for the maximal semisimple quotient of Gk (Proposition 5.3.3). Hence, it sufﬁces to prove that ker as a K -subgroup of GK cannot be deﬁned over a proper subﬁeld containing k. As in the proof of (2), let L ⊂ G be a Levi K-subgroup of R K /K (G ). Thus, we have a k-subgroup inclusion j : R K/k (L) → R K/k (G ) = G. Let h = qL : R K/k (L)K L denote the canonical K-descent of the maximal reductive quotient of R K/k (L)k . Since K/k is purely inseparable, by Theorem 1.6.2(2) the K-subgroup ker h ⊆ R K/k (L)K has ﬁeld of deﬁnition over k equal to K. But hK factors as jK

R K/k (L)K → GK G = LK , so

(ker h)K = jK−1 (ker(GK G )). Thus, if the ﬁeld of deﬁnition over k for the K -subgroup ker ⊆ GK is F ⊆ K then (ker h)K also descends to F (as j is a k-map). Since (ker h)K descends to K, its minimal ﬁeld of deﬁnition over k is therefore contained in F ∩ K. Hence, the K-subgroup ker h ⊆ R K/k (L)K descends to F ∩ K. But we just saw that the minimal ﬁeld of deﬁnition over k for the K-subgroup ker h is K, so F ∩ K = K, which is to say K ⊆ F. Our problem is therefore reduced to showing that K is the ﬁeld of deﬁnition over K (rather than over k) for ker . But factors as the composition GK GK G

(8.1.3)

in which the ﬁrst step is a scalar extension of the quotient map qG : GK G over K and the second step is the canonical quotient map. Since we are now studying ﬁelds of deﬁnition over K, it is equivalent to prove that the kernel of the second step in (8.1.3) (viewed as a K -subgroup of GK ) has minimal ﬁeld of deﬁnition over K equal to K . This is exactly Proposition 7.2.7(1). 2

8.2 Exceptional and exotic pseudo-reductive groups Let k be an imperfect ﬁeld of characteristic 2 or 3. Propositions 7.2.7 and 8.1.5 show that for any ﬁnite extension of ﬁelds K/k and basic exotic pseudoreductive K-group G arising from a triple (G , K /K, G), if G is a k-group

8.2 Exceptional and exotic pseudo-reductive groups

261

that is k-isomorphic to R K/k (G ) then G canonically determines the 5-tuple (K/k, G , G , K /K, G) (note that Aut(K /K) = {1} since K /K is purely inseparable) as well as a canonical isomorphism G R K/k (G ). Since G is absolutely pseudo-simple precisely when K/k is purely inseparable (Corollary 8.1.4), such pseudo-reductive groups provide an explicit description of all absolutely pseudo-simple k-groups G such that G ss is simply connected and G is k exceptional in the sense of Deﬁnition 6.3.3: Theorem 8.2.1 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3} and let G be an absolutely pseudo-simple k-group. Then G is exceptional in the sense of Deﬁnition 6.3.3 with G ss simply connected (so ξG = iG ) if and only if G is k the Weil restriction of a basic exotic pseudo-reductive group over a ﬁnite purely inseparable extension of k. When these two equivalent conditions hold, G is explicitly described as follows. Let K/k be the ﬁeld of deﬁnition of the geometric radical of G, and GKss → GKss the very special K-isogeny. There is a unique proper subﬁeld ss ) is an isomork ⊂ K so that kK p ⊆ k and the k-map ξG : G → R K/k (GK ss phism onto R k/k (G ), where G ⊆ R K/k (GK ) is the basic exotic pseudo-reductive ss , K/k, L) for a Levi k-subgroup L of R ss k-subgroup associated to (GK K/k (GK ) ss whose preimage in R K/k (GK ) is k-smooth. Proof. Since K/k is purely inseparable, the formation of K is compatible with separable algebraic extension on k. Thus, by Galois descent and Proposition 7.2.7(3) we may and do assume k = ks . Choose a maximal k-torus T in G and let = (G, T ). Step 1. First we check that if k/k is a ﬁnite extension and G is a basic exotic pseudo-reductive k-group such that the k-group G is k-isomorphic to R k/k (G) then G is exceptional with G ss simply connected. (Note that all k-groups of the k form R k/k (G) are absolutely pseudo-simple, by Proposition 8.1.2 and Corollary 8.1.4 since k/k is purely inseparable, but we are assuming G to be absolutely pseudo-simple at the outset.) By Proposition A.5.15(2) there is a unique maximal k-torus T in G such that T is the maximal k-torus in R k/k (T), and since k/k is purely inseparable it follows from Example 2.3.2 that naturally T = Tk and moreover = (G, T) via the equality X(T ) = X(T). By Proposition 7.2.7(1), (G, T) is reduced with connected Dynkin diagram having an edge with multiplicity p, so these properties hold for . The natural map Gk → G is a smooth surjection with connected unipotent kernel (Proposition A.5.11), so G ss Gss ; this latter group is simply connected (Proposition 7.2.7(2)). k k To complete the proof that G is exceptional, it remains to analyze properties of the maps ξGc and the ﬁelds Kc /k, where c varies through and Kc /k is the

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ﬁeld of deﬁnition of the geometric radical of the k-subgroup Gc generated by the root groups U±c . The structure of the open Bruhat cell for (G, T) is given in Remark 7.2.8, from which we get the analogous description for (G, T ) by applying R k/k . This explicit description gives a nontrivial ﬁnite extension k /k satisfying k p ⊆ k such that Gc R k/k (SL2 ) for long c and Gc R k /k (SL2 ) for short c. In particular, all maps ξGc are isomorphisms and Kc is equal to k when c is long and k when c is short. But the ﬁelds Kc generate K over k (Proposition 6.3.1(1)), so K = k , and K> = k, where K> is the common subﬁeld Kc for all long roots c. Thus, K> = K. Now we may and do assume that G is exceptional with G ss simply connected. k We need to prove that G is a Weil restriction of the type considered in Step 1, and that more speciﬁcally this description is obtained from G by the intrinsic process given in the second part of the statement of the theorem. Step 2. As in Step 1, for each c ∈ let Gc denote the absolutely pseudo-simple k-subgroup of rank 1 in G generated by the root groups U±c . By Proposition 3.4.1(2), for all c ∈ we have Gc = D (ZG (Tc )) for the codimension-1 k-torus Tc = (ker c)0red in T . Let Kc /k be the ﬁeld of deﬁnition of the geometric radical of Gc . By hypothesis we are in the exceptional case of Proposition 6.3.2, so Kc = K when the root c ∈ is short, and for long c ∈ the ﬁelds Kc are all equal to a common proper subﬁeld k ⊂ K over k satisfying kK p ⊆ k. As a ﬁrst step toward obtaining the desired description of G, we prove that the maps ξGc are isomorphisms. (This is automatic when p = 3, by Theorem 6.1.1, but we will handle both p in a uniform manner.) Since is reduced (by deﬁnition of G being exceptional), the root system of Gc is also reduced. But ξGc is surjective (again, by deﬁnition of G being exceptional), so Gc is standard and kerξGc is central by Proposition 5.3.10. The K-group (Gc )ss K is identiﬁed with the subgroup of GKss generated by the root groups for ±cK (Proposition 6.3.1(2)), and this subgroup is SL2 (rather than PGL2 ) since G ss is assumed k to be simply connected. Therefore, Gc := (Gc )ss Kc is isomorphic to SL2 , and ξGc expresses Gc as a central extension of R Kc /k (Gc ) = R Kc /k (SL2 ) by the unipotent k-group scheme ker ξGc that contains no nontrivial smooth connected k-subgroup schemes. Since SL2 is simply connected, this extension uniquely splits (Example 5.1.4) and consequently ker ξGc = 1 since Gc is smooth and connected. Thus, ξGc : Gc → R Kc /k (Gc ) is indeed an isomorphism for all c. Step 3. Next, we will use a Levi k-subgroup of G to construct a basic exotic pseudo-reductive k-group G such that G R k/k (G ). Let T := TK viewed as a maximal K-torus in G := GKss (so (G , T ) = via the identiﬁcation X(T ) = X(T ), since is reduced and G is perfect). By Theorem 3.4.6, there exists a Levi k-subgroup L in G containing T . Since the

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263

map ξG : G → R K/k (G ) induces the composite map GK → R K/k (G )K → G = GKss that is the canonical quotient map (by the deﬁnition of ξG ), it follows that (ξG )k respects unipotent radicals and induces an isomorphism between maximal reductive quotients. In particular, ξG (L) is a Levi k-subgroup of R K/k (G ) and ξG |L : L → ξG (L) is an isomorphism. But ξG (L) ⊆ ξG (G), so ξG (L) is also a Levi k-subgroup of ξG (G). We may therefore rename ξG (L) as L so that L is a common Levi k-subgroup of ξG (G) and R K/k (G ) that contains T . The composite map LK → R K/k (G )K G is an isomorphism; we will use it to identify G with LK . This identiﬁcation recovers that of TK with the maximal K-torus T of G , so (L, T ) is a k-descent of (G , T ). In particular, (Lk , Tk ) is a k-descent of (G , T ), so Lk is a Levi k-subgroup of R K/k (G ) via the canonical map Lk → R K/k (LK ) = R K/k (G ), so it gives rise to a basic exotic pseudo-reductive k-subgroup G ⊂ R K/k (G ). Explicitly, using Theorem 7.2.3(2) (with k in place of k and K in place of k , and the very special isogeny Lk Lk ), G is generated by the long root groups of Lk relative to Tk and the Weil restrictions from K to k of the short root groups of LK (= G ) relative to TK = T . We shall prove that ξG (G) = R k/k (G ) inside of R K/k (G ) and that ξG : G ξG (G) is an isomorphism. Step 4. For c ∈ (G, T ) = (L, T ), let Lc be the subgroup of L generated by the root groups in L corresponding to the roots ±c. In Step 2 we saw that for each c ∈ , ξG carries Gc isomorphically onto its image in R K/k (Gc ) = R K/k ((Lc )K ), and that this image is abstractly k-isomorphic to R Kc /k ((Lc )Kc ). (Note that Lc SL2 since Lk G ss is simply connected.) Since L is the image k of a Levi k-subgroup of G under ξG , we have the containments Lc ⊆ ξG (Gc ) ⊆ R K/k ((Lc )K ). Rather generally, we claim that if F /F/k is a tower of ﬁnite extensions of k = ks and if H is any connected reductive k-group (such as Lc = SL2 ) then the only intermediate group G such that H ⊆ G ⊆ R F /k (HF ), and which is isomorphic to R F/k (HF ), is the canonical copy of R F/k (HF ) inside of R F /k (HF ). This claim would imply that ξG (Gc ) equals R Kc /k ((Lc )Kc ) inside of R K/k ((Lc )K ) (⊂ R K/k (LK )) for all c ∈ . (That is, if c ∈ < then ξG (Gc ) = R K/k ((Lc )K ), and if c ∈ > then ξG (Gc ) = R k/k ((Lc )k ) inside of R K/k (LK ).)

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Preparations for classiﬁcation in characteristics 2 and 3

To prove the claim, ﬁrst observe that H is embedded as a Levi k-subgroup of G (as it is visibly a Levi k-subgroup of the ambient k-group R F /k (HF )). Choose a k-isomorphism θ : G R F/k (HF ). The composite inclusion ι θ

H → G R F/k (HF ) is a Levi k-subgroup. In particular, the induced composite F-map qHF

ιF

φ : HF −→ R F/k (HF )F HF is an isomorphism. Composing θ with R F/k (φ)−1 then brings us to the case where θ extends the identity map between the given copy of H inside of G and the canonical copy inside of R F/k (HF ). Our problem is therefore reduced to verifying that the only k-homomorphism f : R F/k (HF ) → R F /k (HF ) extending the identity on the k-subgroup H is the canonical inclusion. The map fk respects the unipotent radicals since the composite map fk

R F/k (HF )k −→ R F /k (HF )k Hk is surjective (as the Levi k-subgroup Hk provides a section). It follows that f induces a homomorphism f red between the maximal geometric reductive k

quotients, and the canonical identiﬁcation of these quotients over k with Hk (via the canonical Levi k-subgroups H ) identiﬁes f red with the identity map. k But the canonical inclusion of R F/k (HF ) into R F /k (HF ) has the same property, so by Proposition 1.2.2 it must equal f . Step 5. As G is perfect, it is generated by the k-subgroups Gc for all c ∈ (G, T ) = (G , T ) (Lemma 3.1.5). Thus, by (6.3.1) and Step 4, the ksubgroup ξG (G) ⊆ R K/k (G ) = R K/k (LK ) is generated by the k-subgroups ξG (Gc ) = R Kc /k ((Lc )Kc ) for all c. These latter subgroups generate R k/k (G ) due to the structure of an open Bruhat cell in G with respect to T = TK via (7.2.2) (with (G, k /k) there replaced with (Lk , K/k)). Hence, ξG (G) = R k/k (G ). The same considerations with an open Bruhat cell for G and the isomorphism property of the maps ξGc (with both long and short c) imply that the image of Lie(ξG ) equals the Lie algebra of ξG (G) = R k/k (G ). We conclude that ker ξG is a smooth unipotent normal k-subgroup of G. But G is pseudo-reductive over k,

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265

so ker ξG must be étale and hence central in G since G is connected. Applying Proposition 8.1.2 to G and Z = ker ξG then gives that ker ξG = 1 since G is connected. Thus, the map ξG : G → ξG (G) = R k/k (G ) is an isomorphism as desired. The uniqueness claims in Theorem 8.2.1 (concerning k and a Levi k-subgroup of R K/k (GKss )) follow from Proposition 8.1.5(2). This completes the proof of Theorem 8.2.1. 2 Theorem 8.2.1 justiﬁes the interest in the following deﬁnition. Deﬁnition 8.2.2 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3}. A pseudo-reductive k-group G is exotic (over k) if G R K/k (G ) for a nonzero ﬁnite reduced k-algebra K and a K-group G whose ﬁbers are basic exotic pseudo-reductive groups (in the sense of Deﬁnition 7.2.6). Exotic pseudo-reductive groups are perfect and non-reductive, by Proposition 8.1.2. By Corollary 8.1.4, an exotic pseudo-reductive k-group G is uniquely a direct product of pseudo-simple exotic pseudo-reductive k-groups, and the set of such pseudo-simple factors is in natural bijective correspondence with the set of factor ﬁelds of K. In particular, Gkss

(G ⊗K,σ k)ss

σ :K→k

(product over k-algebra maps σ ) is simply connected and its Aut(k/k)-orbits of simple factors are in bijective correspondence with the set of factor ﬁelds of K. Thus, G is pseudo-simple over k if and only if K is a ﬁeld, it is absolutely pseudo-simple if and only if K/k is purely inseparable, and in the pseudosimple case the connected components of the Dynkin diagram of G ss are all of k the same type. We call this the type of a pseudo-simple exotic pseudo-reductive G (as it is the type which underlies the construction of G over the ﬁeld K). Remark 8.2.3 The root system (over ks ) for an exotic pseudo-reductive k-group is always reduced (even in characteristic 2): in the basic exotic case this is part of Proposition 7.2.7(1), and in general it follows from the behavior of root systems with respect to Weil restriction (Example 2.3.2). The description of the root spaces shows that in the absolutely pseudo-simple case over k = ks the root spaces do not all have the same dimension. Hence, such absolutely pseudo-simple groups are never standard. In Proposition 5.2.2 we described all k-isomorphisms among pairs (G, T ) for maximal k-tori T in a non-commutative standard pseudo-reductive k-group

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Preparations for classiﬁcation in characteristics 2 and 3

G. We now consider the analogous problem in the exotic cases. To that end, let k be an imperfect ﬁeld with characteristic p ∈ {2, 3} and let G be an exotic pseudo-reductive k-group. That is, G R k /k (G ) for a nonzero ﬁnite reduced k-algebra k and a k -group G whose ﬁbers are basic exotic pseudo-reductive groups in the sense of Deﬁnition 7.2.6. By Proposition 8.1.5, the pair (G , k /k) is uniquely determined by G up to k-isomorphism. We wish to describe isomorphisms between exotic pseudo-reductive groups in terms of isomorphisms among the data used to construct such groups. Proposition 5.2.2 motivates our description of all such isomorphisms in Proposition 8.2.4 below, but we require some preparations before we state the description. We will consider 5-tuples (K, K , G , G , G) satisfying: (a) K is a nonzero ﬁnite reduced algebra over an imperfect ﬁeld k with characteristic p ∈ {2, 3}; let {Ki } be its set of factor ﬁelds. (b) G is a K-group whose ﬁber Gi over Ki is a basic exotic pseudo-reductive Ki -group for every i. (c) For each i, (Gi , Ki /Ki ) is the pair that is uniquely associated to Gi (up to unique Ki -isomorphism) as in Deﬁnition 7.2.6, with Gi absolutely simple and simply connected with Dynkin diagram having an edge with multiplicity p. (d) K = Ki , G = Gi , π : G → G is the very special K-isogeny, and G = R K /K (π )−1 (G)

for a suitable Levi K-subgroup G of R K /K (G ). (The K-group G determines G because G is the image of G under R K /K (π ), by Proposition 7.2.7(2).) By Deﬁnition 8.2.2, the pseudo-reductive k-group R K/k (G ) is exotic, and by Proposition 8.1.5 every exotic pseudo-reductive k-group is k-isomorphic to R K/k (G ) for such a pair (G , K/k) that is unique up to canonical k-isomorphism. The description of isomorphisms among k-groups of the form R K/k (G ) is provided by the following result. Proposition 8.2.4 Let (K, K , G , G , G) and (L, L , H , H , H ) be 5-tuples over k satisfying the above properties (a)–(d). Consider a pair (α, ϕ) where α is a k-algebra isomorphism L K that lifts (necessarily uniquely) to a k-algebra isomorphism α : L K and ϕ is a group isomorphism G H over Spec(α ).

8.2 Exceptional and exotic pseudo-reductive groups

267

(1) The induced k-isomorphism R α /k (ϕ) : R K /k (G ) = R K/k (R K /K (G )) → R L /k (H ) restricts to a k-isomorphism σ : R K/k (G ) R L/k (H )

(8.2.1)

between the associated exotic pseudo-reductive k-groups if and only if the isomorphism ϕ : G H over α induced by ϕ satisﬁes R α /α (ϕ)(G) = H inside of R K /K (G ) R L /L (H ), and also if and only if R α /α (ϕ)(G ) = H . Conversely, any k-isomorphism σ : R K/k (G ) R L/k (H ) between exotic pseudo-reductive k-groups arises in this way from a unique such pair (α, ϕ). (2) Let S and T be maximal k-tori in R K/k (G ), and let S and T be the corresponding maximal K-tori in G via Proposition A.5.15(2). For a kautomorphism σ of R K/k (G ) arising from a pair (α, ϕ), the conditions σ (T ) = S and R α /α (ϕ)(T ) = S are equivalent. The same holds for maximal split tori. Proof. Part (2) is a consequence of the bijectivity in Proposition A.5.15(2), so we focus our attention on (1). In particular, by Galois descent we may assume that k is separably closed. Since Weil restriction is compatible with the formation of preimages, if R α /α (ϕ)(G) = H then the map R α /k (ϕ) does induce a k-isomorphism σ as in (8.2.1). Also, the equivalence involving R α /k (ϕ) and R α /α (ϕ) formally implies the one involving R α /k (ϕ) and R α /α (ϕ) (using the end of Proposition 7.2.7(2)). Finally, the necessity in the ﬁrst equivalence in (1) is seen by passing to maximal pseudo-reductive quotients of ﬁbers over L K, so the equivalences in (1) are now all proved. It remains to show that every k-isomorphism σ : R K/k (G ) R L/k (H ) arises from a unique pair (α, ϕ). By Corollary 8.1.4, in the k-group R K/k (G ) = R Ki /k (Gi ) the factors R Ki /k (Gi ) are intrinsically determined: they are the (absolutely) pseudo-simple smooth connected normal k-subgroups. Thus, kisomorphisms between two such k-groups must arise from a collection of isomorphisms between the pseudo-simple factors. This reduces our problem to the case where K and L are ﬁelds, in which case if the k-isomorphism σ exists then Proposition 8.1.5 shows that there is a pair (α, ϕ) giving rise to σ . For the uniqueness of (α, ϕ), since k = ks and K is a ﬁeld we just have to show for ϕ ∈ Aut K (G ) that if R K /k (ϕ) restricts to the identity on G = R K/k (G )

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then ϕ = id G . By Proposition 8.1.5(1), ϕ induces the identity on G /ZG , so indeed ϕ is the identity. 2 Corollary 8.2.5 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3} and G an exotic absolutely pseudo-simple k-group of type F4 when p = 2 and type G2 when p = 3. Every k-automorphism of G is inner. Proof. Consider the unique triple (G , K /K, G) that gives rise to G (see Proposition 8.1.5), so G = R K/k (G ) for the K-subgroup G ⊆ R K /K (G ) that is the preimage of the Levi K-subgroup G ⊆ R K /K (G ). The condition of G being absolutely pseudo-simple says precisely that K is a ﬁeld purely inseparable over k. In particular, K has no nontrivial k-automorphisms. Thus, by Proposition 8.2.4, every k-automorphism of G can be written as R K/k (σ ) for the K-automorphism σ of G induced by R K /K (ϕ) for a unique K -automorphism ϕ of G such that the induced automorphism ϕ of the very special quotient G = G K of G is deﬁned over K. It therefore sufﬁces to prove that any such ϕ is conjugation by a point of the subgroup G (K) ⊆ R K /K (G )(K) = G (K ). Since G is absolutely simple of adjoint type and has no nontrivial diagram automorphisms, by Corollary A.4.7(1) its K -automorphism ϕ is conjugation by a unique point g ∈ G (K ). Thus, ϕ is conjugation on G = G K by the image point g ∈ G (K ). But recall that ϕ descends to a K-automorphism of G. This descent must be conjugation by a point g ∈ G(K) (by again applying Corollary A.4.7(1)). The K -points g, g ∈ G (K ) induce the same conjugation automorphism ϕ of G , so these points are equal (as ZG = 1). This says that the pair (g, g ) ∈ G(K) × G (K ) lies in the subset G(K) ×G (K ) G (K ) = (G ×R

K /K (G

)

R K /K (G ))(K) = G (K).

By construction, conjugation on G by this point in G (K) induces ϕ.

2

Here is an interesting application to the structure of the automorphism schemes that were studied in Theorem 2.4.1. Proposition 8.2.6 Consider a 5-tuple (K, K , G , G , G) as above, a Cartan Ksubgroup C of G , and the associated Cartan k-subgroup C = R K/k (C ) of G = R K/k (G ). Let ZG ,C denote the commutative ﬁberwise maximal smooth K-subgroup of the automorphism scheme Aut G ,C , and deﬁne ZG,C ⊆ Aut G,C similarly. The k-homomorphism R K/k (Aut G ,C ) → Aut G,C ,

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deﬁned by applying R K/k to automorphisms, carries R K/k (ZG ,C ) isomorphically onto ZG,C . Strictly speaking, we have only deﬁned Aut G ,C when the base is a ﬁeld. For a base such as Spec K we deﬁne it ﬁberwise. Proof. We may and do assume k = ks , so the factor ﬁelds Ki of K are purely inseparable over k. In particular, a k-algebra automorphism of K is uniquely determined by its effect on the primitive idempotents of K. But according to Corollary 8.1.4, these idempotents are in natural bijection with the pseudo simple factors Gi of G. Since C = Ci where Ci = C ∩ Gi is a Cartan k-subgroup of Gi , with each Ci = 1, the description of k-automorphisms of G in Proposition 8.2.4(1) implies that those automorphisms that restrict to the identity on C are precisely R K/k (ϕ) for a K-automorphism ϕ of G such that R K/k (ϕ) restricts to the identity on C = R K/k (C ). Passing to k-points shows that it is equivalent to require that ϕ restricts to the identity on C . This proves that the natural map Aut G ,C (K) → Aut G,C (k) is bijective. By construction ZG ,C (K) = Aut G ,C (K), and similarly ZG,C (k) = Aut G,C (k). Thus, the map on k-points induced by the natural map R K/k (ZG ,C ) → ZG,C is Aut G ,C (K) → Aut G,C (k). In general, if f : H → H is a k-homomorphism between (possibly disconnected) smooth k-groups of ﬁnite type then we claim f is an isomorphism if and only if H (F) → H (F) is bijective for all separable extension ﬁelds F/k. In case H and H are connected, hence integral, this was shown near the beginning of the proof of Theorem 1.3.9. In general, we can assume k = ks , so each connected component is geometrically connected and contains a k-point, so the function ﬁeld at any generic point coincides with the function ﬁeld of the identity component. In particular, each generic point of H is the image of a kmorphism h : Spec k(H 0 ) → H , so taking F = k(H 0 ) provides a k-morphism h : Spec k(H 0 ) → H lifting h. Taking h to be the canonical map h0 hitting the generic point of H 0 , the associated map h0 lands in some connected component Hi of H . The resulting k-map f : Hi → H 0 between integral k-schemes is injective on k(Hi )-valued points and admits a lifting h0 of the canonical k-map h0 , so the method of proof in the integral case can be applied to deduce that Hi → H 0 is birational. It follows via translation by the inverse of any k-point of Hi that the k-homomorphism f 0 : H 0 → H 0 is birational, hence an isomorphism, so ker f is ﬁnite étale. But f is injective on k-points with k = ks , so f is an open k-subgroup. Bijectivity on k-valued points then forces f to be an isomorphism, as desired. We conclude that R K/k (ZG ,C ) → ZG,C is an isomorphism provided that it is bijective on F-points for all separable extension ﬁelds F/k. It even sufﬁces to

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check bijectivity on Fs -points, as then passing to Gal(Fs /F)-invariants would give the bijectivity for F-points. Since K ⊗k F is reduced (as F/k is separable), we can therefore rename F as k to reduce to the case F = k which we have already settled. 2 As an application of Proposition 7.2.7, we can reﬁne Proposition 8.2.6 as follows. Let G be the basic exotic pseudo-reductive k-group arising from a triple (G , k /k, G). Let C be a Cartan k-subgroup of G , and let ZG ,C be the maximal smooth k-subgroup of the k-group Aut G ,C . Using the maximal reductive quotient map q : Gk G , let T = q(Ck ) be the associated maximal k -torus of G . The natural action of (ZG ,C )k on Gk restricting to the identity on Ck factors through q because ZG ,C is k-smooth (so we can check the preservation of ker q = Ru,k (Gk )). In this way we get an action of (ZG ,C )k on G that restricts to the identity on T , and the resulting k -homomorphism (ZG ,C )k → Aut G ,T factors through the maximal smooth k -subgroup T /ZG (Theorem 1.3.9). Thus, by the universal property of Weil restriction, we obtain a k-homomorphism ZG ,C → R k /k (T /ZG ).

(8.2.2)

Corollary 8.2.7 The map in (8.2.2) is a closed immersion, and the image of ZG ,C (k) → R k /k (T /ZG )(k) = (T /ZG )(k ) consists of the points whose action on R k /k (G ) preserves each root group of (G , T ), where T is the maximal k-torus in C . An equivalent requirement on such points when k = ks is that they preserve the root groups corresponding to long roots in a basis of the root system. Proof. To prove that (8.2.2) is a closed immersion, it sufﬁces to prove that its scheme-theoretic kernel is trivial (Proposition A.2.1). By Proposition 7.2.7(2), the natural inclusion G → R k /k (G ) corresponds to the maximal reductive quotient map q : Gk → G over k . Thus, if a point of ZG ,C has trivial image in R k /k (T /ZG ) then its action on R k /k (G ) is trivial on G . But ZG ,C acts through its subgroup inclusion into Aut G ,C , so a point of ZG ,C acting trivially on G is itself trivial. This proves that (8.2.2) has trivial kernel, as desired. To identify the image of (8.2.2) on k-points, we ﬁrst observe that for any k-automorphism σ of G , σk induces a unique k -automorphism ϕ of the maximal pseudo-reductive quotient G over k . Since the pseudo-reductive quotient

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map q over k corresponds to the inclusion of G into R k /k (G ), it follows that the k-automorphisms of G are in natural bijective correspondence with the k -automorphisms ϕ of G such that R k /k (ϕ) preserves G inside of R k /k (G ) (as any endomorphism with trivial scheme-theoretic kernel must be an automorphism). Moreover, if σ ∈ Aut(G )(k) restricts to the identity on C then the corresponding ϕ ∈ Aut(G )(k ) restricts to the identity on T = q(Ck ), and the converse holds as well since C ⊆ R k /k (T ). Hence, Aut G ,C (k) corresponds (via Weil restriction) to the points in Aut G ,T (k ) = (T /ZG )(k ) = R k /k (T /ZG )(k) whose natural action on R k /k (G ) restricts to an automorphism of G . But the T /ZG -action on G respects all root groups and is trivial on the Cartan k -subgroup T , so it restricts to an automorphism of any open Bruhat cell attached to (G , T ). Since open Bruhat cells for (G , T ) are direct products (via multiplication) of C and the root groups, it follows that a point in (T /ZG )(k ) comes from ZG ,T (k) if and only if its action on R k /k (G ) preserves the root groups of (G , T ). Finally, for k = ks we check that preservation of the root groups corresponding to long roots in just one basis of the root system is sufﬁcient. Using the natural identiﬁcation of the k-groups W (G , T ) and R k /k (W (G , T )) via Proposition 3.2.7 and the identiﬁcation of these Weyl groups with the Weyl group of the common reduced root system we infer that the root groups of (G , T ) corresponding to every long root are preserved. The short root groups coincide with those of (R k /k (G ), T ) (where T is the maximal k-torus in R k /k (T )), by Theorem 7.2.3(2), so there is nothing to check for the short roots. 2 Over imperfect ﬁelds k of characteristic 2 or 3, certain exotic pseudoreductive groups will uniquely arise as direct factors in the classiﬁcation. A related result will hold in characteristic 2 (assuming [k : k 2 ] = 2) for nonreduced components of the root system. The direct factor result in the exotic case will be treated now, and the method will be applicable when we later study how non-reduced root systems can arise. As a ﬁrst step, we check that the property of a k-group being exotic (in the sense of Deﬁnition 8.2.2) is unaffected by extending the ground ﬁeld to ks . More generally: Proposition 8.2.8 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3}, and F/k a separable extension. A k-group G is exotic over k if and only if GF is exotic over F.

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Proof. If GF is exotic then it is non-commutative and pseudo-reductive, so the same holds for G over k. Thus, throughout this proof we may and do assume that G is non-commutative and pseudo-reductive over k (and hence it is nonsolvable). We now use Proposition 8.1.5 to give a necessary and sufﬁcient criterion for G to be exotic, and the criterion will be visibly insensitive to separable extension on k. Consider the simply connected datum (G , K /k, f ) for the non-solvable smooth connected afﬁne k-group G, with a nonzero ﬁnite reduced k-algebra K and a quotient map f : GK → G /ZG over K with adjoint semisimple target having absolutely simple ﬁbers. There is at most one factorization f

GK

/ G FF FF FF FF f # G /ZG

(8.2.3)

Suppose such a lift f exists and that each ﬁber of G over Spec K has an edge of multiplicity p in its Dynkin diagram; by Proposition 8.1.5(1) this is a necessary condition for G to be exotic. In particular, there is a (ﬁberwise) very special isogeny π : G → G over Spec K . By the universal property of Weil restriction, f corresponds to a k-group homomorphism j : G → R K /k (G ), and again using Proposition 8.1.5(1) we see that if G is exotic then ker j = 1. We now assume ker j = 1, so Lie(G) is identiﬁed with a k-subspace of Lie(R K /k (G )) = Lie(G ). Let K ⊆ K be the maximal k-subalgebra of K such that Lie(G) is a K-submodule of the K-module Lie(G ). Note that K is a nonzero ﬁnite reduced k-algebra. Expressing the target of j as R K/k (R K /K (G )), the map j corresponds to a K-group map GK → R K /K (G ). Let G be the image of this map, and consider the image G of G under the map

h : R K /K (π ) : R K /K (G ) → R K /K (G ), so G ⊆ h−1 (G). By Proposition 8.1.5(2), if G is exotic then the following properties must hold: K p ⊆ K, the ﬁber-degrees of the map Spec K → Spec K are all > 1, G is a Levi K-subgroup of R K /K (G ), G = h−1 (G), and the map G → R K/k (G ) is an isomorphism. Assuming all of these hold, the K-smoothness of h−1 (G) forces G to be the ﬁberwise basic exotic pseudoreductive K-group associated to (G , K /K, G) by Proposition 7.3.1. We can avoid Proposition 7.3.1 by observing the alternative necessary condition that for

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Ks := K ⊗k ks , the Ks -group GKs contains a Levi Ks -subgroup of R K /K (G )Ks , which again implies the same conclusion relating G to (G , K /K, G). But then the isomorphism G R K/k (G ) implies that G is exotic. We can summarize the preceding considerations as an intrinsic characterization of when a non-commutative pseudo-reductive k-group G is exotic. It is necessary and sufﬁcient that the following hold. First, we form the simply connected datum (G , K /k, f ) associated to G. It is necessary that each ﬁber of G over K (all of which are absolutely simple and simply connected) has an edge of multiplicity p in its Dynkin diagram, and that we can lift f as in (8.2.3). Assum ing these hold, we can form the very special K -isogeny π : G → G , and f provides us j : G → R K /k (G ). Necessarily ker j = 1, and then we can deﬁne the k-subalgebra K ⊆ K , the K-group G , and the K-subgroup G ⊆ R K /K (G ) p as above. It is necessary that K ⊆ K (which forces K and K to have com patible decompositions into factor ﬁelds: K = Ki , K = Ki ), each ﬁber of Spec K → Spec K has degree > 1, G is a Levi K-subgroup of R K /K (G ), G = R K /K (π )−1 (G), and GKs contains a Levi Ks -subgroup of R K /K (G )Ks or equivalently a Levi Ks -subgroup of itself. (This ﬁnal condition can be omitted if we are willing to use Proposition 7.3.1, or to introduce the condition that G is pseudo-reductive and invoke the existence of Levi subgroups in pseudoreductive groups over a separably closed ﬁeld via Theorem 3.4.6.) Moreover, when all of these conditions do hold the k-group G is exotic. We just have to check that these conditions interact well with separable extension of the ground ﬁeld. To be precise, since the formation of the simply connected datum commutes with any separable extension of the ground ﬁeld (by Proposition 4.2.1) and any tensor product of ﬁelds F ⊗k K is a ﬁnite product of ﬁelds when F/k is separable and K/k is ﬁnite, it sufﬁces to verify three further compatibilities: (i) the existence or not of f is insensitive to separable extension of the ground ﬁeld, (ii) for a k-algebra A and k-subspace W of an A-module V , the formation of the maximal k-subalgebra F ⊆ A such that W is an F-submodule commutes with any extension on k, (iii) the existence or not of a Levi subgroup in a smooth connected afﬁne group over a separably closed ﬁeld is insensitive to separable extension to another separably closed ﬁeld. In (ii), F is the kernel of the k-linear map A → Homk (W , V /W ) deﬁned by a → (w → aw mod W ).

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Thus, the compatibility in (ii) follows from the injectivity of the natural map L ⊗k Homk (W , W ) → HomL (WL , WL ) for any ﬁeld L/k. Assertion (iii) is an immediate consequence of specialization arguments as in the proof of Proposition 1.1.9(1). For (i), we ﬁrst note that the existence of f is equivalent to the splitting of the f -pullback central extension 1 → ZG → E → GK → 1 with E = GK ×f ,G /ZG G . Since GK is K -smooth with connected ﬁbers and ZG is K -ﬁnite, a necessary and sufﬁcient condition for the existence of a 0 of the identity section splitting is that the ﬁberwise connected component Ered in the underlying reduced scheme Ered maps isomorphically onto GK as a K scheme. But the formation of the underlying reduced scheme of a K -scheme commutes with separable extension on the factor ﬁelds of K , so we are done. 2 To complete the theory of exotic pseudo-reductive groups what remains is to determine their automorphisms that pointwise ﬁx a Cartan subgroup: Example 8.2.9 Let k be an imperfect ﬁeld of characteristic p ∈ {2, 3}, K a nonzero ﬁnite reduced k-algebra, and G = R K/k (G ) for a K-group G whose ﬁbers are basic exotic pseudo-reductive groups all of the same type (which we call the type of G). Let (G , K /K, G) be the triple giving rise to G over Spec K. Let T ⊆ G be a maximal k-torus, C = ZG (T ), and T ⊆ G the unique maximal K-torus such that T is the maximal k-torus in R K/k (T ) (see Proposition A.5.15(2)). Let C = ZG (T ) be the corresponding Cartan K-subgroup of G . The identiﬁcation of G with the maximal reductive quotient of GK carries TK isomorphically onto a maximal K -torus T ⊆ GK . Let Aut G,C be the functor considered in Theorem 2.4.1 (where it was shown to be represented by an afﬁne k-group of ﬁnite type). The action of R K/k (G /ZG ) on G = R K/k (G ) is faithful (use the canonical quotient map R K/k (G )K G ; see Proposition A.5.11(1)), so the group scheme Aut G,C naturally contains R K/k (C /ZG ) as a subgroup. This latter k-group is smooth and connected, so it must lie in the maximal smooth connected k-subgroup ZG,C . Explicitly, Proposition 8.2.6 identiﬁes ZG,C with R K/k (ZG ,C ), and Corollary 8.2.7 determines ZG ,C as a K-subgroup of R K /K (T /ZG ). This computes ZG,C as a k-subgroup of R K /k (T /ZG ). At this point something wonderful happens: in characteristic 3, pseudosimple exotic pseudo-reductive normal k-subgroups always uniquely split off

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as a direct factor. Our proof of this splitting behavior also works over imperfect ﬁelds of characteristic 2 for pseudo-simple exotic pseudo-reductive normal k-subgroups of type F4 , as we show in Theorem 8.2.10 below. Such splitting behavior rests on two properties that exotic absolutely pseudo-simple groups G of types F4 and G2 share in common with GKss : trivial center and only inner automorphisms. Theorem 8.2.10 Let k be an imperfect ﬁeld with characteristic p ∈ {2, 3}. Let G be a non-standard pseudo-reductive k-group. (1) If p = 3 then there is a unique decomposition G = G1 × G2 into commuting k-subgroups such that G1 is a (possibly trivial) standard pseudo-reductive k-group and G2 is an exotic pseudo-reductive k-group in the sense of Deﬁnition 8.2.2. (2) If p = 2 and Gks contains a pseudo-simple normal k-subgroup that is exotic of type F4 then there is a unique decomposition G = G1 × G2 into commuting k-subgroups such that (G1 )ks has no pseudo-simple normal ks subgroups that are exotic with type F4 and G2 is an exotic pseudo-reductive k-group whose pseudo-simple factors are all of type F4 . Theorem 8.2.10(1) and Corollary 6.3.5 yield the classiﬁcation of pseudoreductive groups in characteristic 3. In our later work with generalized standard pseudo-reductive groups, we will handle characteristics 2 and 3 by a uniform method when the root system over ks is reduced (assuming [k : k 2 ] 2 when char(k) = 2). Proof. Since we are claiming uniqueness, by Galois descent and Corollary 5.2.3 and Proposition 8.2.8 (and Corollary 8.1.4) we may and do assume that k is separably closed. The non-standardness hypothesis on G implies that the pseudo-reductive k-group G := D (G) is not standard (Proposition 5.2.1). In particular, G is non-commutative. By Proposition 1.2.6, G is its own derived group. Thus, by Proposition 3.1.8, the collection {Gi }i∈I of pseudo-simple normal k-subgroups of G is pairwise commuting and in natural bijection with the set of simple factors of the maximal semisimple quotient of G . By Proposition k 1.2.7 each Gi is normal in G. Step 1. The ﬁrst part of the argument is devoted to proving that the smooth connected k-subgroup generated by the Gi that are exotic with type F4 or G2 is actually a direct product of such subgroups Gi . Consider the disjoint union decomposition I = I1 I2 in which I2 is the set of i ∈ I such that Gi is exotic of type F4 when p = 2 and type G2 when p = 3. Note that if p = 3 then every pseudo-simple k-group is either standard or exotic (Corollary 6.3.5 and

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Theorem 8.2.1), so when p = 3 we can equivalently say that I1 is the set of i ∈ I such that Gi is standard. (This description is not generally true when p = 2.) By Proposition 5.3.1 non-standardness of G implies that of some Gi , so if p = 3 then the hypothesis on G says I2 = ∅. When p = 2, it is assumed in (2) that I2 = ∅. Let H1 (resp. G2 ) denote the smooth connected normal k-subgroup of G generated by the Gi for i ∈ I1 (resp. i ∈ I2 ). Note that G2 = 1 since I2 is non-empty, and H1 commutes with G2 . By Proposition 3.1.8 applied to G , the Gi for i ∈ I1 (resp. i ∈ I2 ) are the pseudo-simple normal k-subgroups of H1 (resp. of G2 ). Also, observe that H1 is non-commutative if it is nontrivial (i.e., if I1 is non-empty). Hence, by Proposition 5.3.1, the pseudo-reductive k-group H1 is standard. For any i ∈ I2 , the k-group Gi has trivial intersection with the subgroup generated by the Gj for all j ∈ I − {i} since such an intersection is central in Gi yet ZGi = 1 (Corollary 7.2.5). Thus, the natural map i∈I2 Gi → G2 is an isomorphism (so G2 is exotic) and the natural map H1 × G2 → G has trivial kernel. By Proposition 3.1.8, this proves that a perfect pseudo-reductive k-group whose pseudo-simple normal k-subgroups are all exotic of type F4 or G2 is a direct product of such k-subgroups. Step 2. We next carry out the construction of G1 as a subgroup generated by H1 and a piece of a Cartan k-subgroup of G (the idea being to ﬁll out what should be the Cartan k-subgroup of G1 , in case G is not perfect). The quotient G/(H1 ×G2 ) is commutative, or in other words the inclusion H1 × G2 ⊆ G = D (G) is an equality by Proposition 3.1.8. Let T be a maximal k-torus in G. Since the formation of torus centralizers is compatible with quotients of smooth connected afﬁne groups, it follows from the commutativity of G/(H1 × G2 ) that ZG (T ) maps onto G/(H1 × G2 ). Equivalently, G = ZG (T ) · (H1 × G2 ). Note that ZG (T ) is commutative by Proposition 1.2.4, but ZG (T ) ∩ G2 = 1 (e.g., this intersection contains the ksubtorus of T that is maximal in the non-solvable normal k-subgroup G2 of G). We seek a smooth connected k-subgroup Z1 of ZG (T ) which centralizes G2 and such that G = Z1 · (H1 × G2 ). For a suitable such Z1 we will be able to take G1 = Z1 · H1 . The k-subtorus T2 = T ∩ G2 of T is maximal in G2 . The image of (G2 )k in G red is a product among simple factors of the derived group of G red , so k

k

ZG2 (T2 )k and (T2 )k have the same image in G red . In particular, the commutak

tor (ZG2 (T2 ), ZG (T ))k has trivial image in G red , so (ZG2 (T2 ), ZG (T )) = 1 by k Lemma 1.2.1. Equivalently (in view of the commutativity of ZG (T )), we have ZG2 (T2 ) ⊆ ZG (T ).

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Let Z = ZG (T ) and Z2 = ZG2 (T2 ). Every z ∈ Z(k) centralizes T2 and so the conjugation action of z on the normal subgroup G2 ⊆ G is an automorphism of G2 which is trivial on the maximal k-torus T2 . By Corollary 8.2.5 every such automorphism of G2 is inner, so there exists z2 ∈ G2 (k) such that zz2−1 centralizes G2 . In particular, z2 ∈ ZG2 (T2 )(k) = Z2 (k), so Z(k) = C · Z2 (k) with C denoting the subgroup of elements of Z(k) that centralize G2 . The identity component Z1 of the Zariski closure of C in Z = ZG (T ) centralizes G2 , and Z1 (k)Z2 (k) has ﬁnite index in Z(k) = C · Z2 (k) (since C/Z1 (k) is ﬁnite, due to ﬁniteness of the component group of the Zariski closure of C in Z). Hence, (Z/Z1 Z2 )(k) = Z(k)/(Z1 Z2 )(k) is ﬁnite, yet k = ks and Z/(Z1 Z2 ) is smooth and connected, so Z = Z1 Z2 . But Z1 centralizes G2 and Z2 is contained in G2 , so G = Z · (H1 × G2 ) = (Z1 · H1 ) · G2 with Z1 · H1 centralizing G2 . Let G1 = Z1 ·H1 , so G1 is a smooth connected k-subgroup of G that centralizes the normal k-subgroup G2 and satisﬁes G = G1 · G2 . Thus, G1 is normal in G and so is pseudo-reductive over k. The intersection G1 ∩ G2 is central in G2 , so G1 ∩ G2 = 1 since ZG2 = 1. Hence, the natural map G1 × G2 → G is an isomorphism. Step 3. Finally, we prove that the pair (G1 , G2 ) really works, and that it is unique. Among the pseudo-simple normal k-subgroups Gi of G, the ones that lie in G1 are precisely the Gi for i ∈ I1 . It follows once again via Proposition 1.2.7 and the normality of G1 in G that every pseudo-simple normal k-subgroup of G1 is normal in G and hence (by Proposition 3.1.8) is equal to some Gi . If p = 3 then these are standard by deﬁnition of I1 , so G1 is standard by Proposition 5.3.1. This concludes the proof of the existence of the desired decomposition of G. The uniqueness of the decomposition of G is a consequence of the construction of G1 and G2 , as follows. Let G1 × G2 be a decomposition of G with G1 having no exotic pseudo-simple normal k-subgroups of type F4 or G2 and G2 an exotic pseudo-reductive k-group whose pseudo-simple factors are all of type F4 or G2 . Since the scheme-theoretic centers ZG2 and ZG2 are trivial, the decompositions G = G1 × G2 and G = G1 × G2 imply G1 = ZG (G2 ) and G1 = ZG (G2 ). Hence, it sufﬁces to prove that G2 = G2 . By Proposition 3.1.8 the pseudo-simple normal k-subgroups of G2 are among the Gi for i ∈ I2 , so by deﬁintion of G2 being exotic we have G2 = j∈J Gj for some subset J ⊆ I2 . We just have to rule out the possibility J = I2 . If there exists i ∈ I2 − J then Gi centralizes G2 , so Gi is contained in ZG (G2 ) = G1 . But G1 has no exotic pseudo-simple normal k-subgroups of type F4 or G2 and Gi is an exotic pseudo-simple k-group of type F4 or G2 , so we have a contradiction. 2

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Corollary 8.2.11 Let G be a pseudo-reductive group over an imperfect ﬁeld k with char(k) = 3, and let T be a maximal k-torus in G. The k-group G is a standard pseudo-reductive k-group if and only if for every simple smooth connected normal subgroup H ⊆ G ss of type G2 and any two roots for Tk k

arising in Lie(H ), the k-dimensions of the corresponding root spaces in Lie(Gk ) are equal. Proof. The necessity is clear, and by Theorem 8.2.10 the sufﬁciency is immediate from Proposition A.2.10 and the description (7.2.1) of the Lie algebras of exotic pseudo-reductive groups. 2

9 The absolutely pseudo-simple groups in characteristic 2

Let G be an absolutely pseudo-simple group over a ﬁeld k of characteristic 2 and K/k be the ﬁeld of deﬁnition of R(Gk ) ⊂ Gk . The classiﬁcation of possibilities for G will proceed in several stages, inspired by our work in Chapter 6, as follows. Recall that in §6.2 we proved standardness when G ss is either of the k rank-2 types A2 or G2 whose Dynkin diagram has no edge with multiplicity 2 (in contrast with the other rank-2 type, namely B2 = C2 ). In Proposition 6.3.6 these rank-2 cases were used to prove standardness whenever the simple semisimple k-group G ss is of any type with rank at least 2 and has no edge of multiplicity 2 k (i.e., we allow all types aside from Cn (n 2), Bn (n 2), and F4 ). In Deﬁnition 8.2.2 we introduced a class of non-standard pseudo-reductive k-groups with reduced root systems of rank 2. To go further, we need to understand rank-1 cases: type C1 = A1 , where most of the essential new difﬁculties in characteristic 2 arise (e.g., non-reduced root systems; recall from Remark 8.2.3 that exotic pseudo-reductive groups in the sense of Deﬁnition 8.2.2 have reduced root systems). This will rest on the general structural results from §3.4: existence of Levi subgroups and Bruhat decomposition for pseudo-reductive groups over separably closed ﬁelds. As a preliminary step in the classiﬁcation for type A1 , in §9.1 we determine all possibilities for the image of ξG when G ss is of type A1 and k = ks . The existence of a Levi k k-subgroup in G and the Bruhat decomposition for G(k) will enable us to show that the possibilities for the image of ξG are parameterized by the K × -homothety classes of kK 2 -subspaces V ⊆ K that contain 1 and generate K as a k-algebra. If [k : k 2 ] = 2 then the only such V is K. We use this to prove in §9.1 that if [k : k 2 ] = 2 and G has rank 1 and a reduced root system then G is standard. (If [k : k 2 ] 4 then any K/k contained in k 1/2 with [K : k] 4 admits a V = K. Such examples are not covered by the constructions in our main classiﬁcation theorem in characteristic 2.) This step is the source of our restriction to the case [k : k 2 ] = 2 in our classiﬁcation theorem in characteristic 2. 279

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The absolutely pseudo-simple groups in characteristic 2

9.1 Type A1 Let k be a separably closed ﬁeld of characteristic 2 and G a pseudo-simple k-group. Let K/k denote the ﬁeld of deﬁnition of R(Gk ) ⊆ Gk . Assume GKss is of type A1 , which is to say that it is K-isomorphic to SL2 or PGL2 . Our complete description of all such G requires extensive preparations with constructions corresponding to the non-reduced root system of rank 1 (and the assumption [k : k 2 ] = 2), so for now we restrict ourselves to a more limited goal: we seek to describe the image of ξG in R K/k (GKss ) in general, at least after ﬁxing a suitable K-isomorphism of GKss with SL2 or PGL2 , and to prove that G is standard when its root system is reduced and [k : k 2 ] 2. Choose a maximal k-torus T in G (so T GL1 ) and ﬁx a K-group isomorphism from GKss to SL2 or PGL2 such that TK is carried to the diagonal torus D. This isomorphism is unique up to composition with the action of an element of PGL2 (K) normalizing the diagonal torus, so any construction we make that is invariant by such an action is intrinsic to (G, T ). By Proposition 3.2.7, W (G, T )(k) maps isomorphically onto the group W (GKss , TK )(K) of order 2. Thus, the elements of NG (T )(k) − ZG (T )(k) represent the unique nontrivial class in W (G, T )(k). By Proposition 3.4.2, for each nontrivial k-point u in any of the root groups for (G, T ) there is a unique element in NG (T )(k) − ZG (T )(k) of the form ν = u uu where u is a k-point in the opposite root group (and u is then uniquely determined by u), and ν 2 = 1. Fix such a u; the image of u under the map π : G(k) → GKss (K) lies in one of the two root groups for (GKss , TK ). Change the K-isomorphism of GKss with SL2 0 1 so that π(u) or PGL2 by conjugation by the Weyl element −1 0 if necessary lies in the upper triangular root group for D: π(u) = 10 1t for some t ∈ K. The proof of Proposition 3.4.2 shows that π(u) = 1, so t ∈ K × . Thus, we can further change our choice of K-isomorphism from GKss to SL2 or PGL2 by composing t −1 0 0 1

∈ PGL2 (K) to get to the case t = 1. But then 1 0 we see by explicit calculation that π(u ) = −1 1 , so for the resulting k-map

it with conjugation by

iG : G → R K/k (SL2 ) or R K/k (PGL2 ) we have n :=

0 −1

1 = π(ν) ∈ iG (NG (T )(k)[2]). 0

(9.1.1)

By pseudo-reductivity of G, ker iG contains no nontrivial smooth connected unipotent k-subgroups.

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The root system (GKss , TK ) ⊆ X(TK ) = X(T ) Z has the form {±a0 } for a unique nontrivial character a0 of T (or equivalently, of TK ) such that under the chosen isomorphism from (GKss , TK ) to (SL2 , D) or (PGL2 , D) the root group of a0 is the strictly upper triangular matrix group U + in SL2 or PGL2 . We write U − for the corresponding opposite root group over K. We shall use the standard isomorphisms Ga U ± . Consider the smooth connected k-group R K/k (GKss ) that we have identiﬁed with R K/k (SL2 ) or R K/k (PGL2 ). Under this identiﬁcation, T gets identiﬁed via iG with the diagonal k-torus in R K/k (SL2 ) or R K/k (PGL2 ), so the smooth connected k-subgroup H := iG (G) in R K/k (GKss ) contains the diagonal k-torus. By Proposition 4.2.5 the k-group H is pseudo-reductive, and its Cartan k-subgroup containing the diagonal ktorus is Z = H ∩ R K/k (D), so Z(k) = H (k) ∩ D(K). Clearly (H , T ) ⊆ (GKss , TK ) = {±a0 }, so these root systems coincide. Since a0 is non-multipliable in (G, T ), by Proposition 2.3.11 the T -stable root group Ua0 ⊆ G is a vector group such that Lie(Ua0 ) is the root space in Lie(G) for a0 . We will work with the T -stable U(±a0 ) = UG (±a0∨ ), which is strictly larger than U±a0 when a0 is divisible in (G, T ). The image iG (U(±a0 ) ) is a nontrivial smooth connected k-subgroup of R K/k (U ± ) = R K/k (Ga ) that is stable under the natural GL1 -action (due to the equivariance of iG relative to the K-isomorphism TK D GL1 that is necessarily deﬁned over k), so it is a nonzero vector subgroup with respect to the standard k-linear structure on R K/k (Ga ). Let V±a0 ⊆ K be the corresponding nonzero k-subspace. Using conjugation by n we see that Va0 = V−a0 inside of K. Let V ⊆ K denote this common nonzero k-subspace. The above calculation of π(u) also shows that 1 ∈ V. Lemma 9.1.1 The subset V −{0} ⊆ K × is stable under inversion, and V 2 ·V ⊆ V where V 2 denotes the set of squares of elements of V . Proof. Applying Corollary 2.1.9 to the quotient map G H induced by iG and applying Remark 2.3.6 to the inclusion of H into R K/k (GKss ), the root groups UH± for (H , T ) are the smooth connected k-subgroups iG (U(±a0 ) ) ⊆ R K/k (U ± ) = R K/k (Ga ) whose k-points are V±a0 = V ⊆ K. The Bruhat decomposition for H (k) relative to the minimal pseudo-parabolic k-subgroup Z UH− is provided by Theorem 3.4.5, and it says H (k) = UH− (k)nZ(k)UH− (k) Z(k)UH− (k).

(9.1.2)

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Since UH+ ∩ (Z UH− ) = {1} (Corollary 3.3.16), we have UH+ (k) − {1} ⊆ UH− (k)nZ(k)U − (k). Multiplication deﬁnes an open immersion UH+ × Z × UH− → G (Corollary 3.3.16), so it follows from the equality UH+ = n−1 UH− n and left translation by n that the map UH− × Z × UH− → G deﬁned by (u , z, u ) → u nzu is an open immersion. In particular, for u , u ∈ UH− (k) and z ∈ Z(k) the product u nzu ∈ G(k) uniquely determines u , u , and z. Thus, for each nonzero x ∈ V there exist unique y1 , y2 ∈ V and z ∈ Z(k) such that

1 0

x 1 = 1 y1

0 0 · 1 −1

1 1 ·z· 0 y2

0 1

(9.1.3)

in SL2 (K) or PGL2 (K), depending on whether GKss is simply connected or adjoint (of type A1 ). But SL2 (K) ⊆ PGL2 (K) since char(K) = 2, and all terms in (9.1.3) aside from z come from SL2 (K), so even when GKss PGL2 we have that z uniquely arises from SL2 (K). Hence, in all cases z arises from a unique −1 matrix of the form t0 0t with t ∈ K × . We wish to compute y1 , y2 , and t in terms of x. Multiplying out the right side of (9.1.3) now gives

1 0

x ty2 = ty1 y2 − t −1 1

t ty1

in SL2 (K), so t = x and y1 = 1/t = y2 . Hence, 1/x = y1 ∈ V . This proves that the subset V − {0} ⊆ K × is stable under inversion and moreover that if x ∈ V − {0} then

x 0 ∈ H (k) (9.1.4) 0 x−1 inside of R K/k (SL2 ) or R K/k (PGL2 ) (according to whether GKss is simply connected or adjoint). Thus, for any x ∈ V the product

x 0

−1 1 x x · · −1 0 0 1 x 0

0 1 = x 0

x2 x 1

lies in H (k). In other words, x2 x ∈ V for all x, x ∈ V (as the case x = 0 is trivial). This says exactly that V 2 · V ⊆ V . 2 By Lemma 9.1.1 and the property that 1 ∈ V , for the subﬁeld L := k(V ) = k[V ] ⊆ K generated by V over k we have that kL2 ⊆ V ⊆ L with V a kL2 -subspace of K.

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Lemma 9.1.2 The subﬁeld L = k(V ) is equal to K. In particular, V is a nonzero kK 2 -subspace of K. Proof. Since H = iG (G), we have D (H ) = H . We will prove that the Zariski-dense commutator subgroup (H (k), H (k)) in H is contained in SL2 (L) inside of SL2 (K), so H is contained in R L/k (SL2 ) or in the derived group R L/k (SL2 )/R L/k (μ2 ) ⊆ R L/k (PGL2 ) according to whether GKss is simply connected or adjoint. The minimality of K/k will then force L = K as desired. To analyze the commutator subgroup (H (k), H (k)), we use the Bruhat decomposition (9.1.2) and an additional property of the diagonal entries of each z ∈ Z(k) = H (k) ∩ D(K): writing z = diag(t1 , t2 ) for some t1 , t2 ∈ K × (with t2 = t1−1 in the SL2 -case, though we will not use this), since z normalizes UH+ (k) we have (t1 /t2 )V ⊆ V . Hence, t1 /t2 ∈ V ⊆ L since 1 ∈ V . It follows that z normalizes SL2 (L), so the Bruhat description (9.1.2) of H (k) gives that H (k) ⊆ SL2 (L) · Z(k). We conclude that (H (k), H (k)) ⊆ SL2 (L) since Z(k) is commutative. 2 By Lemma 9.1.2, V is a kK 2 -subspace of K containing kK 2 and it generates K as a k-algebra. Conversely, observe that for any kK 2 -subspace V of K, the subset V −{0} is stable under inversion in K × since v −1 = v −2 v with v −2 ∈ K 2 and v in the kK 2 -subspace V . G ss denote the simply connected central cover, and Remark 9.1.3 Let G K μ the kernel of this isogeny. We claim that the case V = K is precisely the case where the map )/R K/k (μ) = D (R K/k (G ss )) ξG : G → R K/k (G K is surjective. To see this, ﬁrst recall that by deﬁnition of ξG , the map iG is the composition of ξG with the inclusion of D (R K/k (GKss )) into R K/k (GKss ). Hence, we always have an inclusion H ⊆ D (R K/k (GKss )) and ξG is surjective precisely when this is an equality. Since D (R K/k (GKss )) has its root groups (relative to T ) equal to the root groups R K/k (U ± ) of R K/k (GKss ) that have dimension [K : k], whereas H has root groups of dimension dimk V , if ξG is surjective then the inclusion of k-vector spaces V ⊆ K is an equality. Conversely, if V = K then H contains the k-subgroups R K/k (U ± ) that are the images of a pair of opposite root groups of ). It follows that ξG is surjective in such cases because G (K) = SL2 (K) R K/k (G is generated by its upper and lower triangular unipotent subgroups. We now aim to show that in the preceding setup (with 1 ∈ V ), the Cartan ksubgroup ZH (T ) is determined by V , so in fact the k-subgroup H in R K/k (SL2 )

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or R K/k (PGL2 ) is determined by V . The description of ZH (T ) will turn out to be given by the following construction. Deﬁnition 9.1.4 Let K/k be a ﬁnite purely inseparable extension with char(k) = 2, and let G0 be the K-group SL2 or PGL2 . Let D ⊆ G0 be the diagonal K-torus, and U + the upper triangular unipotent K-subgroup equipped with its canonical K-isomorphism U + Ga . For a kK 2 -subspace V ⊆ U + (K) = K containing 1 and satisfying k[V ] = K, the smooth connected k-subgroup ∗ ⊆ R K/k (D) VK/k

is the Galois descent of the Zariski closureof the subgroup of SL2 (Ks ) (resp. PGL2 (Ks )) generated by the matrices v0 1/0v ∈ SL2 (Ks ) for v ∈ Vks − {0}. (Note that SL2 (Ks ) ⊆ PGL2 (Ks ) since char(k) = 2.) ∗ ) = V∗ × By construction, (VK/k ks Ks /ks . Also, since ks ⊆ Vks −{0}, the maximal ∗ ∗ does not specify k-torus in R K/k (D) lies in VK/k . Although the notation VK/k whether it is being computed in the SL2 -case or the PGL2 -case, the context will always make the intended meaning clear. Our interest in Deﬁnition 9.1.4 is due to the following result. ∗ , and the k-subgroup Proposition 9.1.5 The Cartan k-subgroup ZH (T ) is VK/k R(Hk ) ⊆ Hk has ﬁeld of deﬁnition over k equal to K. In particular, the k-subgroup H ⊆ R K/k (G0 ) is uniquely determined by V .

Proof. Without loss of generality we can assume that k = ks . We ﬁrst check ∗ ⊆ Z (T ), or equivalently that V ∗ ⊆ H . The subset V − {0} ⊆ K × that VK/k H K/k is stable under inversion, so we just apply the standard identity

t 0

0

t −1

1 = 0

1 1 1 t−1

0 1 1 0

−1/t 1

1 t(1 − t)

0 1

(9.1.5)

expressing a diagonal matrix in SL2 as a product of upper and lower triangular unipotent matrices. (Note that t(1 − t) = t − t 2 , and V contains K 2 .) Since ∗ V is a K 2 -subspace of K, it also follows from the deﬁnition of VK/k as a Zariski-closure that it normalizes the smooth connected k-subgroup UV ⊆ R K/k (U + ) = R K/k (Ga ) corresponding to V ⊆ K. Hence, it makes sense to deﬁne the following subgroups of H (k): ∗ (k), U = UV (k), P = Z U . Z = VK/k

In view of the Bruhat decomposition of R K/k (G0 )(k) = G0 (K) (i.e., SL2 (K) or PGL2 (K)), or even of H (k), the map U ×P → H (k) deﬁned by (u, p) → unp

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285

is injective and we have the disjoint union containment UnP ∪ P ⊆ H (k). It is easy to check that H := UnP ∪ P is stable under inversion and multiplication, so it is a subgroup. It contains the opposite “root groups” U and nUn−1 , yet these generate the pseudo-simple H by Lemma 3.1.5, so H is Zariski-dense in H . The functor R K/k carries the open cell of (G0 , D) to the “open cell” of (R K/k (G0 ), T ) as in Corollary 3.3.16, which in turn meets H in the “open cell” of (H , T ). This latter open locus is n−1 UV n × ZH (T ) × UV , so its lefttranslate by n meets H in UnP. Hence, UnP is Zariski-dense in H , so the closed locus UV n × ZH (T ) × UV in the n-translate of the open cell of R K/k (G0 ) is the Zariski-closure of UnP (as the formation of closure is local in any topo∗ × U , so we conclude that logical space). But this latter closure is UV n × VK/k V ∗ ZH (T ) = VK/k . Let L0 denote the standard k-structure on G0 , so by Proposition A.5.11 the k-subgroup L0 in R K/k (G0 ) is a Levi k-subgroup. It is contained in H since k ⊆ V and L0 (k) is generated by its upper and lower triangular unipotent subgroups. By Lemma 7.2.4, L0 is also a Levi k-subgroup of H and the canonical map π : HK → G0 is a K-descent of the maximal reductive quotient of Hk . Hence, our remaining task is to show that the ﬁeld of deﬁnition over k for ker π ⊆ HK is equal to K. Equivalently, if F ⊆ K is a subﬁeld containing k such that ker π descends to an F-subgroup R ⊆ HF then we wish to prove F = K. Since the Levi k-subgroup L0 in H yields a Levi F-subgroup (L0 )F in HF , this F-subgroup maps isomorphically onto the quotient HF /R. Hence, the K-map π descends to an F-map π0 : HF (L0 )F . (The point is that the descent π0 of π has a target that is the canonical F-descent of the target G0 of π over K.) Passing to Lie algebras of root groups relative to both the standard maximal Ftorus of HF (arising from the natural Levi F-subgroup (L0 )F ) and the diagonal F-torus of the target (L0 )F of π0 , we get an abstract F-linear map F ⊗k V → F that descends the canonical K-linear map K ⊗k V → K induced by Lie(π ). Since π is induced by the canonical quotient map R K/k (G0 )K → G0 that on Lie algebras is the natural multiplication map K ⊗k g0 → g0

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The absolutely pseudo-simple groups in characteristic 2

(with g0 := Lie(G0 )), it follows that the map K ⊗k V → K induced by Lie(π ) is c ⊗ v → cv. Hence, the existence of π0 implies that F · V ⊆ F inside of K, and in particular V ⊆ F. But F is a subﬁeld of K containing k, so F contains k[V ] = K. 2 In view of Proposition 9.1.5, it is natural to wonder if there are any further restrictions to be found on the V which arise from G as above having K as the (minimal) ﬁeld of deﬁnition over k for the subgroup Ru (Gk ) ⊆ Gk . The answer is negative: Proposition 9.1.6 Let k be a ﬁeld of characteristic 2, K/k a ﬁnite purely inseparable extension, G0 one of the two K-groups SL2 or PGL2 , and U ± Ga the root groups relative to the diagonal K-torus D. Let V ⊆ K be a kK 2 -subspace containing 1 such that k[V ] = K, and let V ⊆ R K/k (Ga ) be the corresponding k-subgroup. The smooth connected k-subgroup H = HV ,K/k ⊆ R K/k (G0 ) generated by the k-subgroups UV± ⊆ U ± corresponding to V is absolutely pseudo-simple, the natural map HK → G0 is a K-descent of the maximal reductive quotient of Hk , and K/k is the ﬁeld of deﬁnition for Ru (Hk ). Moreover, H ∩ R K/k (U ± ) = V . Proof. We may and do assume k = ks . Let L0 denote the standard k-structure on G0 , and T the standard k-structure on D (so T is a maximal k-torus in L0 ). As we saw in the proof of Proposition 9.1.5, L0 is a Levi k-subgroup of R K/k (G0 ) that is also contained in H , so H is pseudo-reductive with L0 as a Levi k-subgroup and HK → G0 is a K-descent of the maximal reductive quotient of Hk . In particular, the ﬁeld of deﬁnition F/k for Ru (Hk ) is contained in K/k. Since V is contained in the k-points of the root groups of (H , T ), the method of determination of the ﬁeld of deﬁnition over k for Ru (Hk ) in the proof of Proposition 9.1.5 applies verbatim to HV to show that F = K. The method of determination of the Cartan k-subgroup in the proof of Proposition 9.1.5 applies ∗ and its verbatim to H = HV ,K/k to prove that the centralizer of T in H is VK/k root groups relative to T coincide with V inside of R K/k (U ± ) = R K/k (Ga ). In particular, (9.1.5) implies that H is generated by its pair of opposite root groups relative to T , so it follows from (the proof of) Proposition 3.1.8 that H is pseudo-simple. 2 Remark 9.1.7 In Proposition 9.1.6, if k is a ﬁnite purely inseparable extension of a subﬁeld k0 and k0 [V ] = K then for H := HV ,K/k the absolutely pseudo-simple k-subgroup R k/k0 (H ) in R K/k0 (G0 ) has geometric unipotent radical whose ﬁeld of deﬁnition F over k0 is K. Indeed, we can assume k = ks

9.1 Type A1

287

(since F/k0 is purely inseparable), and F is certainly contained in K. The calculation with Lie algebras of root groups in the proof of Proposition 9.1.5 can be adapted using the diagonal torus of the standard k0 -structure on G0 to show that the multiplication map K ⊗k0 V → K carries F ⊗k0 V into F. Hence, F contains k0 [V ] = K, providing the reverse inclusion. Corollary 9.1.8 Let k be a separably closed ﬁeld of characteristic 2, and let L0 be either of the k-groups SL2 or PGL2 . Fix a ﬁnite extension K/k. (1) Every pseudo-simple k-subgroup H ⊆ R K/k ((L0 )K ) for which K is the ﬁeld of deﬁnition of R(Hk ) over k is PGL2 (K)-conjugate to HV ,K/k for a kK 2 subspace V ⊆ K containing 1 such that k[V ] = K, and every such V arises in this way. If V is such a subspace then so is vV for every v ∈ V − {0}, and HV ,K/k and HV ,K/k are isomorphic as k-groups if and only if V = cV for some c ∈ K × (in which cases necessarily c ∈ V − {0}). (2) If G is a pseudo-simple k-group such that G ss (L0 )k and the ﬁeld of deﬁnik tion over k of R(Gk ) is equal to K then the natural map G → R K/k ((L0 )K ) has image that is PGL2 (K)-conjugate to HV ,K/k for some V as above. The map ξG is surjective if and only if V = K. Proof. All of the assertions follow from Proposition 9.1.5 and Proposition 9.1.6 (along with Remark 9.1.3) except for the ﬁnal part of (1) concerning variation of V . That is, we have to show that if H := HV ,K/k and H := HV ,K/k are k-isomorphic then V = vV inside of K for some v ∈ V − {0}, and that for any v ∈ V − {0} the kK 2 -subspace vV of K contains 1 and generates K as a k-algebra. Consider an arbitrary isomorphism f : H H as k-groups. The isomorphism fK induces an isomorphism between the maximal reductive quotients HK /Ru,K (HK ) and HK /Ru,K (HK ) over K. These quotients are each identiﬁed with (L0 )K , so we obtain a K-automorphism of (L0 )K . Such an automorphism is given by the action of a unique element g ∈ PGL2 (K), and it is clear that the g-action on R K/k ((L0 )K ) carries H over to H . Thus, it remains to study when there is a PGL2 (K)-conjugacy between H and H . Let T be the diagonal k-torus in L0 , so T is a maximal k-torus in each of H and H . By conjugacy of maximal k-tori in H , we can modify g so that it preserves T . Modifying g by a representative of the nontrivial element of W (L0 , T )(k) if necessary brings us to the case that g centralizes T . But T (k) has Zariski-dense image in the diagonal K-torus of PGL2 , so g lies c 0maximal in this diagonal K-torus. That is, g = 0 1 for some c ∈ K × . The action of g on R K/k (U ± ) = R K/k (Ga ) is multiplication by c±1 , so intersecting these root groups with H and H gives V = cV and V = c−1 V . Since 1 ∈ V , we have

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The absolutely pseudo-simple groups in characteristic 2

1 = cv for some v ∈ V − {0}. Hence, c = 1/v. Since V − {0} is stable under inversion, we have c ∈ V − {0}. Conversely, apply conjugation by 0c 01 ∈ PGL2 (K) to HV ,K/k for c ∈ V − {0}. This conjugate is an absolutely pseudo-simple k-subgroup H of R K/k ((L0 )K ) containing the standard diagonal k-torus T as a maximal k-torus and having root groups (relative to T ) that correspond to the common kK 2 subspace cV = c−1 V in K which contains 1 since c ∈ V − {0}. It is clear that K/k is the ﬁeld of deﬁnition for Ru (Hk ), so the proof of Lemma 9.1.2 may be used to deduce that k[cV ] = K. 2 The following corollary shows that the case K 2 ⊆ k is the essential one. Corollary 9.1.9 Let K/k be a ﬁnite purely inseparable extension of ﬁelds with characteristic 2, and let V ⊆ K be a kK 2 -subspace containing 1 such that k[V ] = K. Let G0 denote one of the K-groups SL2 or PGL2 , and let D denote its diagonal K-torus. Then ∗ ∗ HV ,K/k = R kK 2 /k (HV ,K/kK 2 ), VK/k = R kK 2 /k (VK/kK 2)

inside of R K/k (G0 ) and R K/k (D) respectively. Proof. By Galois descent, we may and do assume k = ks . Consider the k-subgroup G := R kK 2 /k (HV ,K/kK 2 ) ⊆ R kK 2 /k (R K/kK 2 ((L0 )K )) = R K/k ((L0 )K ). This is pseudo-simple over k since kK 2 /k is purely inseparable and HV ,K/kK 2 is pseudo-simple over kK 2 , and the minimal ﬁeld of deﬁnition over k for R(Gk ) is K by Remark 9.1.7. The canonical k-subgroup L0 in R K/k ((G0 )K ) is contained in G, and relative to its diagonal k-torus the root groups of G have Lie algebras equal to the common k-subspace V ⊆ K containing 1. Hence, we can apply Corollary 9.1.8(1) to deduce that G = HV ,K/k . It then follows from two applications of Proposition 9.1.6 (relative to K/k and K/kK 2 ) that ∗ ∗ VK/k = G ∩ R K/k (D) = R kK 2 /k (HV ,K/kK 2 ∩ R K/kK 2 (D)) = R kK 2 /k (VK/kK 2 ).

2 Corollary 9.1.8 enables us to give a complete classiﬁcation result in the A1 -case over a separably closed ﬁeld of characteristic 2 when the root system is reduced and [k : k 2 ] = 2:

9.1 Type A1

289

Proposition 9.1.10 Let G be an absolutely pseudo-simple group over a ﬁeld k of characteristic 2 with [k : k 2 ] = 2, and let K/k be the ﬁeld of deﬁnition over k for the subgroup R(Gk ) ⊆ Gk . Assume that GKss is of type A1 and that Gks has a reduced root system. Then ξG is surjective and hence G is standard. If G ss SL2 then the natural map ξG : G → R K/k (GKss ) is an isomorphism. k

The reducedness hypothesis in Proposition 9.1.10 is satisﬁed whenever G ss PGL2 (by Theorem 2.3.10, since (PGL2 , GL1 ) consists of nonk divisible elements in X(GL1 )). In §9.3 we will construct G with G ss SL2 for k which Gks has a non-reduced root system (and so G is not standard). Proof. We may and do assume k = ks . Let T be a maximal k-torus in G, so G is generated by the G(k)-conjugates of T (since G = D (G)). Since [k : k 2 ] = 2, the only ﬁnite extensions of the separably closed ﬁeld k n are the ﬁelds k 1/2 for n 0. But K is such a ﬁeld, so [K : K 2 ] = 2. The case K = k is trivial, so we can assume K = k. Hence, kK 2 = K 2 . Upon identifying GKss with SL2 or PGL2 in a suitable manner carrying TK over to the diagonal K-torus, the image of ξG is equal to HV ,K/k for some K 2 -subspace V ⊆ K containing 1 such that k[V ] = K. The inclusion K 2 ⊆ V is strict (because k[V ] = K), so V = K. It follows from Remark 9.1.3 that ξG is surjective, so the reducedness hypothesis on (G, T ) implies that G is standard (Proposition 5.3.10). In the SL2 -case the surjective map ξG : G → R K/k (GKss ) with central unipotent kernel Z is a (uniquely) split extension by Z due to Proposition 5.1.3 and Example 5.1.4 (since GKss is simply connected). This forces Z to be a direct factor of G and thus to be smooth and connected, so it is trivial by the pseudo-reductivity of G over k. 2 We conclude this section with a result that gives some more instances of standardness in the absolutely pseudo-simple case when (G, T ) is reduced. This will be used as a step towards the complete classiﬁcation in characteristic 2 when [k : k 2 ] = 2. Since standardness has been proved in the rank 1 case when the root system is reduced (Proposition 9.1.10), as well as away from the types Bn (n 2), Cn (n 2), and F4 in higher rank (without any hypotheses on [k : k 2 ] or on the root system), we may as well assume GKss is one of these remaining higher-rank types. In other words, it has an edge with multiplicity 2 in its Dynkin diagram. Let G be an absolutely pseudo-simple group over a separably closed ﬁeld k of characteristic 2 such that [k : k 2 ] = 2, and let K/k be the ﬁeld of definition of the geometric radical of G. Let T be a maximal k-torus in G, and let := (Gks , Tks ). For each c ∈ , let Gc ⊆ G be the smooth connected

290

The absolutely pseudo-simple groups in characteristic 2

k-subgroup generated by Uc and U−c as in Proposition 3.4.1, so Gc is absolutely pseudo-simple of rank 1 with Sc := T ∩ Gc as a maximal k-torus. In particular, (Gc , Sc ) ⊆ . We let Kc /k be the ﬁeld of deﬁnition over k for the geometric radical of Gc , so Kc ⊆ K by Proposition 6.3.1. Proposition 9.1.11 Using notation and hypotheses as above (in particular, [k : k 2 ] = 2), assume that is reduced and GKss has an edge in its Dynkin diagram with multiplicity 2. (1) For all long roots a ∈ , the subﬁeld Ka ⊆ K is equal to a common subﬁeld K> ⊆ K independent of a. The subﬁeld is either K or K 2 , and the k-group G is standard if and only if K> = K. (2) If G is not standard and G ss is simply connected then G is an exotic pseudok reductive group. Proof. Choose an a ∈ . Since is reduced, the K-group (Ga )ss K is identiﬁed with the K-subgroup Ga in G = GKss generated by the ±aK -root groups (Proposition 6.3.1). Since (Ga , Sa ) ⊆ , it is a reduced root system. Thus, by Proposition 9.1.10, ξGa : Ga → R Ka /k (Ga ) is surjective, so we may apply Proposition 6.3.2(2) to get that Ka = K for short a and that for long a the ﬁelds Ka are equal to a common K> containing K 2 . Since we are assuming [k : k 2 ] = 2 it follows that [K : K 2 ] = 2, so in the chain of inclusions K 2 ⊆ K> ⊆ K one must be an equality. By the explicit standard description of Ga from Proposition 9.1.10, we see that [Ka : K] is the dimension of the a-root group for each long root of . But in the absolutely pseudo-simple standard case, all root groups have the same dimension. Hence, K> = K when G is standard (as K = Ka for short a ∈ ). Conversely, assume K> = K. The proof of standardness in Theorem 6.3.4 assumed ξGa is a surjection for all a. This surjectivity has been proved to hold, so the standardness conclusion follows. To prove (2), since we observed at the outset of this proof that (Ga )ss k (G ss )a , if G ss is simply connected then this rank-1 group is SL2 (rather than k k PGL2 ). Hence, by Proposition 9.1.10 the map ξGa is an isomorphism for all a, so the hypotheses in Theorem 8.2.1 are satisﬁed, from which it follows that G is exotic. 2

9.2 Root groups and birational group laws Let k be a ﬁeld of characteristic 2. Consider an absolutely pseudo-simple kgroup G, and let K/k denote the ﬁnite purely inseparable ﬁeld of deﬁnition

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291

over k for Ru (Gk ) = R(Gk ) ⊆ Gk . Let T be a maximal k-torus in G, so TK is identiﬁed with a maximal K-torus T in the connected absolutely simple and semisimple quotient G := GKss = GKred . By Theorem 2.3.10, we have (GK s , TK s ) ⊆ (Gks , Tks ) ⊆ (GK s , TK s )

1 · (GK s , T ), 2

with (GK s , TK s ) equal to the set of non-multipliable roots in (Gks , Tks ). Assume that (Gks , Tks ) is non-reduced, so G is not reductive and hence K = k. Since this root system is also irreducible (as G is absolutely pseudo-simple), it is the unique non-reduced irreducible root system of its rank n 1, so the root datum R(GK s , TK s ) consisting of non-multipliable roots (and non-divisible coroots) is the reduced root datum of a simply connected group of type Cn for some n 1; i.e., GK s Sp2n . Our aim is to ﬁnd all examples when [k : k 2 ] = 2. They arise from certain birational group laws arising out of special properties of Sp2n in characteristic 2. In this section we undertake a preliminary study of root groups in any such G, assuming k = ks (no hypotheses on [k : k 2 ]), and we review the theory of birational group laws over a ﬁeld. In §9.3 we will construct (without assuming k = ks ) absolutely pseudo-simple k-groups of any rank n 1 with a nonreduced root system and any purely inseparable quadratic extension as the ﬁeld of deﬁnition over k for the geometric radical. In §9.4 we completely classify the possibilities when [k : k 2 ] = 2: the rank n 1 and the ﬁeld of deﬁnition K/k for the geometric radical will determine the k-isomorphism class (and all possibilities do occur). In particular, we will prove that such G have no nontrivial k-forms (for the étale topology over k)! Now make no hypotheses on [k : k 2 ], but assume k = ks . Choose a nondivisible a ∈ (G, T ) and consider the associated root group Ua = U(a) ⊆ G. This is a vector group admitting a T -equivariant linear structure, and the set of T -weights on its Lie algebra is {a} when a is non-multipliable and {a, 2a} when a is multipliable. The map ξG : G → R K/k (G ) cannot kill Ua over k since G is pseudoreductive, so the smooth quotient map GK G carries (Ua )K onto the 1-dimensional root group Ua for some a ∈ (G , T ) such that a kills the codimension-1 torus (ker a)0red in T = TK . We have a = a when a is non-multipliable and a = 2a when a is multipliable. Note that in the non-multipliable case, the linear structures on the vector groups Ua and Ua linearizing the respective actions of T and T are unique. We conclude that if a is not multipliable then ξG : G → R K/k (G ) carries Ua into R K/k (Ua ) via a nonzero linear map a .

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The absolutely pseudo-simple groups in characteristic 2

Our main interest is in the multipliable roots a, so consider such an a. By Proposition 2.3.11 we can choose a (non-canonical) linear structure on Ua = U(a) linearizing the T -action, so the resulting weight space decomposition Ua = Va × U2a is a T -equivariant splitting of the canonical quotient map Ua Ua /U2a . In fact, the choice of linear structure on Ua amounts to precisely a choice of T -equivariant splitting of this quotient map. Via this choice (which we will later improve to a ZG (T )-equivariant splitting when [k : k 2 ] = 2 and ) by a map [K : k] = 2), ξG carries Ua into R K/k (U2a (v, w) → qa (v) + 2a (w),

(9.2.1)

) where 2a is as above for the non-multipliable root 2a and qa : Va → R K/k (U2a is a 2-linear map (i.e., qa is additive and is equivariant with respect to squaring in the linear structure: qa (c.v) = c2 qa (v)). The map (9.2.1) is injective on k-points, since otherwise by scaling appropriately we would get inﬁnitely many k-points in the kernel, contradicting that (ker ξG )(k) must be ﬁnite (as G is pseudo-reductive over k and ker ξG is a unipotent normal k-subgroup scheme). In particular, qa and 2a are nonzero.

Lemma 9.2.1 Assume [K : k] = 2. Then dim U2a = 1, 2 dim Ua /U2a [k : k 2 ], and the natural map ξG : ZG (T ) → R K/k (T ) is surjective. In particular, if [k : k 2 ] = 2, then dim Ua /U2a = 2. Proof. Choose a T -equivariant lift Va ⊆ Ua of Ua /U2a as above and let 2a and ) = 2, the nonzero linear map has a qa be as in (9.2.1). Since dim R K/k (U2a 2a 1-dimensional image, as otherwise it would be surjective on k-points, contradicting that (9.2.1) is injective and Va = 0. Hence, U2a is 1-dimensional and (K)/ (U (k)) qa on k-points deﬁnes an injective 2-linear map Va (k) → U2a 2a 2a between k-vector spaces, with 1-dimensional target. Since qa becomes k-linear (K)/ (U (k)) is viewed as a k-vector space through its k 2 -vector when U2a 2a 2a space structure and the squaring isomorphism k k 2 , the injectivity of qa implies that (K)/2a (U2a (k)) = [k : k 2 ]. 1 dim Va (k) dimk 2 U2a

Let Ta = (ker a)0red and let Ga = D (ZG (Ta )), so Ga is a pseudo-simple k-subgroup in which the maximal torus is 1-dimensional and the root groups for ±a and ±2a coincide with those for G (so the root system is non-reduced). In particular, Ga is not reductive, so its geometric radical has nontrivial ﬁeld of deﬁnition over k. By two applications of Proposition A.4.8 (for the subgroup ZG (Ta ) in G and the subgroup Ga in ZG (Ta )), this geometric radical is deﬁned

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293

over the quadratic extension K. Hence, the ﬁeld of deﬁnition over k for the geometric radical is exactly K. Since G is simply connected of type Cn for some n 1, the lattice X∗ (T ) = X∗ (T ) is generated by the multipliable coroots in R(G, T ). We can therefore rename Ga as G to reduce to the case where G is of type C1 , so G SL2 and (G , T ) is the set {±2a} of non-multipliable roots in (G, T ). The K-isomorphism G SL2 can be chosen to carry T = TK over to the diagonal torus D, and this map of K-tori is deﬁned over k (relative to the natural k-structures on TK and D) since k = ks . In particular, a is a generator of X(T ), so we may use a to identify T with GL1 and hence the action of T on Va is through ordinary scaling relative to the unique T -equivariant linear structure on Va . By Remark 9.1.3, the map ξG : G → R K/k (G ) = R K/k (SL2 ) is surjective. Thus, the Cartan k-subgroup C := ZG (T ) is carried onto R K/k (D) (Proposition A.2.8). In particular, dim C 2, so C is strictly larger than its maximal torus T . We now assume dim Va = 1 and seek a contradiction. The Cartan ksubgroup. C is a commutative pseudo-reductive k-group. Since C acts on Va ( Ua /U2a ) via conjugation, under the 1-dimensionality hypothesis this action is through a k-homomorphism h : C → GL1 . But the maximal k-torus T = GL1 acts on Va through ordinary scaling, so h splits T off as a direct factor of the commutative k-group C. The pseudo-reductivity of C then forces C = T , contradicting the earlier deduction that C = T . 2 We now assume [k : k 2 ] = 2 and [K : k] = 2. Since k is separably closed, up to unique isomorphism k 1/2 is the only quadratic extension of k, so K = k 1/2 . The linear structure on Ua /U2a is determined by the T -action (through a), so the ZG (T )-action on Ua /U2a is linear. This can be described in more concrete terms as follows. Since a is non-divisible in X(T ) = X(T ), the quotient map a : T → GL1 splits off as a direct factor. Hence, R K/k (a) is surjective, so the diagram

χa : ZG (T )

ξG

/ R K/k (T )

R K/k (a)

/ R K/k (GL1 )

(9.2.2)

consists of surjections and hence the composite map χa is surjective. Lemma 9.2.2 Assume [k : k 2 ] = 2 and [K : k] = 2. The ZG (T )-action on Ua /U2a uniquely factors through the quotient R K/k (GL1 ) of ZG (T ) via χa , and the resulting action of R K/k (GL1 ) on Ua /U2a arises from a unique K-vector space structure on Ua (k)/U2a (k) extending its natural k-vector space structure.

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The absolutely pseudo-simple groups in characteristic 2

In particular, there is a k-group isomorphism Ua /U2a R K/k (Ga ) carrying the ZG (T )-action over to the natural scaling action through χa . Proof. Let Wa = Ua /U2a , equipped with the unique linear structure linearizing the action of T . The image Z of ZG (T ) in GL(Wa ) GL2 is a smooth connected commutative k-subgroup that contains the central k-torus as the image of T via a. Thus, dim Z 2. We claim Z is actually the 2-dimensional group R K/k (GL1 ) arising from a uniquely determined K-vector space structure on Wa extending the k-vector space structure. To prove this it is enough to check that some smooth connected k-subgroup of ZG (T ) maps onto a copy of the 2-dimensional R K/k (GL1 ) inside of GL(Wa ) arising from such a K-vector space structure, so as in the proof of Lemma 9.2.1 we may reduce to the case where G SL2 . In particular, the map a : T → GL1 is an isomorphism. If dim Z = 2 then Z would be the central GL1 , but then the resulting quotient map ZG (T ) GL1 which restricts to the isomorphism a : T GL1 on T would split T off as a direct factor of ZG (T ). However, ZG (T ) is a commutative pseudo-reductive group, so splitting off its maximal k-torus T as a direct factor would force ZG (T ) = T . This is impossible since ξG carries ZG (T ) onto the k-group R K/k (T ) that is not a torus. Hence, ZG (T ) has 2-dimensional image Z in GL(Wa ). By dimension reasons Z is not a torus, since ZG (T ) has a 1-dimensional maximal k-torus T . Consider the commutative k-subalgebra A of End k (Wa (k)) generated by the image of ZG (T )(k), so A is 2-dimensional as a k-vector space. By construction, the action of ZG (T ) factors through the 2-dimensional unit group A× := R A/k (GL1 ), and dimension and connectedness considerations force the map ZG (T ) → A× to be surjective. In particular, A× = Z is not a torus, so A = k × k. Hence, either A is non-reduced or A is a quadratic extension of k. In other words, A k[x]/(x2 ) or A K. The quotient map ZG (T ) A× carries T isomorphically onto the maximal k-torus GL1 in A× , so GL1 cannot split off as a direct factor of the k-group A× since T does not split off as a direct factor of ZG (T ). This rules out the case A k[x]/(x2 ), so we conclude that A K as k-algebras. Note that this latter isomorphism is unique. The ZG (T )-action on Wa is the composition of a quotient map ZG (T ) A× = R K/k (GL1 ) extending a : T GL1 followed by the multiplicative scaling deﬁned by a K-vector space structure on Wa extending the k-linear structure. It is clear that this K-vector space structure is uniquely determined. It remains to show that the k-homomorphism ZG (T ) R K/k (GL1 ) just constructed from the T -equivariant action of ZG (T ) on Wa coincides with χa . More generally, if two k-homomorphisms C ⇒ C between smooth

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295

connected commutative afﬁne k-groups coincide on the maximal k-torus of C and if C is pseudo-reductive over k then these homomorphisms are equal (Proposition 1.2.2). 2 Here is a crucial reﬁnement of Lemma 9.2.2. Lemma 9.2.3 Assume [k : k 2 ] = 2 and [K : k] = 2. There is a ZG (T )equivariant section to Ua Ua /U2a . In particular, the ZG (T )-action on Ua factors through the quotient map ZG (T ) R K/k (T ). Proof. The map ξG carries Ua into the 2-dimensional vector group ) R R K/k (U2a K/k (Ga ) and restricts to a linear isomorphism 2a from U2a ). Letting W = R onto a line L in R K/k (U2a K/k (U2a )/L denote the quotient line, ξG induces a 2-linear map qa : Wa := Ua /U2a W of k-groups. These ﬁt into a commutative diagram of exact sequences 0

/ U2a 2a

0

/ L

/ Ua ξG

/ R K/k (U ) 2a

/ Wa

/ 0

qa

/ W

(9.2.3) / 0

and the vertical map qa is compatible with the conjugation action of ZG (T ) ) = R on Wa via χa , and the action of R K/k (GL1 ) on R K/k (U2a K/k (Ga ) via 2 c.x = c x. Squaring on R K/k (GL1 ) is precisely the norm character valued in the k-subgroup GL1 , and the bottom exact sequence in (9.2.3) is GL1 -equivariantly split (as for any exact sequence of linear representation spaces for a torus). Hence, when the bottom exact sequence in (9.2.3) is viewed as a ZG (T )equivariant diagram via ZG (T ) → R K/k (GL1 ), it admits a splitting. Thus, to deduce the desired equivariant splitting for the top exact sequence it sufﬁces to prove that the top sequence is the pullback of the bottom along qa . This pullback property is immediate from the commutativity of the diagram, the exactness of horizontal parts, and the fact that the vertical map 2a along the left side is an isomorphism. 2 The following result is the culmination of our efforts at describing the root group structure in G under some restrictions on k and K/k. Proposition 9.2.4 Let G be an absolutely pseudo-simple group over a separably closed ﬁeld k with characteristic 2 such that [k : k 2 ] = 2, and let T be a maximal k-torus in G. Assume that (G, T ) is non-reduced, and that the ﬁeld of deﬁnition K/k of R(Gk ) in Gk is k 1/2 . Let G = GKss .

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The absolutely pseudo-simple groups in characteristic 2

(1) The map ξG : G → R K/k (G ) is surjective. (2) For any α ∈ K − k and multipliable a ∈ (G, T ), there exists G , an R a K-isomorphism U2a a K/k (GL1 )-equivariant isomorphism Ua /U2a R K/k (Ga ), a k-isomorphism U2a Ga , and a ZG (T )equivariant splitting of Ua Ua /U2a such that the resulting identiﬁcations ) with R of Ua with R K/k (Ga ) × Ga and R K/k (U2a K/k (Ga ) carry ξG |Ua to the map R K/k (Ga ) × Ga → R K/k (Ga ) deﬁned functorially by (x, y) → NK/k (x) + yα = x2 + yα

(9.2.4)

Proof. By Lemma 9.2.1, ξG carries ZG (T ) onto the Cartan k-subgroup R K/k (T ). Fix a positive system of roots in (G, T ), so its non-multipliable roots constitute a positive system of roots in (G , T ). By using the resulting open Bruhat cell in G and its analogue in G (see Corollary 3.3.16), we can pass to derived subgroups of centralizers of suitable k-tori to reduce the proof of surjectivity of ξG to the rank 1 case (using the same argument as in the proof of Lemma 9.2.1). In this case the surjectivity of ξG was shown in the proof of Lemma 9.2.1. This settles (1). To prove (2), ﬁrst choose a K-basis of (Ua /U2a )(k), a k-basis of U2a (k), and a ZG (T )-equivariant lift Va ⊆ Ua of the quotient Ua /U2a . By Lemma 9.2.2, the k-group Va admits a unique K-linear structure relative to which the ZG (T )-action is scaling through χa (i.e., for any z ∈ ZG (T ) and v ∈ Va we ) denote the restriction have zvz −1 = χa (z)v). Letting qa : Va → R K/k (U2a −1 2 of ξG to Va , we have qa (zvz ) = χa (z) qa (v), so in terms of the K-linear structure we have qa (cv) = c2 qa (v) for all v ∈ Va and all c ∈ R K/k (GL1 ) (not just c ∈ GL1 ). By Zariski-density, the same holds for all c ∈ R K/k (Ga ). Identify Va with R K/k (Ga ) as k-groups via a K-basis {v0 } of Va (k), and with G as K-groups via the K-basis {q (v )} of U (K); we have identify U2a a a 0 2a ) qa (v0 ) = 0 since the map in (9.2.1) is injective. Thus, qa : Va → R K/k (U2a 2 becomes the endomorphism of R K/k (Ga ) deﬁned by x → x , as this holds on the Zariski-dense open R K/k (GL1 )-orbit of v0 . In other words, qa becomes the norm map NK/k : R K/k (Ga ) → Ga . The injectivity of the map in (9.2.1) then implies that the nonzero linear map 2a : U2a → R K/k (Ga ) is an isomorphism from U2a (k) onto a k-line in K distinct from k. The chosen k-basis vector of U2a (k) is then carried over to some α0 ∈ K −k, so 2a is identiﬁed with the map Ga → R K/k (Ga ) deﬁned by t → tα0 . This yields (9.2.4) with α0 . By changing the k-basis of U2a (k) we can get such a description using any k × -multiple of α0 . Now choose α ∈ K − k such that α0 /α ∈ k, and we seek to make suitable choices so that the effect on (9.2.4) is to replace α0 with α. Changing the splitting Va of Ua Ua /U2a and the K-basis of Va (k) amounts to the operation

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297

(x, y) → (cx, y + bx2 ) on the source in (9.2.4), with c ∈ K × and b ∈ k × . (K) amounts to the operation x → c x on the Changing the K-basis of U2a × target in (9.2.4) with c ∈ K . Hence, for any α ∈ K × it sufﬁces to check that there exist c, c ∈ K × and b ∈ k × such that NK/k (x) + yα = c (NK/k (cx) + (y + bx2 )α0 ) = c (c2 + bα0 )NK/k (x) + c yα0 ?

for all x ∈ R K/k (Ga ) and y ∈ Ga ; the triple (c, c , b) will turn out to even be unique. Setting y = 0 forces c = (c2 + bα0 )−1 (which makes sense since c2 ∈ k and b ∈ k × ), and setting x = 0 forces c α0 = α. Hence, it is necessary and sufﬁcient that c2 + bα0 = α0 /α. Since α0 /α ∈ K − k = k × k × α0 = K 2 × k × α0 and α ∈ k × , there is a unique choice for c ∈ K × and b ∈ k × (which determines c ). 2 The next step is to reverse the above procedures by constructing, for each imperfect ﬁeld k of characteristic 2, an absolutely pseudo-simple k-group G of any rank n 1 such that the ﬁeld of deﬁnition of its geometric radical is a speciﬁed purely inseparable quadratic extension K/k. (We do not require [k : k 2 ] = 2 or that k = ks .) Inspired by Proposition 9.2.4, the construction will depend on a choice of element α ∈ K − k. When [k : k 2 ] = 2, Weil restriction to k of such examples over ﬁnite purely inseparable extensions of k will be shown to account for all absolutely pseudo-simple k-groups with a non-reduced root system. How can we construct such G with a non-reduced root system? The exotic constructions given in §§7.1–7.2 have reduced root systems (as in the standard case). To construct absolutely pseudo-simple groups with a non-reduced root system we need a new technique. Building on ideas of Tits, the technique will be Weil’s theory of birational group laws over a ﬁeld, so the groups we construct will only be known in a very indirect manner as algebraic varieties; the most important example with rank n 1 will be an extension of R k 1/2 /k (Sp2n ) by a non-commutative unipotent group scheme directly spanned by 2n copies of the k-group R k 1/2 /k (α2 ), where α2 = ker FGa /k . We now digress to review some basic deﬁnitions and results concerning birational group laws, and refer the reader to [BLR, Ch. 5] for a complete treatment. Deﬁnition 9.2.5 For any ﬁeld k, a birational group law on a smooth separated k-scheme X of ﬁnite type is a rational map m:X ×X

/ X

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The absolutely pseudo-simple groups in characteristic 2

such that the universal translation rational maps deﬁned by

X ×X

/ X ×X

ρ : (x, y) → (x, m(x, y)), λ : (x, y) → (m(x, y), y) are birational (so m is dominant) and the rational composition law m is associative in the sense of rational maps (i.e., m(x, m(y, z)) = m(m(x, y), z) as rational maps from X 3 to X ; such composite rational maps make sense since m is dominant). Note that there is no axiom concerning inversion or an identity element. (The case X = ∅ admits a birational group law.) To give an example, let G be a smooth (possibly disconnected) k-group of ﬁnite type and X ⊆ G a dense open subscheme. The k-scheme X is smooth and separated (since G is), and the group law mG on G deﬁnes a birational group law on X . These birational group laws satisfy some additional properties. To formulate such properties, we require the following concept. For any smooth k-scheme Y of ﬁnite type, an open subscheme U ⊆ Y × Y is Y -dense if it has dense intersection with all ﬁbers of both projection maps Y × Y ⇒ Y . (In case Y is geometrically connected over k, and hence is geometrically irreducible, it is equivalent to say that U meets all ﬁbers of both projection maps Y × Y ⇒ Y .) The property of Y -density is insensitive to ground ﬁeld extension, and when k is algebraically closed it sufﬁces to check ﬁbers over Y (k). If (X , m ) is a birational group law on a smooth separated k-scheme X of ﬁnite type and X ⊆ X is a dense open subscheme then for the ratio/ X induced by m the pair (X , m) is a birational nal map m : X × X group law. Consider a birational group law (X , m) which arises as a dense open subscheme of a smooth k-group G of ﬁnite type. The open subscheme U := m−1 G (X )∩(X ×X ) in X ×X is the domain of deﬁnition of the rational map / X , and it is X -dense since for any x ∈ X (k) the open overlaps m:X ×X (x−1 · Xk ) ∩ Xk and (Xk · x−1 ) ∩ Xk inside of Gk are dense in Xk . Moreover, the universal translation morphisms ρ : U → X × X and λ : U → X × X are open immersions (induced by the universal translation isomorphisms G × G G × G deﬁned by (g, g ) → (g, gg ), (gg , g )) and each has X -dense image. This motivates the following reﬁnement of the concept of a birational group law. Deﬁnition 9.2.6 A strict birational group law on a smooth separated k-scheme X of ﬁnite type is a birational group law m on X such that there is an open subscheme U ⊆ dom(m) which is X -dense and for which the k-morphisms

9.3 Construction of absolutely pseudo-simple groups

299

ρ : U → X ×X and λ : U → X ×X respectively deﬁned by u → (p1 (u), m(u)) and u → (m(u), p2 (u)) are open immersions with X -dense images in X × X . Since U is required to be X -dense, the X -density conditions on the images of ρ and λ amount to the condition that m(U ) = X . By [BLR, 5.2/2], for any birational group law (X , m ) over a ﬁeld there is a dense open subscheme X ⊆ X such that the induced birational group law (X , m) on X is strict. The main result of Weil concerns strict birational group laws: Theorem 9.2.7 (Weil) If (X , m) is a strict birational group law over a ﬁeld k and X = ∅ then up to unique k-isomorphism there is a dense open immersion of X into a smooth k-group G of ﬁnite type such that the group law G × G → G restricts to the rational map m on X × X . Proof. This is [BLR, 5.2/3] in case k is separably closed. By uniqueness up to unique isomorphism, the general case follows from the separably closed case by Galois descent (since a smooth k-group of ﬁnite type is necessarily quasi-projective, so the descent is effective). 2

9.3 Construction of absolutely pseudo-simple groups with a non-reduced root system Our experience with the previous exotic constructions suggests that it may be possible to make non-standard pseudo-reductive groups by “thickening” some root groups and Cartan subgroups in certain connected semisimple groups. The earlier constructions carried out that idea by exploiting very special isogenies. But examples with non-reduced root systems will arise from an entirely different source: divisibility of roots in the character lattice. The only type for which this happens is simply connected type Cn (n 1), or in other words (forms of) Sp2n , in which case the long roots are precisely the roots that are divisible in the character lattice (with the convention for n = 1 that both roots are viewed as long, as they are divisible by 2 in the character lattice). To see how this might give rise to some new examples, consider a purely inseparable quadratic extension K/k of an arbitrary imperfect ﬁeld k with characteristic 2. The pseudo-reductive k-group G = R K/k (Sp2n ) has as a Cartan k-subgroup R K/k (D), where D is the n-dimensional diagonal maximal K-torus in Sp2n . Letting D0 denote the k-split maximal k-torus in R K/k (D), the root groups for (G, D0 ) are R K/k (Ua ) for the root groups Ua of (Sp2n , D), with a ∈ (Sp2n , D) = (G, D0 ) (Example 2.3.2). But observe that if a is a long

300

The absolutely pseudo-simple groups in characteristic 2

root, so a/2 ∈ X(D), then the k-group R K/k (D) acts on R K/k (Ua ) through the homomorphism R K/k (a) : R K/k (D) → R K/k (GL1 ) which lands in the image GL1 of the squaring endomorphism of R K/k (GL1 )! Thus, inspired by the structure of the open Bruhat cell in pseudo-reductive groups (Corollary 3.3.16), it makes sense to try to make a new absolutely pseudo-simple k-group of rank n in which R K/k (D) remains as a Cartan ksubgroup and the short root groups (equipped with their R K/k (D)-action) remain unchanged but the long root groups R K/k (Ua ) are expanded to R K/k (Ua ) × Ga with R K/k (D) acting in a modiﬁed manner: it acts through the GL1 -valued R K/k (a) on Ga and it acts through R K/k (a/2) on R K/k (Ua ). (By the classiﬁcation of non-reduced irreducible root systems, the only possibility in rank n is to introduce a/2 into the root system for each long root a in (Sp2n , D).) This idea will really work, but one surprising feature is that such groups are not constructed directly as afﬁne schemes (in contrast with our construction of exotic pseudo-reductive groups as Weil restrictions of ﬁber products). To actually construct an absolutely pseudo-simple group with a non-reduced root system of rank n 1, we will need to use some special properties of Sp2n in characteristic 2. Roughly speaking, the fact which gets things off the ground is a result for GLn that generalizes the following trivial observation for n = 1. √ Let K/k be a purely inseparable quadratic extension, with α = c a primitive element for some non-square c ∈ k × . For any k-algebra R, each ζ ∈ RK has the form ζ = u + vα for unique u, v ∈ R, and we will call u the real part of ζ and v the imaginary part of ζ ; these concepts depend on α, but that will not matter. For any ζ ∈ R× K with real part u ∈ R and imaginary part v ∈ R (so u2 + cv 2 ∈ R× ), we have v u 1 + 2 · α, = 2 ζ u + cv 2 u + cv 2

(9.3.1)

so the real part of 1/ζ involves the imaginary part of ζ only through its square. It is this property which has to be generalized to GLn over k. Choose n 1 and ﬁx the following notation for working with Sp2n over an arbitrary base ring (or over F2 -algebras, which sufﬁces for our needs). We view a b Sp2n as a subgroup of GL2n and denote its points as c d , where a, b, c, d ∈ Mat n are n × n matrices. The notation t g denotes the transpose of any N × N matrix g for any N 1, and 1n denotes the identity in GLn . We deﬁne Sp2n to

9.3 Construction of absolutely pseudo-simple groups be the automorphism group of the symplectic form

0 1n −1n 0

301

, which is to say

t 0 1n 0 1n g= = g ∈ GL2n g −1n 0 −1n 0

a b = ∈ GL2n t a · c = t c · a, t b · d = t d · b, c d

Sp2n

t

a · d − c · b = 1n . t

Let Dn ⊆ GLn be the diagonal torus and let Un± ⊆ GLn denote the standard upper and lower triangular unipotent subgroups. The split torus in Sp2n given by

−1 0 t t ∈ D D := n 0 t has rank n, so it is maximal in Sp2n . Likewise, letting Symn ⊆ Mat n denote the space of symmetric n × n matrices, the closed subscheme

t −1 u mu + m ∈ Sym ⊆ Sp2n , u ∈ U U := n n 0 u is a subgroup because

t −1 t −1 u1 m1 u1 u2 0 u1 0

m2 u2 u2

=

t

(u1 u2 )−1 0

(t u1−1 m2 u1−1 + m1 )(u1 u2 ) u 1 u2

and t

Since

u−1 0

t

u−1 0

mu u

−1 =

t

mu 1 = n u 0

u 0

(−t umu)u−1 . u−1

m 1n

t

u−1 0

0 , u

clearly U is smooth with connected unipotent ﬁbers of dimension n(n − 1)/2 + n(n + 1)/2 = n2 . It is also easy to check that U is normalized by D. Let Bn± := D Un± ⊆ GLn denote the standard upper and lower triangular Borel subgroups, so

t −1 b mb + m ∈ Symn , b ∈ Bn B := D U = 0 b

302

The absolutely pseudo-simple groups in characteristic 2

is a Borel subgroup of Sp2n with unipotent radical U . This corresponds to the “standard” positive system of roots + n in n := (Sp2n , D). The matrix entries of u ∈ Un+ and m ∈ Symn are the root groups for (Sp2n , D) contained in B (i.e., root groups for the roots in + n ). More explicitly, for 1 i < j n we deﬁne uij : Ga → GLn to carry x to the matrix in Un+ whose ij-entry is x and whose other off-diagonal entries vanish (these are the root groups for (GLn , Dn ) contained in Un+ ), so the closed subgroup [u]ij : Ga → U deﬁned by [u]ij : x →

t

uij (x)−1 0

0 uij (x)

(9.3.2)

is normalized by D, and more speciﬁcally for t := diag(t1−1 , . . . , tn−1 , t1 , . . . , tn ) ∈ D we have t[u]ij (x)t −1 = [u]ij (ti tj−1 x). Likewise, for 1 i < j n we deﬁne mij : Ga → Symn to carry x to the symmetric matrix whose ij and ji entries are x and all other entries vanish, and for 1 i n we deﬁne mii : Ga → Symn to carry x to the symmetric matrix whose ii-entry is x and whose other entries vanish. Then the closed subgroup [m]ij : Ga → Sp2n deﬁned by [m]ij : x →

1n 0

mij (x) 1n

(9.3.3)

for 1 i j n is normalized by D with t as above satisfying t[m]ij (x)t −1 = [m]ij ((ti tj )−1 x). Thestandard representative for the longest Weyl element in W (Sp2n , D) is

0 1n ; this matrix normalizes D via inversion. Explicitly, we have the w := −1 n 0 conjugation formula

w

t

u−1 0

mu u −1 w = u −mu

0 t u−1

(so one can check directly that w acts on the root groups for D in Sp2n in the required manner). Remark 9.3.1 Inspection of the preceding descriptions of root groups shows that the long roots in + n (equivalently, the roots that are divisible in X(D)) correspond precisely to the root groups [m]ii (for which the Un+ -component is 1n ).

9.3 Construction of absolutely pseudo-simple groups

303

Now ﬁx an imperfect ﬁeld k of characteristic 2, a purely inseparable quadratic extension K/k, and an element α ∈ K − k. We will create a new absolutely pseudo-simple k-group with rank n, a non-reduced root system, and K/k as the ﬁeld of deﬁnition over k for its geometric radical. Consider the pair (Sp2n , D) over K. We will deﬁne a birational group law on a variant of the open Bruhat cell of R K/k (Sp2n ) in which, for each long root a ∈ (Sp2n , D), the a-root group R K/k (Ua ) = R K/k (Ga ) is expanded to R K/k (Ga ) × Ga via the homomorphism R K/k (Ga ) × Ga → R K/k (Ga ) deﬁned by (x, y) → x2 + yα. This map is R K/k (D)-equivariant when letting R K/k (D) act on the source through the GL1 -valued R K/k (a) on the factor Ga over k and via the R K/k (GL1 )-valued R K/k (a/2) on the factor R K/k (Ga ). Remark 9.3.2 The element c := α 2 ∈ k × is a non-square in k, so c is transcendental over F2 and hence we have an embedding F2 (x) → k via x → c which √ identiﬁes k ⊗F2 (x) F2 ( x) with K. This will enable us to use scalar extension to reduce some construction problems for general K/k to the special case of √ F2 ( x)/F2 (x), which has the advantage that K 2 = k. For the K-group Sp2n , choose the positive system of roots + n ⊆ (Sp2n , D) corresponding to the Borel K-subgroup B = D U as above. Deﬁne + > and + , with the + to respectively be the sets of long roots and short roots in < n convention for n = 1 that we declare all roots to be long, which is to say + < = ∅. This ensures that in all cases, the long roots are the ones that are + divisible in X(D). Fix enumerations of + > and < , so we have the direct spanning expression

Ua ×

a∈+ >

Ua U

a∈+ <

as K-schemes via multiplication, where Ua denotes the root group for a ∈ (Sp2n , D). The Ua for a ∈ + n are naturally identiﬁed with Ga over K via the root group parameterizations [u]ij (1 i < j n) and [m]ij (1 i j n) deﬁned in (9.3.2) and (9.3.3). + Using the enumerations of + > and < ﬁxed above, deﬁne the product scheme U =

a∈+ >

Ua ×

Ua

a∈+ <

over k, where Ua := R K/k (Ga ) for a ∈ + < and Ua := R K/k (Ga ) × Ga for a ∈ + . For long a (so a/2 ∈ X(D)) we also deﬁne Ua/2 to be the factor R K/k (Ga ) > in Ua . Finally, deﬁne the map of pointed k-schemes f : U → R K/k (U ) as the

304

The absolutely pseudo-simple groups in characteristic 2

direct product of k-maps fa : Ua → R K/k (Ua ) respecting origins as follows: if a is short then fa is the identity map of R K/k (Ga ), and if a is long then fa is the map R K/k (Ga ) × Ga → R K/k (Ga ) given by (x, y) → x2 + yα. (Note that squaring on R K/k (Ga ) is valued in Ga over k.) The formation of f is compatible with any extension k /k such that K ⊗k k is a ﬁeld, and f is injective on k-points. The map f depends on the choice of enumeration of + n , but this choice will be ﬁxed and so we omit it from the notation. The map f also depends on α, but that choice will also be ﬁxed for some time. (The k-groups we eventually construct will be independent of the choice of enumeration, up to k-isomorphism, and will be independent of the choice of α when [k : k 2 ] = 2.) Proposition 9.3.3 There is a unique group law on the pointed k-scheme U making f : U → R K/k (U ) a homomorphism, and on Ua for each a ∈ + n this group law is the usual additive one. There is also a unique action of R K/k (D) on U making f equivariant with respect to the natural action of R K/k (D) on R K/k (U ). Proof. Since k is an inﬁnite ﬁeld and U and Rk/k (D) are rational varieties, the k-points are Zariski-dense. But f is injective on k-points, so uniqueness follows. Likewise, if there is to be a group law then the distinguished point must be the identity and the restriction fa of f to each Ua (via the distinguished point in all other factors) will automatically be the usual additive group law. To prove existence of a group law on U making f a homomorphism, as well as the existence of an R K/k (D)-action on U making f an R K/k (D)-equivariant map, by Remark 9.3.2 it sufﬁces to treat the case of the ﬁelds k = F2 (x) and √ √ K = k( x) (with α = x). The special feature of this case, and the only aspect of it which we shall use, is that [k : k 2 ] = 2, so K 2 = k. Since f is injective on k-points and carries the “origin” of its source to the identity point of its target, to make the group law it sufﬁces to ﬁll in the top arrows to make commutative diagrams of k-schemes U ×U

?

/ U

f ×f

R K/k (U ) × R K/k (U )

μ

U f

/ R K/k (U )

f

?

R K/k (U )

/ U

i

f

/ R K/k (U )

9.3 Construction of absolutely pseudo-simple groups

305

where the bottom maps μ and i are multiplication and inversion in the k-group law of R K/k (U ). Consider the composite maps h = μ ◦ ( f × f ) : U 2 = U × U → R K/k (U ), g = i ◦ f : U → R K/k (U ) along the left and bottom sides. Let ha and ga respectively denote the composition of h and g with projection to R K/k (Ua ) for each a ∈ + n . To factor h and g through f means that ha and ga factor through fa for all a. For short a the map fa is an isomorphism, so there is nothing to do. For long a, the problem is to show that the maps ha : U 2 → R K/k (Ga ), ga : U → R K/k (Ga ) have the respective forms h21 + αh2 and g12 + αg2 for some maps h1 : U 2 → R K/k (Ga ), h2 : U 2 → Ga , g1 : U → R K/k (Ga ), g2 : U → Ga . In other words, this says that the real parts of ha and ga are respectively the squares of k-scheme maps U 2 → R K/k (Ga ) and U → R K/k (Ga ). For any k-scheme X , to give a k-map X → R K/k (Ga ) is the same as to give a K-map XK → Ga , so an equivalent formulation of our problem is that when the real parts of ha and ga are pulled back to UK2 and UK respectively then each resulting pullback is a square in the function ﬁeld over K. But U 2 and U are rational varieties over k, and k = K 2 , so a rational function on U 2 or U becomes a square after scalar extension to K precisely when it involves all variables only through their squares. By the deﬁnition of the maps f , h = μ ◦ ( f × f ), and g = i ◦ f , it now sufﬁces (for the construction of the group law on U ) to show that for each a ∈ + > the real part of the a-components μa and ia of the multiplication and inversion maps μ and i on R K/k (U ) involve the real and imaginary parts of short root factors and the imaginary parts of the long root factors only through their squares. To verify this, we compute the multiplication law μ and inversion i on R K/k (U ) as follows. With the shorthand

1 (m, u) := n 0

m 1n

t

u−1 0

0 u

306

The absolutely pseudo-simple groups in characteristic 2

we have (m1 , u1 )(m2 , u2 ) = (t u1−1 m2 u1−1 + m1 , u1 u2 ), (m, u)−1 = (−t umu, u−1 ) where m1 , m2 , m are points of R K/k (Symn ) and u1 , u2 , u are points of R K/k (Un+ ). By Remark 9.3.1, the long root groups correspond to precisely the diagonal entries along the “Symn -component”, so our problem is to check that the real part of the diagonal of t u1−1 m2 u1−1 + m1 (resp. −t umu) depends on the real and imaginary parts of all matrix entries of m1 , m2 , u1 , u2 (resp. m, u) apart from the real parts of the diagonals of m1 , m2 (resp. m) only through their squares. The formation of the diagonal is additive, so we reduce to the general claim that for any u ∈ R K/k (Un+ ) and m ∈ R K/k (Symn ), the real part of the diagonal of t umu only involves the real part of the diagonal of m and the squares of the real and imaginary parts of the entries of u. By the following lemma, the real part of the diagonal of t umu involves the entries of u only through their squares, which in particular are Ga -valued (i.e., “real-valued”), so since (x + yα)2 = x2 + cy2 we are then done with the construction of the group law on U . Lemma 9.3.4 Let M = (mij ) be a symmetric n × n matrix over an F2 -algebra R, and choose g = (gij ) ∈ GLn (R). The diagonal entries of t gMg only involve the gij through their squares. In other words, as algebraic functions on the R-group GLn they factor through the relative Frobenius morphism of GLn . Proof. Let ·, ·M denote the symmetric bilinear form on afﬁne n-space over R associated to M (i.e., mij vi wj ). Since R is an F2 -algebra, the associated quadratic form qM encodes precisely the diagonal entries mii : for any point v = (v1 , . . . , vn ) in afﬁne n-space, mii vi2 . qM (v) = v, vM = This involves each vi only through its square. Since qM (gv) = q(t gMg) (v), we are done. 2 Finally, we deﬁne an action of R K/k (D) on the k-scheme U making f an R K/k (D)-equivariant map. Since f is a k-homomorphism that is injective on k-points, it would then be automatic that the action of R K/k (D) on U respects the group law just constructed on U since the action of R K/k (D) on R K/k (U ) respects the group law on R K/k (U ). For each a ∈ + < we have Ua = R K/k (Ua ) via f , so we make R K/k (D) act on such a Ua through its usual action on R K/k (Ua ) (via R K/k (a)). For a ∈ + > the action of R K/k (D) on R K/k (Ua ) is through the R K/k (GL1 )-valued Weil restriction of a ∈ X(D), but since a is a

9.3 Construction of absolutely pseudo-simple groups

307

long root it lies in 2X(D). Hence, R K/k (a) is valued in [2]R K/k (GL1 ) = GL1 , so R K/k (a) deﬁnes a scaling action of R K/k (D) on Ga . Likewise, R K/k (a/2) deﬁnes a scaling action of R K/k (D) on R K/k (Ga ). By using these to make R K/k (D) act on Ua = R K/k (Ga ) × Ga , since R K/k (a) is GL1 -valued it is easy to check that f |Ua is R K/k (D)-equivariant, so we are done (as f is deﬁned as a direct product of R K/k (D)-equivariant maps of pointed k-schemes). 2 By Proposition 9.3.3 we can deﬁne a k-group B := R K/k (D) U , and naturally extend f to a homomorphism B → B which we will denote again by f . Let w denote a formal symbol which we imagine as a nontrivial 2-torsion k-point normalizing R K/k (D) via inversion (much as w ∈ Sp2n (K) has order 2 in characteristic 2 and acts on D via inversion). Theorem 9.3.5 There is a unique strict birational group law on U w × B making the map f : U w × B → R K/k (U wB) deﬁned by (u, b) → f (u)wf (b) compatible with the birational group laws. Proof. As in the proof of Proposition 9.3.3, the uniqueness in the theorem follows from the injectivity of f on k-points. Hence, once again we can use Remark 9.3.2 to reduce the proof of existence to the special case [k : k 2 ] = 2. Since B and U have been endowed with k-group structures making f a khomomorphism and we aim to make w to be of order 2 and normalize R K/k (D) via inversion, the main issue in the construction is to decide which points g of U should satisfy wgw−1 ∈ U w × B. To ﬁgure this out, we ﬁrst examine the birational group law on the product scheme R K/k (U wB) = R K/k (U )w × R K/k (D) × R K/k (U ) in which w has order 2 and normalizes R K/k (D) via inversion. The domain of deﬁnition V for the birational group law R K/k (U wB) × R K/k (U wB)

/ R K/k (U wB)

consists of points (u1 wt1 u1 , u2 wt2 u2 ) such that wt1 u1 u2 t1−1 w−1 ∈ R K/k (U wB). (In particular, (w, w) is not contained in V (k).) This is a strict birational group law, since it is an open locus in a smooth connected k-group. Let V = ( f × f )−1 (V ) be the preimage of V in (U w × B)2 , so V is (U w × B)-dense since V is R K/k (U wB)-dense. Using the group laws of B and B and the w-conjugation and w-conjugation on R K/k (D) via inversion, we

308

The absolutely pseudo-simple groups in characteristic 2

have a commutative diagram / Uw×Uw×B

(U w × B) × (U w × B) f ×f

f

R K/k (U wB) × R K/k (U wB)

/ R K/k (U w × U w × B)

(9.3.4)

in which the horizontal maps are surjective. Explicitly, the bottom row is (u1 wt1 u1 , u2 wt2 u2 ) → (u1 w, t1 (u1 u2 )t1−1 w, t1−1 t2 u2 ) and the top row is similar except with w replacing w. Letting U ⊆ U be the open locus deﬁned by U = {u ∈ U | wuw−1 ∈ U wB}, V is the full preimage of R K/k (U w × U w × B) in R K/k (U wB)2 along the bottom side of (9.3.4). Deﬁne the open U := f −1 (R K/k (U )) ⊆ U , so V is the full preimage of U w × U w × B along the top side of (9.3.4). The key point is to show that U = ∅ (so U = ∅) and that the diagram of k-schemes ?

U f

/ Uw×B f

/ R K/k (U wB)

R K/k (U )

(9.3.5)

(using u → wuw−1 on the bottom) can be uniquely ﬁlled in to be commutative. The uniqueness is clear since f is injective on k-points. Grant non-emptiness of U and the existence in (9.3.5) for a moment. We can use such existence along with (9.3.4) to ﬁll in a commutative diagram of k-schemes V f ×f

μα

/ Uw×B f

V

μ

/ R K/k (U wB)

9.3 Construction of absolutely pseudo-simple groups

309

where μ is the birational group law on R K/k (U wB). (The map μα depends on α, as f does.) By equivariance for left U -translation and right B-translation μα is surjective, and consideration on k-points (using that f is injective on k-points and μ is surjective and associative in the sense of dominant rational maps) implies that μα is automatically associative in the sense of dominant rational maps. Thus, μα would be a birational group law whose universal left and right translation morphisms ρα , λα : V ⇒ (U w ×B)2 have U w ×B-dense images provided that these universal translation morphisms are birational. Hence, to verify that we have constructed a strict birational group law, it just remains to prove that ρα and λα are open immersions. We will address this ﬁnal point at the end, after we have proved U = ∅ and ﬁlled in (9.3.5) to a commutative square. To analyze (9.3.5), we need to describe U ⊆ U and the components of wgw−1 in U wB for any point g of U (i.e., any g ∈ U such that wgw−1 ∈ U wB). This will be expressed in terms of the Bruhat decomposition for GLn associated to the choice of Dn and Un+ (equivalently, Bn+ ), so to be explicit we deﬁne the standard open Bruhat cell of GLn to be Un+ × Bn− = Un+ × Dn × Un− . Lemma 9.3.6 For g=

t

u0−1 0

m 0 u0 u0

∈ U,

wgw−1 lies in U wB (i.e., g ∈ U ) if and only if m0 u0 lies in the standard open Bruhat cell of GLn (so m0 ∈ GLn ), in which case

t −1

t −1 b −u−1 m0 t u−1 b u −m−1 0 u w wgw−1 = , 0 b 0 u where u t b−1 is the decomposition of m0 u0 in Un+ Bn− (with b ∈ Bn+ , u ∈ Un+ ). In particular, U = ∅. u 0 0 Proof. We compute wgw−1 = −m u t u−1 , and an arbitrary point of U wB is 0 0

t

u−1 0

mu w u

t

b−1 0

0

m b −mu t b−1 = b −u t b−1

−mum b + t u−1 b . −um b

Hence, the necessary and sufﬁcient conditions on the unknown u ∈ Un+ , b ∈ Bn+ , and m, m ∈ Symn are: (i) mu t b−1 = −u0 , (ii) mum b = t u−1 b, (iii) u t b−1 = m0 u0 , and (iv) um b = −t u0−1 . Condition (iii) encodes precisely the Bruhat constraint on m0 u0 ∈ GLn , and when this holds then (i) becomes t −1 −1 m = −m−1 0 and (iv) becomes m = −u m0 u . When these all hold, one checks by direct calculation that (ii) always holds. 2

310

The absolutely pseudo-simple groups in characteristic 2

Consider any symmetric m0 ∈ R K/k (GLn ) and any u0 ∈ R K/k (Un+ ) such that m0 u0 lies in the Weil restriction of the standard open Bruhat cell of GLn . By Lemma 9.3.6, this says exactly that the point t g :=

u0−1 0

m0 u0 u0

∈ R K/k (U )

lies in R K/k (U ), so wgw−1 ∈ R K/k (U wB) = R K/k (U )w × R K/k (DU ). Due to Remark 9.3.1, our interest will be in the diagonal part of the R K/k (Symn )components of both R K/k (U )-components of wgw−1 . These two points of t −1 in the pre −1 R K/k (Symn ) were seen to be m = −m−1 0 and m = −u m0 u ceding proof, where ub is the decomposition of m0 u0 in R K/k (Un+ Bn− ) with u ∈ R K/k (Un+ ) and b ∈ R K/k (Bn− ). To ﬁll in (9.3.5) it remains to prove: Lemma 9.3.7 Choose g ∈ R K/k (U ). Writing g=

t

u0−1 0

m 0 u0 u0

as usual, let ub denote the Bruhat decomposition of m0 u0 ∈ R K/k (Un+ )R K/k (Bn− ). Identify R K/k (U ) with R K/k (Ga )n(n+1)/2 via the matrix entries of m0 ∈ R K/k (Symn ) and u0 ∈ R K/k (Un+ ), and so identify R K/k (U ) with a dense open W ⊆ R K/k (Ga )n(n+1)/2 . t −1 is viewed −1 When the real part of each diagonal entry of −m−1 0 or −u m0 u as an algebraic function on W over k, then its dependence on the real and imaginary parts of u0 , as well as its dependence on the real and imaginary parts of m0 apart from the real diagonal part, is only through their squares. If n = 1 then Un+ = 1, so in this case the assertion for −u−1 m0 t u−1 is trivial 2 2 and for −m−1 0 it is exactly the fact that x/(x − cy ) involves y only through its square. Proof. Since (x + yα)2 = x2 + cy2 , Lemma 9.3.4 takes care of the case of real parts of the diagonal entries of u−1 m0 t u−1 (even though u is a somewhat mysterious function of (m0 , u0 ) ∈ W ). Thus, we are reduced to proving that if m0 ∈ R K/k (GLn ) is symmetric then the real part of each diagonal entry of −m−1 0 ∈ R K/k (Sym n ), viewed as a rational function (over k) of the real and imaginary parts of m0 ∈ R K/k (Symn ), involves the imaginary part of m0 and the non-diagonal real part only through the squares of their entries. (It sufﬁces to work with rational functions instead of regular functions since R K/k (Symn ) is normal and connected, even an afﬁne space.)

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Since we are working over a ﬁeld k of characteristic 2, a rational function of several variables over k depends on a given variable only through its square if and only if the corresponding partial derivative vanishes. Hence, it is equivalent to prove that the rational endomorphism m0 → −m−1 0 of R K/k (Symn ) has total differentials in the imaginary symmetric direction and the real symmetric off-diagonal direction that take values in the space of symmetric matrices whose diagonal has vanishing real part (i.e., its diagonal is “pure imaginary”). Letting ε = (εij ) denote a “generic” symmetric n × n matrix over k for which εij εi j = 0 for all i, j, i , j , the total differential in the imaginary symmetric direction is computed by determining the “ﬁrst-order” part of −(m0 +εα)−1 . Clearly −1 −1 −(m0 + εα)−1 = −m−1 0 + m0 εm0 α,

so the total differential in the imaginary symmetric direction at any m0 is the −1 mapping M → m−1 0 Mm0 α on points M of Sym n (so the entries of M are −1 “real”). We therefore have to check that the diagonal of m−1 0 Mm0 α has van−1 2 × ishing real part. Since α = c ∈ k , it is equivalent to say that m−1 0 Mm0 has −1 diagonal that is “real”. The matrix m0 is its own transpose, as m0 is symmetric, −1 so by Lemma 9.3.4 we see that m−1 0 Mm0 involves the R K/k (Ga )-entries of m−1 0 only through their squares. But squaring on R K/k (Ga ) is valued in the subgroup Ga of “real” points, so since M has real entries we are done with the analysis of the imaginary parts. Now consider the analogous derivative calculation in the real symmetric offdiagonal directions. This time we take ε as above to have vanishing diagonal entries, and since −1 −1 −(m0 + ε)−1 = −m−1 0 + m0 εm0 ,

the total differential in the real symmetric off-diagonal direction at any m0 is −1 the mapping M → m−1 0 Mm0 on the space of points M ∈ Sym n with vanishing diagonal. Our problem is to prove that the real part of the diagonal of −1 m−1 0 Mm0 vanishes; we will show that the entire diagonal vanishes. A symmetric n × n matrix T in characteristic 2 has vanishing diagonal precisely when the associated quadratic form qT as in the proof of Lemma 9.3.4 vanishes. Since m−1 −1 0 is an invertible symmetric matrix (with entries in R K/k (Ga )), qm−1 0 Mm0 is obtained from qM = 0 by an invertible linear change of variables, whence qm−1 Mm−1 = 0. 2 0

0

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The absolutely pseudo-simple groups in characteristic 2

To complete the proof of Theorem 9.3.5, we have to show that the universal translation morphisms ρα , λα : V ⇒ (U w × B)2 are open immersions. We will treat the case of the universal left translation, and the case of right translation will go the same way. We will show that λα is injective on R-points for any k-algebra R. In other words, if x, y ∈ (U w × B)(R) such that (x, y) ∈ V (R) then we claim that the ordered pair (x, μα (x, y)) of points in (U wB)(R) determines y. Granting this for a moment, let us deduce that λα is an open immersion. The map λα has ﬁnite ﬁbers (as follows just from the assumed injectivity of λα on geometric points), so since V and (U w × B)2 are geometrically irreducible (over k) of the same dimension it follows that λα is dominant and generically ﬁnite. Hence, the induced map Spec k(V ) → Spec(k((U w × B)2 )) at generic points is ﬁnite and a functorial monomorphism. But if F /F is a ﬁnite extension of ﬁelds such that Spec F → Spec F is a monomorphism then the projections Spec(F ⊗F F ) ⇒ Spec F coincide, so F = F. Thus, λα is birational. By Zariski’s Main Theorem [EGA, IV3 , 8.12.10], any birational morphism of ﬁnite type between separated integral normal noetherian schemes is an open immersion if it has ﬁnite ﬁbers. This implies that λα is an open immersion, conditional on λα being a functorial injection. To prove the injectivity on R-points for every k-algebra R, ﬁrst observe that inversion on Sp2n preserves U wB = U wDU (since w normalizes D), so inversion on R K/k (Sp2n ) preserves R K/k (U wB). Letting i denote the resulting involution of R K/k (U wB), we wish to lift i to an involution I of U w × B with enough properties to undo μα on the entirety of V . This is made precise by Lemma 9.3.8 below, from which the functorial injectivity of λα follows. 2 Lemma 9.3.8 There is a unique endomorphism I of U w × B over k ﬁtting into a commutative diagram of k-schemes Uw×B

I

f

R K/k (U wB) and I is an involution ﬁxing (w, 1).

/ Uw×B f

i

/ R K/k (U wB)

9.3 Construction of absolutely pseudo-simple groups

313

Moreover, for any point (x, y) of V ⊆ (U w × B)2 , the point (I (x), μα (x, y)) also lies in V and μα (I (x), μα (x, y)) = y. Proof. We use the same technique as in the proof of Theorem 9.3.5, except that matters will be much simpler because we do not have to contend with domains of deﬁnition and we can write things down explicitly. Uniqueness follows as usual from the injectivity of f on k-points, and for the same reason I is automatically an involution ﬁxing (w, 1) if it exists since i is an involution ﬁxing w (i.e., w2 = 1, as we are in characteristic 2). Since w normalizes D via inversion, in U wB = U wDU the formula for inversion is i(uwtu ) = u −1 wtu−1 . Hence, we deﬁne I on U w × B = U w × (R K/k (D) U ) by I (uw, tu ) := (u −1 w, tu−1 ). This clearly works. By deﬁnition, V is the full preimage of the open locus V of points in R K/k (U wB)2 whose product in R K/k (Sp2n ) is contained in R K/k (U wB). Thus, to check if (I (x), μα (x, y)) ∈ V for any (x, y) ∈ V it sufﬁces to apply f × f and check that (i(x ), μ(x , y )) ∈ V for any (x , y ) ∈ V . But this is clear since μ(i(x ), μ(x , y )) = y ∈ R K/k (U wB). We conclude that the expression μα (I (x), μα (x, y)) makes sense as a map from V to U w × B. To check that it agrees with pr 2 : V → U w × B, we are faced with the task of comparing two k-morphisms V ⇒ U w × B. Since V is a rational variety it sufﬁces to compare on k-points. Thus, by applying f we are reduced to the analogous agreement for two maps V ⇒ R K/k (U wB), in which case the agreement is obvious by working within the k-group R K/k (Sp2n ). 2 By Weil’s theorem on the unique extension of strict birational group laws to actual groups (Theorem 9.2.7), we can now make the following general deﬁnition. Deﬁnition 9.3.9 Let k be an imperfect ﬁeld of characteristic 2, K/k a purely inseparable quadratic extension, and α ∈ K − k any element. For each n 1, deﬁne Gα,n to be the smooth connected k-group containing (U w × B, μα ) as a dense open locus. In principle, Gα,n may seem to depend on the initial choice of + n and the enumeration of its long and short roots. Later we will see that (up to kgroup isomorphism) these choices do not matter when [k : k 2 ] = 2 (see Theorem 9.4.3(1)). Note that it is not obvious from the deﬁnition if Gα,n is even afﬁne. As a ﬁrst step, we show that Gα,n is a pseudo-reductive group (in particular, afﬁne) of rank n, with a non-reduced root system. In fact, we prove more:

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The absolutely pseudo-simple groups in characteristic 2

Theorem 9.3.10 Let k be an imperfect ﬁeld of characteristic 2, K/k a purely inseparable quadratic extension, and α ∈ K − k any element. Fix n 1 and let Gα = Gα,n and = U w × B. (1) The smooth connected k-group Gα is absolutely pseudo-simple. (2) The k-group R K/k (D) is identiﬁed with a Cartan k-subgroup of Gα via t → (w, t)(w, 1), and if D0 ⊆ R K/k (D) denotes the k-split maximal k-torus then the subset (Gα , D0 ) ⊆ X(D0 ) = X(D) is the non-reduced and irreducible root system obtained from (Sp2n , D) by adding in a/2 for every long root a. The resulting positive system of roots in (Gα , D0 ) corresponding to + n determines the open Bruhat cell · (w, 1) with U as the smooth connected unipotent k-subgroup generated by the root groups for positive roots. (3) The ﬁeld of deﬁnition over k for the geometric radical of Gα is K, and (Gα )ss K Sp2n . (4) The map ξGα is surjective and the k-group ker ξGα is connected with under lying k-scheme a∈(n )> R K/k (ker FUa /K ) where the product is taken in − any order that puts (+ n )> before (n )> or the other way around. In particular, (ker ξGα )(k) = 1. Proof. We may and do assume k = ks . Step 1. The ﬁrst thing we shall prove is that Gα is pseudo-reductive of rank n. The surjective map f : → R K/k (U wDU ) between respective dense opens in Gα and R K/k (Sp2n ) is compatible with the birational group laws on these opens. Thus, f uniquely extends to a k-homomorphism ξα : Gα → R K/k (Sp2n ). The map ξα is dominant and hence surjective, and its restriction to is the map f that is injective on k-points. In particular, (ker ξα )(k) is ﬁnite, so ker ξα contains no nontrivial smooth connected k-subgroups (as k = ks ). By Theorem A.3.9, there is a central extension 1 → Z → Gα → H → 1 with H afﬁne and Z a semi-abelian variety. Since ξα is surjective, it must carry the central k-subgroup Z into the center of R K/k (Sp2n ). The center of Sp2n is identiﬁed with μ2 , so the center of R K/k (Sp2n ) is identiﬁed with R K/k (μ2 ) (Proposition A.5.15(1)). For any k-scheme Y , the k-maps Y → R K/k (μ2 ) correspond bijectively to the K-maps YK → μ2 . In particular, if Y is smooth then the only such map is the constant map to 1. This forces Z ⊆ ker ξα , so Z = 1. Hence, Gα is afﬁne. Likewise, Ru,k (Gα ) = 1 since R K/k (Sp2n ) is pseudo-reductive over k, so Gα is pseudo-reductive. Since ker ξα contains no

9.3 Construction of absolutely pseudo-simple groups

315

nontrivial k-tori, we also conclude that Gα has the same rank as its quotient R K/k (Sp2n ), namely n. Step 2. Now we compute the maximal reductive quotient of (Gα )k and the map ξD (Gα ) . Consider the quotient map ψ : (Gα )K R K/k (Sp2n )K Sp2n . We claim that this is a descent of the maximal reductive quotient of (Gα )k . As a preliminary step, consider the exact sequence 1 → (ker ψ)/(ker ψ)0 → (Gα )K /(ker ψ)0 → Sp2n → 1. The term on the left is an étale normal k-subgroup of the connected middle term, so this exact sequence is a central extension and the middle term must be a connected semisimple group with an étale isogeny onto Sp2n . Since Sp2n is simply connected, it follows that ker ψ is connected. The connected K-group scheme ker ψ cannot contain any nontrivial K-tori since the rank of (Gα )K coincides with the rank of its quotient Sp2n by ker ψ. Thus, to show that ψ descends the maximal reductive quotient of k we just need to check that ker ψ is smooth, or equivalently that ψ is a smooth map. To verify this it is enough to check smoothness on some dense open of (Gα )K , so we restrict ψ to K . The restriction ψ|K is a direct product of copies of the following three surjective maps: the two natural maps q : R K/k (GL1 )K → GL1 , q : R K/k (Ga )K → Ga (the latter corresponding to short roots in n ) and the map h

q

(R K/k (Ga ) × Ga )K (R K/k (Ga ))K Ga where h(x, y) = x2 +yα. By surjectivity, we just have to check that ker q , ker q, and ker(q ◦ h) are smooth. For q and q we may apply Proposition A.5.11(1) (or check directly), and for q◦h the R-points of the kernel (for a K-algebra R) consist of points (x, α −1 m(x)2 ) the where m : K ⊗k R → R is the natural multiplication map using the K-algebra structure on R. Hence, ker(q ◦ h) R K/k (Ga ) and so this kernel is also smooth. Since ψ is a K-descent of the maximal reductive quotient of (Gα )k , we conclude that the radical of (Gα )k has dimension dim Gα − dim Sp2n = dim − dim Sp2n > 0 (so it is nontrivial) and that its ﬁeld of deﬁnition over k (as a subgroup of (Gα )k ) is contained in K. But K has no proper subﬁelds strictly containing k, so K is the

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The absolutely pseudo-simple groups in characteristic 2

ﬁeld of deﬁnition of the geometric radical of Gα . The restriction ψ|D (Gα )K is a K-descent of the maximal reductive quotient of D (Gα )k , by Proposition A.4.8, so the minimal ﬁeld of deﬁnition over k for the geometric radical of D (Gα ) is equal to K or k. In the latter case D (Gα ) Sp2n , and this is impossible for dimension reasons since R K/k (Sp2n ) is a quotient of Gα and hence is also a quotient of D (Gα ) (as R K/k (Sp2n ) is perfect, by Corollary A.7.11). Thus K/k is the ﬁeld of deﬁnition over k of the geometric radical of D (Gα ) and ξα |D (Gα ) = ξD (Gα ) . In particular, ξD (Gα ) is surjective, so D (Gα ) has rank n as well. Thus, all maximal k-tori of Gα are contained in D (Gα ). Step 3. The Bruhat cell of Gα (in the sense of Corollary 3.3.16) will now be related to . Consider the translate · (w, 1)−1 ⊆ Gα . This is carried by ξα into the open Bruhat cell R K/k (UDU − ) where U − := wU w−1 , and this restriction of ξα is injective on k-points. The loci (U w, 1) · (w, 1)−1 and (w, R K/k (D)) · (w, 1)−1 are respectively carried into R K/k (U ) and R K/k (D), with the latter map being inversion and the former being a direct product of evident maps that are additive. It then follows that R K/k (D) viewed inside Gα via t → (w, t) · (w, 1)−1 acquires its usual commutative k-group structure that is normalized by (w, 1) via inversion, and in particular the maximal k-torus D0 of R K/k (D) is a maximal k-torus of Gα . Likewise, it follows that for each −1 a ∈ + n the smooth connected subscheme Ua of Gα via u → (uw, 1) · (w, 1) generates a smooth connected k-subgroup that contains Ua with ﬁnite index since (ker ξα )(k) is ﬁnite. By connectedness it follows that each Ua is thereby identiﬁed with a k-subgroup of Gα , and its k-group structure is the expected additive one (as we check using the k-homomorphism f whose restriction to Ua = (Ua w, 1)(w, 1)−1 is injective on k-points). We can run through this argument using B in place of U , and again exploit the injectivity of f |(w,1)−1 on k-points to infer that each k-subgroup Ua is normalized by the k-subgroup R K/k (D) with the expected action induced by the group law of Gα (depending on whether a is long or short). In a similar manner, it follows that the k-subgroups Ua directly span U = (U w, 1) · (w, 1)−1 in the group law of Gα . Since the birational group law is deﬁned so that (w, 1) normalizes R K/k (D) via inversion, we see via weight space considerations in the Lie algebras that (i) R K/k (D) = ZGα (D0 ), (ii) (Gα , D0 ) is the expected non-reduced and irreducible root system of rank n containing , (iii) the positive system of roots in (Gα , D0 ) corresponding to + n has as root groups the Ua for the non-divisible positive roots a (i.e., for long a ∈ + n we really have that Ua is the a/2-root group in the sense of

9.3 Construction of absolutely pseudo-simple groups

317

Deﬁnition 2.3.13, with its Lie algebra spanned by weight spaces for a/2 and a) and has the evident factor Ga of Ua as the a-root group for long a. We conclude that (w, 1)−1 is the open Bruhat cell of Gα corresponding to D0 and + n. Step 4. Finally, we show that Gα is perfect, so it is (absolutely) pseudo-simple over k and ξGα = ξα , and hence ξGα is surjective. We also compute ker ξGα . Fix a maximal k-torus T in Gα , so T ⊆ D (Gα ). By forming commutators of T against points in root groups for (Gα , T ), we see that D (Gα ) contains all root groups of (Gα , T ). Since the image R K/k (Sp2n ) of ξα is its own derived group, so D (Gα ) maps onto R K/k (Sp2n ) via ξα , we see that the Cartan k-subgroups of D (Gα ) map onto those of R K/k (Sp2n ) and hence have dimension at least dim R K/k (D). We conclude that an open Bruhat cell in Gα associated to T is contained in D (Gα ), so D (Gα ) = Gα . Since Gα has been proved to be absolutely pseudo-simple, connectedness of ker ξGα follows from Theorem 5.3.8. By construction of ξα , the restriction of ξGα to is the map f that is deﬁned by an explicit direct product of k-scheme maps. It follows that ξGα has the following properties: (w, 1) is carried to w, ZGα (D0 ) is carried isomorphically onto R K/k (D), and U is carried onto R K/k (U ) with kernel having underlying k-scheme H := a∈(+n )> R K/k (ker FUa /K ) where the product can be taken in any order (as U is directly spanned in any order by positive root groups). Applying a (w, 1)-conjugation to handle the unipotent ksubgroup U = (w, 1)U (w, 1)−1 “opposite” to U , we see that ker ξGα meets the open cell in the k-scheme (not k-group!) direct product H × H where H is the (w, 1)-conjugate of H . To prove that this coincides with ker ξGα , we ﬁrst observe that (by construction) ker ξGα meets U , U , and ZGα (D0 ) in H , H , and {1} respectively. Thus, if multiplication identiﬁes the underlying scheme of ker ξGα with the direct product of such intersections then we will be done. Choose λ ∈ X∗ (D0 ) such that (n )λ>0 = (+ n )> . Hence, Lie algebra and connectedness considerations imply that the containments U ⊆ UGα (λ), U ⊆ UGα (−λ), and ZGα (D0 ) ⊆ ZGα (λ) are equalities. Since the normal k-subgroup scheme N := ker ξGα in Gα is clearly normalized by the GL1 -action through λ, we see that UN (−λ) = H , UN (λ) = H , and ZN (λ) = 1. The problem is therefore to prove that the multiplication map UN (−λ) × PN (λ) → N is an isomorphism. By Proposition 2.1.12(1), it sufﬁces to prove that the smooth connected afﬁne k-group (Nk )red is solvable. In fact, it contains no nontrivial k-tori (as ξGα has trivial kernel on the maximal k-torus D0 , hence on all tori over any extension of k), so it is unipotent and hence solvable. 2

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The absolutely pseudo-simple groups in characteristic 2

9.4 Classiﬁcation of absolutely pseudo-simple groups with a non-reduced root system We now turn our attention to proving some ﬁner properties of the k-groups Gα,n , and then using such properties to prove that if [k : k 2 ] = 2 then up to isomorphism there exists exactly one absolutely pseudo-simple k-group of each rank n 1 with any desired nontrivial purely inseparable ﬁnite extension K/k as the ﬁeld of deﬁnition of its geometric radical. The dependence on K/k will proceed in two steps: ﬁrst the case K = k 1/2 , and then the general case using arguments with Weil restriction. Proposition 9.4.1 Let k be an imperfect ﬁeld of characteristic 2, K/k a purely inseparable quadratic extension, and α ∈ K − k any element. Fix n 1 and let G = Gα,n . Let T be a k-split maximal k-torus in G. (1) The map ξG : G → R K/k (Sp2n ) is bijective on k-points if [k : k 2 ] = 2. (2) The center ZG is trivial. (3) Let + be a positive system of roots in (G, T ), and let be the set of simple positive roots. For each a ∈ , let χa be as in (9.2.2). Then is a basis of X(T ) and the natural map a∈ χa : ZG (T ) → a∈ R K/k (GL1 ) is an isomorphism. Although (1) identiﬁes G(k) with Sp2n (K) via ξG when [k : k 2 ] = 2, the kernel of ξG has positive dimension (equal to the number of long roots for Sp2n ; i.e., 2n). Thus, although one may think of G(k) as a “k-structure” on Sp2n (K) when [k : k 2 ] = 2, the actual k-group G is quite different from R K/k (Sp2n ). Proof. We may and do assume k = ks . First we prove (3), and then use it to deduce (1) and (2). To prove (3), we may use any choice of T , so we take D0 as in Theorem 9.3.10(2) and we can arrange that + has + n as its subset of nonmultipliable roots. This identiﬁes ZG (T ) with R K/k (D), where D is the diagonal maximal K-torus of Sp2n . The root groups for (G, D0 ) and the action of R K/k (D) on them were described explicitly in Theorem 9.3.10(2). In particular, it follows that is a basis of X(D) = X(D0 ), and the map χa : ZG (T ) → R K/k (GL1 ) is identiﬁed with the projection to the a-factor of R K/k (D). This proves (3). Since ZG ⊆ a∈ ker χa (due to Lemma 9.2.2), the explicit description of ZG (T ) in (3) shows that ZG = 1. This proves (2), and to prove (1) we ﬁrst note that (ker ξG )(k) is a ﬁnite normal subgroup of G(k), so it is also a normal étale k-subgroup of G (as G(k) is Zariski-dense in G). By connectedness of G it follows that (ker ξG )(k) is central in G, whence it is trivial (as ZG = 1). Thus, ξG carries G(k) isomorphically onto a subgroup of GKss (K) = Sp2n (K) that we want to be the entire group when [k : k 2 ] = 2.

9.4 Classiﬁcation of absolutely pseudo-simple groups

319

We saw in the proof of Theorem 9.3.10 that the restriction of f : G → R K/k (Sp2n ) to the open Bruhat cell (U w × B)(w, 1)−1 is a direct product of k-scheme morphisms, each of which is one of the following three types: the identity map between Cartan k-subgroups, an isomorphism between root groups for short roots of Sp2n , and the homomorphism R K/k (Ga ) × Ga → R K/k (Ga ) deﬁned by (x, y) → x2 + yα. But this latter map is bijective on k-points since K 2 = k (as [k : k 2 ] = 2). Hence, the image of ξG on k-points contains the Kpoints of an open Bruhat cell of Sp2n . Since K is inﬁnite, for each g ∈ Sp2n (K) the two subsets g(K) and (K) meet for any non-empty open ⊆ Sp2n (such as an open Bruhat cell). Thus, Sp2n (K) is the image of ξG on k-points when [k : k 2 ] = 2. 2 We next check that Proposition 5.1.3 applies to the groups Gα,n : Proposition 9.4.2 Let k be an imperfect ﬁeld of characteristic 2, and K/k a purely inseparable quadratic extension. Let α ∈ K − k be any element. For any n 1 and any commutative k-group scheme C of ﬁnite type that contains no nontrivial smooth connected k-subgroup, every central extension 1 → C → E → Gα,n → 1 is uniquely split. Proof. Fix a maximal k-torus T in Gα,n and a positive system of roots in + in (Gα,n , T ). Let be the set of simple positive roots. Since Gα,n is perfect, by Proposition 5.1.3 we just have to ﬁnd a dense open Z ⊆ ZGα,n (T ) and ﬁnitely many k-scheme maps hi from Z into root groups of (Gα,n , T ) such that hi : Z → Gα,n is the canonical inclusion. By Proposition 9.4.1(3), we have ZGα,n (T ) = a∈ R K/k (GL1 ) via a∈ χa . The elements of are all short in the non-reduced irreducible root system (Gα,n , T ) of rank n. In any basis of the root system of type Cn there is one long root (which is divisible by 2 in X(T )) and there are n − 1 short roots. Hence, in there is one multipliable root and there are n − 1 non-multipliable roots. If a ∈ is not multipliable then the direct product (via multiplication) of the ±a-root groups and the a-factor of ZGα,n (T ) is an open Bruhat cell of R K/k (SL2 ). (To be precise, when working in U w × B we get a translate of that open cell by a representative for the nontrivial Weyl element.) Hence, the desired generic formula for the a-factor of ZGα,n (T ) in the non-multipliable case was given at the end of Example 5.1.4.

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The absolutely pseudo-simple groups in characteristic 2

It remains to treat the unique a ∈ that is multipliable (and hence not divisible). Consider (Gα,n )a := D (ZGα,n (Ta )), where Ta = (ker a)0red . This is an absolutely pseudo-simple group of rank 1, and its root system is nonreduced because its root groups are those of Gα,n for roots that are nonzero rational multiples of a (i.e., ±a and ±2a). Its open Bruhat cell with respect to the maximal k-torus T ∩ (Gα,n )a and the positive system of roots {a, 2a} is easily computed inside of the known open Bruhat cell of (Gα,n , + ) (see Proposition 3.2.3), and from this we see that (Gα,n )a is a solution to the birational group law problem that deﬁned Gα,1 . Hence, we are reduced to the case of rank 1. We now have + = {a, 2a}, so U±a = R K/k (Ga ) × Ga in Gα,1 . Letting c = α 2 ∈ k × , the open subset R K/k (GL1 ) ⊆ R K/k (Ga ) represents the functor of points x + yα (for x, y ∈ Ga ) such that x2 + cy2 ∈ GL1 . We deﬁne the open locus ⊆ R K/k (GL1 ) by imposing the additional condition y ∈ GL1 . Thus, for points of we can write x + yα = yα(1 + (x/yc)α), so it is equivalent to work with points of the form yα(1 + zα) with y ∈ GL1 and z ∈ Ga . That is, is parameterized by pairs (y, z) ∈ GL1 × Ga . The key identity is

1 x−a (−1/t)xa (t)x−a (−1/t) = −1/t 0 = −1/t

0 1

t . 0

1 0

t 1

1 −1/t

0 1

Setting t = −yα for y ∈ k × gives the formula

0 1/yα

−yα = x−a ((yc)−1 α)xa (−yα)x−a ((yc)−1 α). 0

(9.4.1)

Now consider the value 1 + zα 1 = = t= 1 + zα 1 + cz 2

1 1 + zα

2 +

z · α = f (z)2 + g(z)α 1 + cz 2

for z ∈ k, where f (z) = (1 + zα)−1 and g(z) = z/(1 + cz 2 ). This gives the formula

0 1/(1 + zα) −(1 + zα) 0 (9.4.2) = x−a (−1 − zα)xa ( f (z)2 + g(z)α)x−a (−1 − zα).

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Multiplying (9.4.1) against (9.4.2) yields that diag(yα(1 + zα), 1/yα(1 + zα)) is equal to x−a ((1/yc)α)xa (−yα)x−a ((1/yc)α)x−a (−1 − zα) × xa ( f (z)2 + g(z)α)x−a (−1 − zα). Using the identiﬁcations of the root groups U±a with R K/k (Ga ) × Ga , we are led to consider the open locus Z := {(y, z) ∈ GL1 × Ga | 1 + cz 2 ∈ GL1 } inside of = GL1 × Ga and the maps h1 (y, z) = h3 (y, z) = (0, 1/yc) ∈ U−a , h2 (y, z) = (0, −y) ∈ Ua , h4 (y, z) = h6 (y, z) = (−1, −z) ∈ U−a , h5 (y, z) = ( f (z), g(z)) ∈ Ua . To check that hi : Z → Gα,1 is the canonical inclusion we can assume k = ks and it sufﬁces to work with k-points. Applying the map ξGα,1 that is injective on k-points, we can use the identities (9.4.1) and (9.4.2) just established in SL2 (K) to conclude the argument. 2 Theorem 9.4.3 Let k be a ﬁeld of characteristic 2 such that [k : k 2 ] = 2. Fix n 1. (1) Up to k-isomorphism there is a unique absolutely pseudo-simple k-group G of rank n with a non-reduced root system such that K := k 1/2 is the ﬁeld of deﬁnition over k for its geometric radical, and GKss Sp2n . (2) The center ZG is trivial, the k-group G contains a k-split maximal k-torus, the map ξG : G(k) → GKss (K) = Sp2n (K) is bijective, and all k-automorphisms of G are inner. (3) For any commutative k-group scheme C of ﬁnite type that contains no nontrivial smooth connected k-subgroups, any central extension of G by C is uniquely split over k. Proof. By Theorem 9.3.10, Proposition 9.4.1, and Proposition 9.4.2, there exists a G satisfying all of the asserted properties except possibly the property of having only inner automorphisms: Gα,n for any α ∈ K − k. To prove the uniqueness in (1), we ﬁrst treat the case k = ks and then use a Galois cohomology argument to deduce the general case from this. Fix a maximal k-torus T in G. Since k = ks , by Theorem 3.4.6 there exists a Levi k-subgroup L ⊆ G containing T . Hence, L Sp2n over k and the natural

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map LK → GKred = GKss is an isomorphism. Fix an isomorphism of k-split pairs (L, T ) (Sp2n , D0 ), and let w ∈ NL (T )(k) correspond to the standard representative w ∈ Sp2n (k) that has order 2 (as char(k) = 2) of the longest Weyl element. Hence, the isomorphism LK GKss identiﬁes GKss with Sp2n over K carrying the maximal K-torus TK over to the diagonal K-torus D and the point w ∈ GKss (K) of order 2 over to w ∈ Sp2n (K). This identiﬁes (Sp2n , D) with the set of non-multipliable roots in (G, T ), so our “standard” choice + n of positive system of roots in (Sp2n , D) as in the construction of Gα,n gives rise to a positive system of roots + in (G, T ). Note that the set of non-multipliable positive roots in + is identiﬁed with + n , and this latter set was used to index factors in the deﬁnition of U in the construction of Gα,n . However, in the description of the open Bruhat cell for (Gα,n , T ) corresponding to + we label the positive unipotent “factors” by the non-divisible positive roots. Using the obvious bijection between the set of non-multipliable positive roots and the set of non-divisible positive roots in any positive system of roots in a root system, we can regard the index set for the factors of U and the index set for the positive unipotent “factors” in the open Bruhat cell (both corresponding to a common choice of positive system of roots) as the same. For example, the set of non-divisible elements of + is endowed with a speciﬁc enumeration according to the choice made in + n in the construction of Gα,n . Let U + and U − denote the smooth connected unipotent k-subgroups of G generated by the positive and negative root groups respectively. For w ∈ NG (T )(k) as above that represents the longest Weyl element in W (G, T )(k) and has order 2, the map ξG : G → R K/k (Sp2n ) carries w to w ∈ Sp2n (K). But ξG is surjective (Proposition 9.2.4(1)), so it carries U + onto R K/k (U ) and ZG (T ) onto R K/k (D) (Corollary 2.1.9). Hence, ξG restricts to a surjection f : U + wZG (T )U + R K/k (U wDU ) from the w-translate of the open Bruhat cell of (G, T , + ) to the w-translate of the open Bruhat cell of (R K/k (Sp2n ), D0 , + n ). View f as a map between strict birational group laws, and observe that U + is directly spanned in any order by the root groups Ua for non-divisible a ∈ + . We choose the ordering according to the enumeration of the non-divisible elements of + as speciﬁed in the preceding discussion. Since f (w) = w and w2 = 1, by Lemmas 9.2.2, 9.2.3, and Proposition 9.2.4 (for a ﬁxed choice of α ∈ K − k) it follows from the uniqueness aspects of Proposition 9.3.3 and Theorem 9.3.5 that the strict birational group law on U + wZG (T )U + is the

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one whose solution is Gα,n . Thus, G Gα,n as k-groups by the uniqueness of solutions to strict birational group laws! Having proved (1) when k = ks , before we carry out the proof for general k we prove one further property when k = ks : all k-automorphisms φ of G are inner. In view of the uniqueness of G over k = ks , to prove this it sufﬁces to analyze a single speciﬁc case; we will work with G = Gα,n for some α since in these cases we have described the structure of an open Bruhat cell explicitly. Composing with an inner automorphism, we can assume that φ preserves some maximal ktorus T . Composing with conjugation by a suitable element of NG (T )(k) allows us to arrange in addition that the action of φ on (G, T ) preserves a choice of positive system of roots + (Proposition 3.2.7). This corresponds to a positive system of roots in the set (GKss , TK ) of non-multipliable roots of (G, T ). Observe that (by Lemma 1.2.1) if two k-automorphisms of a pseudo-reductive group coincide on the maximal geometric reductive quotient over k then the automorphisms are equal. Hence, it sufﬁces to ﬁnd some g ∈ G(k) such that g-conjugation and φ induce the same automorphisms of GKss . Let ss = {a1 , . . . , an−1 , an } ⊆ (GKss , TK ) be the simple positive roots for GKss with an the unique long simple root, so an /2 ∈ X(TK ) = X(T ) and := {a1 , . . . , an−1 , an /2} is a basis of X(T ). Since φ preserves + it preserves the subset ss . It must then ﬁx the elements of ss pointwise. Indeed, this is trivial if n = 1, and otherwise holds because the Dynkin diagram admits no nontrivial automorphisms. Hence, for each 1 j n the automorphism φ induced by φ on G = GKss acts on each root group Ua j Ga via scaling by some λj ∈ K × . For j = n something special happens: the map ξG : G → R K/k (G ) carries the long root group Uan into R K/k (Ua n ) by a nonzero linear map and Uan is 1-dimensional over k (due to the explicit construction of G = Gα,n ). Since ξG intertwines φ and R K/k (φ ), it follows that the scaling action by λn ∈ K × on the 1-dimensional K-vector space Ua n (K) preserves the 1-dimensional k-subspace Uan (k). Hence, λn ∈ k × = (K × )2 . In view of the above explicit basis of X(T ), there exists a unique g ∈ T (K) = R K/k (TK )(k) such that aj (g) = λj ∈ K × for j < n and (an /2)(g) = 1/2 λn ∈ K × . The description of Cartan k-subgroups in Gα,n in Proposition 9.4.1(3) then identiﬁes such a g with an element of ZG (T )(k) such that the conjugation by g on GKss and the action of φ have the same effect on all simple positive root groups of (G , TK ). Thus, composing φ with conjugation by g −1 on G brings us to the case where φ is an automorphism of GKss that preserves T , ﬁxes + pointwise inside of X(T ), and is the identity on all simple positive root groups. In general, for any split connected semisimple group H over a ﬁeld F and any F-automorphism f of H , if f preserves an F-split maximal F-torus S, ﬁxes

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a positive system of roots + ⊆ (H , S) pointwise, and is the identity on all simple positive root groups then f = id. Indeed, by passing to the simply connected central cover of H we can assume H is simply connected, and for rank 1 this is a simple calculation (since Aut F (SL2 ) = PGL2 (F)), so on the rank-1 connected semisimple subgroups of H generated by U±a for simple roots a we see that f induces the identity. In particular, f is the identity on the maximal F-torus S. It follows from the Isomorphism case of the Isogeny Theorem (Theorem A.4.10) that f is “conjugation” by a unique F-point t of the image S of S in the adjoint quotient. However, the simple positive roots in + are a basis of the character lattice of the maximal torus S in the adjoint quotient of H , so t = 1. Finally, we return to the general case of (1) (i.e., allow k = ks ) and also prove that all k-automorphisms of G are inner. Since ZG = 1, by Galois descent the property of k-automorphisms being inner is a consequence of its validity in the settled case of a separably closed ground ﬁeld. To verify the uniqueness of G up to k-isomorphism in general it is equivalent to ﬁx one example over k and to show that it has no nontrivial k-forms for the étale topology over k (as any two examples do become isomorphic over ks , by what we proved above). To handle this problem of analysis of k-forms, we may and do work with the example G := Gα,n for some α because then the k-group G has a k-split maximal k-torus. As every automorphism of Gks is inner, the triviality of the center implies that Aut ks (Gks ) = G(ks ) as groups with Gal(ks /k)-action. But the natural map G(ks ) → GKss (Ks ) is bijective (Proposition 9.4.1(1)), so the set of k-forms of G for the étale topology over k is in bijection with the Galois cohomology set H1 (ks /k, Aut ks (Gks )) = H1 (ks /k, G(ks )) = H1 (Ks /K, GKss (Ks )). Since G has a k-split maximal k-torus, GKss has a K-split maximal K-torus. Thus, GKss Sp2n . But H1 (Ks /K, Sp2n ) = 1 since non-degenerate symplectic spaces of dimension 2n over K are unique up to K-isomorphism. 2 In the preceding considerations we have generally focused attention on cases for which the ﬁeld of deﬁnition K/k of the geometric radical is quadratic over k. In our eventual classiﬁcation when [k : k 2 ] = 2, all cases over k with a given nontrivial purely inseparable ﬁnite extension K/k will be obtained from such “quadratic” cases over K 2 by applying R K 2 /k . To make the link between the case of general K/k and the “quadratic cases” (over K 2 ), we need to prove that if [k : k 2 ] = 2 then the property of having a non-reduced root system is not destroyed by passing from the absolutely pseudo-simple k-group G of

9.4 Classiﬁcation of absolutely pseudo-simple groups

325

rank n 1 to the absolutely pseudo-simple K 2 -group GK 2 /Ru,K 2 (GK 2 ) of rank n. Before we state such a result, we set up some notation, beginning with the following convenient shorthand. Deﬁnition 9.4.4 For a ﬁeld F and a smooth connected afﬁne F-group G, the maximal pseudo-reductive quotient G/Ru,F (G) is denoted G pred . If F /F is an pred extension ﬁeld then GF denotes (GF )pred . Assume [k : k 2 ] = 2, so the purely inseparable ﬁnite extensions of k are −j the ﬁelds k 2 for j 0. In particular, if K/k is a nontrivial purely inseparable ﬁnite extension then k ⊆ K 2 . Let G be an absolutely pseudo-simple k-group of rank n 1 such that Gks has a non-reduced root system. Let K/k be the ﬁeld of deﬁnition of the geometric radical of Gk . Then GKss is simply connected (it is pred isomorphic to Sp2n ). Observe that the maximal pseudo-reductive quotient GK 2 of GK 2 is absolutely pseudo-simple over K 2 of rank n and that the natural map pred

(GK 2 )k Gkss is its maximal reductive quotient over k. This quotient map cannot be an isomorphism, as otherwise Ru,K 2 (GK 2 ) would be a K 2 -descent of the geometric radical of G, contradicting the minimality of K/k as a ﬁeld of deﬁnition for that radical. Hence, the natural quotient map pred

pred

(GK 2 )K GK

= GKss

is a K-descent of the maximal reductive quotient over k and is not an isomorphism. Since [K : K 2 ] = 2, it follows that the (minimal) ﬁeld of deﬁnition over K 2 pred for the (nontrivial!) geometric radical of GK 2 is equal to K, and ξGpred is the pred

K2

natural map GK 2 → R K/K 2 (GKss ) since GKss is simply connected. To classify absolutely pseudo-simple G with a non-reduced root system, it is convenient to pred reduce to the study of GK 2 over K 2 in place of G over k because [K : K 2 ] = 2 whereas [K : k] may be large. The success of this strategy rests on: Proposition 9.4.5 In the preceding setup, if the rank of G is 1 and Gks has a pred non-reduced root system then so does (GK 2 )Ks2 . This result is also true for any rank, but to handle higher rank we need the complete classiﬁcation of rank 1 cases; up to now we have only handled the case [K : k] = 2 (in Theorem 9.4.3(1)).

326

The absolutely pseudo-simple groups in characteristic 2 pred

Proof. Since Ks2 = K 2 ⊗k ks , we may and do assume that k = ks . Let H = GK 2 and S = TK 2 . We assume that (H , S) is reduced and seek a contradiction. Since G has rank 1, so does H . The non-reduced root system (G, T ) is {±a, ±2a}, where the non-multipliable root 2a is in the root system (GKss , TK ) with GKss SL2 , so a is a generator of X(T ). Proposition 9.1.10 then implies (due to the reducedness hypothesis on (H , S)) that H is standard. Thus, by the ﬁnal assertion of Theorem 5.3.8, the map ξH : H → R K/K 2 (GKss ) is an isomorphism. The connected semisimple K-group GKss is simply connected of type C1 and 2aK is a root for TK that is divisible in the character lattice. The cocharacter (2aK )∨ : GL1 → TK has trivial kernel and carries μ2 into (even isomorphically onto) the center of GKss . Letting μ ⊆ T be the image of (2a)∨ on μ2 over k, the roots ±2a are trivial on μ but the roots ±a are nontrivial on μ because a, ±(2a)∨ =

1 · 2a, ±(2a)∨ = ±1. 2

The preceding considerations motivate us to consider the scheme-theoretic centralizer G μ := ZG (μ). Since {±2a} is the set of roots of G that are trivial on μ, by Proposition A.8.14 the identity component (G μ )0 is a pseudo-reductive kgroup with root system {±2a} and the natural map ((G μ )0 )red → G ss is an étale k k unipotent isogeny (i.e., its kernel is unipotent). A connected reductive group (such as G ss ) has no nontrivial connected étale group cover with a unipotent k

kernel, so the étale unipotent isogeny ((G μ )0 )red → G ss is an isomorphism. k k The formation of the maximal reductive quotient of a smooth connected afﬁne k-group is compatible with the formation of derived subgroups (by Proposition A.4.8), so we conclude that the natural map D ((G μ )0 )ss = D (((G μ )0 )red ) → Gkss k k is an isomorphism. The above argument proves that D ((G μ )0 ) is absolutely pseudo-simple with maximal geometric semisimple quotient G ss SL2 , and that this quotient map k can be descended to K as D ((G μ )0 )K → GK → GKss . It therefore follows from Proposition 9.1.10 (since n = 1) that D ((G μ )0 ) = R K /k (G ) for an intermediate ﬁeld k ⊆ K ⊆ K and a K -descent G of GKss .

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327

More precisely, the composite map D ((G μ )0 ) R K /k (G ) → R K/k (GK ) = R K/k (GKss )

(9.4.3)

is the restriction of ξG to D ((G μ )0 ). We now prove that K = K, and use that to get a contradiction (showing that H has a non-reduced root system after all, at least in the case of rank 1). Assume K = K, so K ⊆ K 2 (since every proper subﬁeld of K containing k is contained in K 2 ). The K 2 -group D ((G μ )0 )K 2 = R K /k (G )K 2 therefore admits GK 2 as its maximal pseudo-reductive quotient over K 2 . In particular, any pseudo-reductive quotient of D ((G μ )0 )K 2 has dimension at most dim G = dim GKss . To get a contradiction (and so to prove K = K) we will construct a pseudo-reductive quotient over K 2 with larger dimension. The natural quotient map pred

ξH

q : GK 2 GK 2 = H R K/K 2 (GKss ) is the maximal pseudo-reductive quotient over K 2 , and this quotient map carries μK 2 into the central K 2 -subgroup R K/K 2 (μK ). (Here we use that μK is central in GKss .) By Proposition A.8.14(1), q restricts to a surjective map μ

(GK 2 )0 R K/K 2 (GKss ). The target is its own derived group since GKss SL2 , so we get a surjective homomorphism μ

π : D ((GK 2 )0 ) R K/K 2 (GKss ). μ

This is a pseudo-reductive quotient of D ((GK 2 )0 ) whose dimension exceeds dim GKss , completing the proof that K = K. Since K = K, the composite map (9.4.3) is an isomorphism D ((G μ )0 ) R K/k (GKss ) induced by ξG . Hence, we have constructed a k-subgroup of G that maps isomorphically onto R K/k (GKss ) under ξG , so ker ξG is a direct factor of G as a k-scheme. This forces ker ξG to be smooth and connected, yet it is a unipotent normal k-subgroup scheme of G. Thus, by pseudo-reductivity of G we conclude that ker ξG = 1. But we have also shown that ξG is surjective, so it is an isomorphism from G onto R K/k (GKss ). This is impossible because G is assumed to have a non-reduced root system whereas the Weil restriction of a connected reductive group (through a ﬁnite extension of separably closed ﬁelds) always has a reduced root system (Example 2.3.2). Hence, the initial pred hypothesis that GK 2 has a reduced root system is false. 2

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The absolutely pseudo-simple groups in characteristic 2

As an application of Proposition 9.4.5, we can make more examples of absolutely pseudo-simple G such that the root system is non-reduced. The novelty is that we can arrange that the geometric radical has ﬁeld of deﬁnition over k equal to any desired nontrivial purely inseparable ﬁnite extension of k, assuming [k : k 2 ] = 2. Example 9.4.6 Let k be a ﬁeld of characteristic 2 with [k : k 2 ] = 2, and let k /k be a ﬁnite purely inseparable extension of ﬁelds. As we vary k /k, the extension k 1/2 /k varies through the nontrivial purely inseparable ﬁnite extensions of k. Consider G = R k /k (G ) for an absolutely pseudo-simple k -group G of rank 1 with a non-reduced root system and k 1/2 /k as the ﬁeld of deﬁnition of the geometric radical of G . We claim that G is absolutely pseudo-simple of rank 1 over k and it satisﬁes the following additional properties: (i) its root system is non-reduced, (ii) the ﬁeld of deﬁnition over k for its geometric radical is k 1/2 , (iii) for any commutative k-group scheme Z of ﬁnite type that contains no nontrivial smooth connected k-subgroup, every central extension of G by Z over k is uniquely split. We may and do assume k = ks . The k-group G is pseudo-reductive since is pseudo-reductive (Proposition 1.1.10), and G(k) = G (k ) is its own commutator subgroup (Theorem 9.4.3(2)), so G is (absolutely) pseudo-simple. Each maximal k-torus T in G is the maximal k-torus in R k /k (T ) for a unique maximal k -torus T in G (Proposition A.5.15(2)), so G has rank n since k /k is purely inseparable and G has rank n. (Explicitly, G is identiﬁed with pred Gk and T is thereby identiﬁed with Tk .) The identiﬁcation of Lie(G) with the Lie algebra over k underlying Lie(G ) (Corollary A.7.6) thereby yields (G, T ) = (G , T ), so (G, T ) is non-reduced. This proves (i). To verify (ii), ﬁrst note that the natural map Gk → G has a smooth connected unipotent kernel (Proposition A.5.11) and G has k 1/2 /k as the ﬁeld of deﬁnition over k for its geometric (unipotent) radical. Thus, the ﬁeld of deﬁnition over k for the geometric radical of G is a subﬁeld of k 1/2 . To prove it is equal to k 1/2 , we just have to rule out the possibility that it could be contained in k . If such a containment holds then Ru,k (Gk ) would be a k -descent of the geometric unipotent radical of Gk , so its image in G would be a k -descent of the geometric unipotent radical of G . This contradicts the property that the minimal ﬁeld of deﬁnition over k for the latter unipotent radical is k 1/2 , so (ii) is proved. Finally, we check (iii). Since G is perfect, by Proposition 5.1.3 we just have to group-theoretically describe a Cartan k-subgroup ZG (T ) generically in terms of root groups of (G, T ), for a ﬁxed choice of maximal k-torus T of G. Let T G

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329

be the unique maximal k -torus of G such that T ⊆ R k /k (T ), so ZG (T ) = R k /k (ZG (T )) by Proposition A.5.15(1). The identiﬁcation of (G, T ) and (G , T ) and the unique characterization of root groups (Proposition 2.3.11) imply that for each a ∈ (G, T ) and its corresponding root a ∈ (G , T ), the root group Ua in G is R k /k (Ua ). Hence, we can apply Weil restriction to the rational maps constructed in the proof of Proposition 9.4.2. For any ﬁeld k of characteristic 2 such that [k : k 2 ] = 2, Example 9.4.6 shows that for every nontrivial ﬁnite purely inseparable extension K/k, there exists an absolutely pseudo-simple k-group G of rank 1 such that its root system is non-reduced and the ﬁeld of deﬁnition over k for its geometric radical is K/k. This existence result can be strengthened to a uniqueness result, giving a classiﬁcation of all such G: Proposition 9.4.7 Let k be a ﬁeld of characteristic 2 with [k : k 2 ] = 2 and let G be an absolutely pseudo-simple k-group of rank 1 with a non-reduced root system. Let K/k be the ﬁeld of deﬁnition of the geometric radical of Gk . pred

(1) The natural map !G : G → R K 2 /k (GK 2 ) is an isomorphism, so G has a k-split maximal k-torus and GKss Sp2 . (2) The center ZG is trivial, the natural map G(k) → GKss (K) is bijective, the map ξG is surjective, and all k-automorphisms of G are inner. (3) The k-group G is determined up to isomorphism by K/k, and each nontrivial purely inseparable ﬁnite extension K/k arises from some such G. (4) If Z is a commutative k-group scheme of ﬁnite type containing no nontrivial smooth connected k-subgroup, then every central extension of G by Z over k is uniquely split over k. Proof. By Proposition 9.4.5, the uniqueness result in Theorem 9.4.3 applies to pred G := GK 2 over K 2 . Thus, G := GKss GKss Sp2 , ZG = 1, G contains a K 2 -split maximal K 2 -torus, and ξG : G (K 2 ) → GKss (K) is bijective. Assume for a moment that (1) is proved. Then (3) and (4) follow from the discussion in Example 9.4.6, and we have ZG = R K 2 /k (ZG ) (Proposition A.5.15(1)). Hence, ZG = 1 and ξG : G(k) → G (K) is an isomorphism. The surjectivity of ξG is then a formal consequence of its surjectivity on ks -points. To complete the veriﬁcation of (2), granting (1), we have to check that every automorphism φ of G is inner. Letting φ denote the K 2 -automorphism of pred G = GK 2 induced by φK 2 , the map !G is equivariant with respect to φ and R K 2 /k (φ ). By Theorem 9.4.3(2) the automorphism φ is inner, say conjugation by some γ ∈ G (K 2 ). But we are assuming (1) is known, so γ arises from some g ∈ G(k). Hence, composing φ with conjugation by g −1 reduces us

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The absolutely pseudo-simple groups in characteristic 2

to the case where φ is the identity map. In this case we will show φ is the identity, completing the veriﬁcation of (2). Consider the endomorphism f : x → φ(x)x−1 of the pointed k-scheme (G, 1). Since φ = id, the image of fk is contained in Ru,K 2 (GK 2 ) ⊗K 2 k ⊆ Ru (Gk ), so by the pseudo-reductivity of G it follows that f = 1 (Lemma 1.2.1). Hence, φ = id as desired. It remains to prove (1), for which we can assume k = ks . Since kK 2 = K 2 and [K : K 2 ] = 2, the only kK 2 -subspace of K that contains 1 and generates K as a kK 2 -algebra is K itself. Hence, by Remark 9.1.3 and the isomorphism G SL2 , the map ξG : G → R K/k (G ) is surjective. Letting T be a maximal k-torus of G we have := (G, T ) = {±a, ±2a} for a generator a of X(T ) and we identify TK 2 with a maximal K 2 -torus T of G and identify TK with a maximal K-torus T of G . Thus, (G , T ) = (G, T ) and a identiﬁes T , T , and T with GL1 (over k, K 2 , and K respectively). We denote the root groups ). in G (resp. G , resp. G ) as U±a and U±2a (resp. U±a and U±2a , resp. U±2a The surjectivity of ξG implies that the induced map ZG (T ) → R K/k (T ) between Cartan k-subgroups is surjective, but it factors as the composition of maps ZG (T )

!G

R K 2 /k (ξG )

/ R K 2 /k (ZG (T ))

/ R K/k (T )

between Cartan k-subgroups. Proposition 9.4.1(3) implies that ξG is an isomorphism between Cartan K 2 -subgroups, so !G (ZG (T )) = R K 2 /k (ZG (T )). The quotient map GK 2 G carries (U±a )K 2 onto U±a equivariantly with respect to TK 2 T , so weight space considerations imply that the induced maps (U±2a )K 2 → U±2a , (U±a /U±2a )K 2 → U±a /U±2a are nonzero. Hence, !G induces nonzero maps U±2a → R K 2 /k (U±2a ), U±a /U±2a → R K 2 /k (U±a /U±2a )

(9.4.4)

that are equivariant with respect to !G : ZG (T ) R K 2 /k (ZG (T )). ξG

a

The isomorphism ZG (T ) R K/K 2 (T ) R K/K 2 (GL1 ) determines the Klinear structure on the 2-dimensional vector group Ua /U2a over K 2 . Thus, the surjectivity of !G between Cartan k-subgroups implies that the image of second map in (9.4.4) on k-points is a nonzero linear subspace of the 1-dimensional K-vector space (U±a /U±2a )(K 2 ) and that the image of the ﬁrst map in (9.4.4) on k-points is a nonzero linear subspace of the 1-dimensional K 2 -vector space U±2a (K 2 ). Hence, both maps in (9.4.4) are surjections with kernels that are

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331

linear subspaces relative to the linear structures determined by T , so these kernels are smooth and connected. But ker !G ⊆ ker ξG , so ker !G contains no nontrivial smooth connected k-subgroup. Thus, the ﬁrst map in (9.4.4) is an isomorphism, the map U±a → R K 2 /k (U±a ) is surjective, and for the preimage V± of R K 2 /k (U±2a ) under this map we see that the quotient V± /U±2a is identiﬁed with the linear kernel of the second map in (9.4.4). In particular, V± is smooth and connected with π± : V± → R K 2 /k (U±2a ) a surjective map that is split by U±2a ⊆ V± . It follows that ker π± is a direct factor of V± , so it is smooth and connected. But ker π± ⊆ ker !G and ker !G has no nontrivial smooth connected k-subgroups, so ker π± = 1. Summarizing, we have shown that the maps in (9.4.4) are isomorphisms. Combining this with the surjectivity of !G between Cartan k-subgroups, we conclude via consideration of open Bruhat cells (Corollary 3.3.16) that !G is surjective and (ker !G )0 ⊆ ZG (T ). Since G is generated by its maximal k-tori (as G = D (G)), yet we showed that every such torus centralizes (ker !G )0 , it follows from Lemma 5.3.2 that ker !G ⊆ ZG . In other words, !G expresses G as a central extension of R K 2 /k (G ) by a k-group scheme of ﬁnite type that contains no nontrivial smooth connected k-subgroups. Such an extension is uniquely split (Example 9.4.6), so ker !G is a direct factor of G and thus is smooth and connected. This forces ker !G = 1, so !G is an isomorphism. 2 Proposition 9.4.7(1) provides what we need to generalize Proposition 9.4.5 to arbitrary rank: Proposition 9.4.8 Let G be an absolutely pseudo-simple group over an imperfect ﬁeld k of characteristic 2 such that [k : k 2 ] = 2. Assume that the root system of Gks is non-reduced, and let K/k be the ﬁeld of deﬁnition of the geopred metric radical of G. The maximal pseudo-reductive quotient GK 2 of GKs2 has s a non-reduced root system. Proof. We may and do assume k = ks . Since the case of rank 1 is settled (Proposition 9.4.5), we may assume that the rank of G is equal to some n 2. pred As in the case of rank 1, we assume that H := GK 2 has a reduced root system and we seek a contradiction. The ﬁrst step, once again, is to deduce that H is standard (so ξH is an isomorphism). To do this, ﬁrst observe that the maximal geometric semisimple quotient H ss is naturally isomorphic to G ss and has the k

k

quadratic extension K/K 2 as its minimal ﬁeld of deﬁnition (as a quotient of Hk ). This quotient is simply connected of type Cn with n 2, so the root system (H , S) = (GKss , TK ) (S = TK 2 ) is of type Cn and Proposition 9.1.11 can be applied. It follows that for all long roots a ∈ (H , S) the geometric radical of

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the K 2 -subgroup Ha generated by the ±a-root groups of (H , S) has minimal ﬁeld of deﬁnition over F := K 2 equal to a common subﬁeld F> ⊆ K over F, and that F> = K if and only if H is standard. Thus, to prove the standardness of H we just have to rule out the possibility that F> = K 2 . For c ∈ (G, T ), consider the smooth connected k-subgroup Gc ⊆ G generated by the root groups Uc and U−c . By Proposition 3.4.1 this is absolutely pseudo-simple of rank 1 and if c is not divisible in (G, T ) then Gc = D (ZG (Tc )) for Tc = (ker c)0red . If c is a root which is neither multipliable nor divisible, then Gc has a reduced root system with U±c as the root groups for the maximal k-torus T ∩ Gc . The reducedness also holds if c is divisible (as then Proposition 3.4.1(3) applies). Let Kc /k be the ﬁeld of deﬁnition of the geometric radical of Gc and let Gc = (Gc )ss Kc . By Proposition 6.3.1, as we vary c through (G, T ) the associated subﬁelds Kc in K/k generate K over k and the map (Gc )K → GKss factors through a K-subgroup inclusion of the quotient (Gc )K . The K-subgroup (Gc )K ⊆ GKss is generated by a pair of opposite roots groups (±cK -root groups if c is not multipliable, and ±2cK -root groups if c is multipliable), so it is SL2 since GKss is simply connected. If c is not multipliable, then by Proposition 9.1.10 the map ξGc is an isomorphism and Gc R Kc /k (SL2 ). Now we ﬁx a multipliable root c. Then Gc has a non-reduced root system, so Kc = k and hence k ⊆ Kc2 . The classiﬁcation in Proposition 9.4.7(1) gives that Gc R Kc2 /k (H ) for such c, with H the unique (up to isomorphism) absolutely pseudo-simple Kc2 -group of rank 1 with a non-reduced root system and Kc /Kc2 as the ﬁeld of deﬁnition of its geometric radical. But G2c ⊆ Gc , and by inspection of the structure of Gc we then see that K2c = Kc2 . The condition F> = K 2 says precisely that H2c is reductive, so we assume this and seek a contradiction. In view of the explicit description of Gc and H2c in Proposition 3.4.1 in terms of derived groups and torus centralizers in G and H respectively, it follows from Proposition A.8.14(2) that there is a natural étale unipotent isogeny pred

(Gc )K 2 → H2c . So the reductivity of H2c implies that Ru,K 2 ((Gc )K 2 ) is a K 2 -descent of the geometric radical of Gc (as reductivity is insensitive to isogenies). The reductivity of H2c thus implies that Kc ⊆ K 2 , so K2c = Kc2 ⊆ K 4 . We will show that this is impossible. The Weyl group W (G, T )(k) = W (GKss , TK )(K) acts transitively on the set of non-multipliable roots of a given length (Proposition 3.2.7), so we get a

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pair of subﬁelds K< , K> ⊆ K over k (the common value of Ka for short nonmultipliable a and long non-multipliable a respectively). We can therefore do the exact same calculation with Cartan subgroups and root groups inside of 2 ⊆ R K/k (GKss ) as at the end of the proof of Proposition 6.3.2 to infer that K< 4 2 K> ⊆ K< . But we saw above that K> ⊆ K , so K< ⊆ K . For a multipliable 1/2 1/2 root c the associated ﬁeld Kc is K2c = K> ⊆ K 2 , so this contradicts the fact that the ﬁelds Kc for all c ∈ (G, T ) generate K. Hence, H is indeed standard since we have assumed that it has a reduced root system, so H R K/K 2 (Sp2n ). In particular, Hc = R K/K 2 (SL2 ) for all c ∈ (H , S). Now that H is known to be standard, consider a multipliable root c ∈ (G, T ). The explicit description of Gc in Proposition 3.4.1(2) implies (by Proposition A.8.14(2)) that there is an étale unipotent isogeny pred

(Gc )K 2 → H2c . The left side therefore inherits reducedness of its root system from the right side. But Gc has a non-reduced root system (as c is multipliable), so it follows from Proposition 9.4.5 that Kc ⊆ K 2 and thus K> = K2c = Kc2 ⊆ K 4 . However, we 2 ⊆ K , so once again K ⊆ K 2 for all a ∈ (G, T ), have already seen that K< > a a contradiction. 2 Example 9.4.9 Using Proposition 9.4.8 in place of Proposition 9.4.5, the construction and justiﬁcation in Example 9.4.6 work as written in any rank n 1. To be precise, let k be an imperfect ﬁeld with characteristic 2 such that [k : k 2 ] = 2, and let k /k be a ﬁnite purely inseparable extension. For any absolutely pseudosimple k -group G of rank n 1 having a non-reduced root system and k 1/2 /k as the ﬁeld of deﬁnition of its geometric radical (e.g., G = Gα,n for any α ∈ k 1/2 − k ; cf. Theorem 9.4.3(1)), the k-group G = R k /k (G ) is absolutely pseudo-simple of rank n with a non-reduced root system and k 1/2 /k as the ﬁeld of deﬁnition of its geometric radical. Moreover, for any commutative k-group scheme Z of ﬁnite type such that Z contains no nontrivial smooth connected k-subgroups, every central extension of G by Z is (uniquely) split over k. We can now classify all absolutely pseudo-simple groups with a non-reduced root system, provided that the imperfect ground ﬁeld k of characteristic 2 is “almost perfect” (i.e., [k : k 2 ] 2). The case [k : k 2 ] = 1 is perfect k, over which pseudo-reductive groups are connected reductive and there is nothing new for us to do. The case [k : k 2 ] = 2 is: Theorem 9.4.10 Let k be a ﬁeld of characteristic 2 with [k : k 2 ] = 2 and let G be an absolutely pseudo-simple k-group of rank n 1 with a non-reduced root system. Let K/k be the ﬁeld of deﬁnition of the geometric radical of Gk .

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(1) The natural map !G : G → R K 2 /k (GK 2 ) is an isomorphism, so G has a k-split maximal k-torus and GKss Sp2n . (2) The center ZG is trivial, the natural map G(k) → GKss (K) is bijective, the map ξG is surjective with connected kernel whose underlying k-scheme is R K/k (ker FGa /K )2n , and all k-automorphisms of G are inner. (3) The k-group G is determined up to isomorphism by K/k and n, and for every nontrivial purely inseparable ﬁnite extension K/k and every n 1 the pair (K/k, n) arises from some such G. (4) If Z is a commutative k-group scheme of ﬁnite type containing no nontrivial smooth connected k-subgroup, then every central extension of G by Z over k is uniquely split. Proof. Arguing as in the proof of Proposition 9.4.7, but replacing Proposition 9.4.5 with Proposition 9.4.8 and replacing Example 9.4.6 with Example 9.4.9, pred to prove the theorem it sufﬁces to prove part (1). Since GK 2 has a non-reduced root system and a K 2 -split maximal K 2 -torus (Proposition 9.4.8 and Theorem 9.4.3(2)), GKss Sp2n by Theorem 9.4.3(1) and we can now assume k = ks . By deﬁnition, the map !G induces an isomorphism between maximal geometric semisimple quotients, which is to say that it identiﬁes the reduced root data formed by the non-multipliable roots and non-divisible coroots. But in these smaller root data it is precisely the long roots that are divisible in the non-reduced root data (and likewise the short coroots that are multipliable). Hence, !G identiﬁes the non-reduced root data as well. That is, for a maximal pred k-torus T in G, the root data R(G, T ) and R(GK 2 , TK 2 ) are naturally identiﬁed pred

via !G . This allows us to speak of root groups for (G, T ) and (GK 2 , TK 2 ) labeled by a common root. The strategy of the proof that !G is an isomorphism is to study its behavior on subgroups generated by pairs of opposite roots. Step 1. For each c ∈ (G, T ) consider the smooth connected k-subgroup Gc ⊆ G generated by the root groups Uc and U−c . By Proposition 3.4.1 this k-subgroup is absolutely pseudo-simple of rank 1, and it has a reduced root system precisely when c is not multipliable in (G, T ). The natural map hc : (Gc )k → G ss factors through (Gc )ss via a subgroup inclusion, due to k k Proposition 6.3.1(2), yet this semisimple subgroup of rank 1 is centralized by the k-ﬁber of Tc = (ker c)0red (with Tc = T2c when c is multipliable). The map hc induces an isomorphism (Gc )ss (G ss )c when c is non-multipliable and k k (Gc )ss (G ss )2c when c is multipliable. The group G ss Sp2n is simply conk k k nected, so these rank-1 semisimple subgroups are all isomorphic to SL2 (rather than PGL2 ).

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Let Kc /k be the ﬁeld of deﬁnition of the geometric radical of Gc , so by Proposition 6.3.1(1) each Kc is a subﬁeld of K and they generate K over k. The Weyl group W (G, T )(k) coincides with that of the reduced and irreducible root system (GKss , TK ) of non-multipliable roots in (G, T ) (by Proposition 3.2.7), so NG (T )(k) acts transitively on the set of non-multipliable roots of a given length (short or long). Hence, we may and do write K< (resp. K> ) to denote the common subﬁeld Kc ⊆ K for the non-multipliable c that are short (resp. long). For a multipliable root c, 2c is a long root, so W (G, T )(k) acts transitively on the set of such c and hence we may write K>/2 to denote the common subﬁeld of K attached to such c. Step 2. A key step is to determine all of the ﬁelds Kc in terms of K: Lemma 9.4.11 We have K> = K 2 and K< = K>/2 = K. Equivalently, Kc = K 2 for divisible roots c, and Kc = K for all non-divisible c. Proof. Since (Gc )ss SL2 for all c, with Gc having a reduced root system k precisely when c is non-multipliable, we can describe Gc concretely in all cases as follows. In case c is non-multipliable, it follows from Proposition 9.1.10 that there is an isomorphism ξGc : Gc R Kc /k (SL2 ). If instead c is multipliable then Kc = k (as Gc cannot be reductive since its root system is non-reduced), and the classiﬁcation of absolutely pseudo-simple groups of rank 1 with a nonreduced root system and an arbitrary ﬁeld of deﬁnition for its geometric radical (Proposition 9.4.7(1)) gives an isomorphism !Gc : Gc R Kc2 /k (Gαc ,1 ) for any choice of αc ∈ Kc − Kc2 . For multipliable c we have U±2c ⊆ U±c , so G2c ⊆ Gc . More speciﬁcally, the ±2c-root groups for (G, T ) coincide with those for the pair (Gc , T ∩ Gc ), and by inspection of the root groups of the Kc2 -group Gαc ,1 we conclude that the ±2c-root groups of Gc have k-dimension [Kc2 : k]. Yet 2c is not multipliable, so G2c R K2c /k (SL2 ), so its root groups have dimension [K2c : k]. Every ﬁnite 2 extension of k is determined by its k-degree, so K>/2 = K> , which is to say 1/2

K>/2 = K> . Now for any non-multipliable c, c ∈ (G, T ) we study how the Cartan ksubgroup of Gc containing the maximal k-torus T ∩ Gc acts on the c -root group of (Gc , T ∩Gc ). By the same technique that was used in the proof of Proposition ∨ 6.3.2, we get that (Kc× )c ,c ⊆ Kc× . If we choose a positive system of roots in (G, T ) and consider the associated positive system of non-multipliable roots then for adjacent vertices {a, b} in the Dynkin diagram with a long and b short we have a, b∨ = −2 and b, a∨ = −1. Taking c = a and c = b thereby

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2 ⊆ K . Thus, gives K> ⊆ K< , whereas taking c = b and c = a gives K< > 1/2

K> ⊆ K< ⊆ K> = K>/2 . But we have already noted that the ﬁelds Kc for c ∈ (G, T ) generate K over k, so this forces K>/2 = K, K> = K 2 , and K< ∈ {K, K 2 }. Hence, to prove K< = K it sufﬁces to rule out the possibility K< = K 2 . Suppose K< = K 2 . Fix a short non-multipliable c ∈ (G, T ); such a root is non-divisible and the hypothetical equality Kc = K 2 implies Gc R K 2 /k (SL2 ). pred

pred

Thus, (Gc )K 2 SL2 , so (Gc )K 2 has 1-dimensional root groups. But Gc = D (ZG (Tc )) for the non-multipliable and non-divisible c (Proposition 3.4.1(2)), so Proposition A.8.14(2) and a passage to derived groups provides an étale unipotent isogeny pred

pred

(Gc )K 2 → (GK 2 )c .

(9.4.5)

pred

It follows that (GK 2 )c has 1-dimensional root groups. Let us compare this dimension result with an explicit description of the pred non-multipliable short root groups in GK 2 . The absolutely pseudo-simple K 2 pred

group GK 2 of rank n has K/K 2 as the ﬁeld of deﬁnition of its geometric radical, and its root system is non-reduced (Proposition 9.4.8). Thus, this K 2 -group is classiﬁed by Theorem 9.4.3(1): it is Gα,n for any α ∈ K − K 2 . By an analysis of its construction, its non-multipliable short root groups are 2-dimensional over K 2 , which contradicts the computation of its dimension via comparison with Gc . 2 Step 3. Continuing with the proof of Theorem 9.4.10, now we return to a pred study of the natural map !G : G → R K 2 /k (GK 2 ) that we have already seen induces an isomorphism between the root data. By an analysis of this map on Gc for all roots c, we will prove that !G is surjective as a k-homomorphism and also induces an isomorphism between root groups associated to a common root. Granting this for a moment, we now explain how to deduce that !G is an isomorphism. The isomorphism condition between root groups implies that Lie(ker !G ) has no nontrivial weights for the action by T , whence T centralizes (ker !G )0 (Corollary A.8.11). But G is generated by its maximal k-tori (as G = D (G)), so (ker !G )0 is central in G and thus ker !G is also central (Lemma 5.3.2). The assumed surjectivity of !G then implies that G is a central extension of pred R K 2 /k (GK 2 ) by the k-subgroup scheme ker !G . But ker !G is unipotent (as

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(ker !G )K 2 ⊆ Ru,K 2 (GK 2 )) and hence contains no nontrivial smooth connected k-subgroups due to the pseudo-reductivity of G. Thus, the central extension !G

pred

1 → ker !G → G → R K 2 /k (GK 2 ) → 1 is split, due to Example 9.4.9. It follows that ker !G is identiﬁed with a direct factor of the smooth connected k-group G, so this kernel is smooth and connected. But ker !G contains no nontrivial smooth connected k-subgroups, so ker !G = 1 and hence !G is an isomorphism (assuming it is surjective and restricts to an isomorphism between root groups for a common root). Step 4. To prove that !G is surjective we will use the known structure of pred GK 2 Gα,n (with α ∈ K − K 2 ), and to prove that it is an isomorphism between c-root groups for all c ∈ (G, T ) we will use the known structure of Gc from the proof of Lemma 9.4.11 (and this lemma identiﬁes the ﬁeld Kc in each case). We initially focus on the non-multipliable c, so the map ξGc is an isomorphism (as we noted in the proof of Lemma 9.4.11). Since K< = K and K> = K 2 by Lemma 9.4.11, we have isomorphisms ξGc : Gc R K/k ((Gc )ss K ) for short non-multipliable c, ξGc : Gc R K 2 /k ((Gc )ss ) for long c. K2 If the non-multipliable c is short (i.e., not divisible) then (9.4.5) can therefore be written as an étale (and hence central) unipotent isogeny R K/K 2 (SL2 ) → pred

(GK 2 )c , and this must then be an isomorphism since SL2 (K) has trivial center. If instead c is long (i.e., divisible) then the description of Gc provided by Proposition 3.4.1(3) involves an additional centralizer against the 2-torsion in a speciﬁc maximal torus, and the same description applies to (GKss2 )c . Hence, Proposition A.8.14(2) and a passage to derived groups provides a natural étale unipotent isogeny pred

pred

= (Gc )K 2 → (GK 2 )c (Gc )ss K2 that is again necessarily an isomorphism. Summarizing, for short non-multipliable c ∈ (G, T ) we get a composite isomorphism ξGc

pred

Gc R K/k ((Gc )ss K ) R K 2 /k ((GK 2 )c )

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The absolutely pseudo-simple groups in characteristic 2

and for long c ∈ (G, T ) we get a composite isomorphism ξGc

pred

Gc R K 2 /k ((Gc )ss ) R K 2 /k ((GK 2 )c ). K2 pred

pred

By deﬁnition, (GK 2 )c contains the c-root group of GK 2 , and Weil restriction naturally commutes with the formation of root groups due to their unique characterization (Proposition 2.3.11). Hence, the target of the above isomorphism pred contains the c-root group of R K 2 /k (GK 2 ). But it follows from the deﬁnitions pred

that composing this isomorphism with the inclusion of R K 2 /k ((GK 2 )c ) into pred

R K 2 /k (GK 2 ) recovers !G |Gc , so for each non-multipliable c we have proved that !G restricts to an isomorphism between c-root groups. Step 5. Now suppose that c is multipliable, so it is not divisible and Kc = K. pred In particular, the absolutely pseudo-simple K 2 -group (Gc )K 2 of rank 1 has a non-reduced root system (by Proposition 9.4.5), and we have seen (in the proof of Lemma 9.4.11) that there is an isomorphism pred

!Gc : Gc R K 2 /k ((Gc )K 2 ). pred

By non-divisibility of c, the groups Gc and (GK 2 )c are respectively described in Proposition 3.4.1(2) as the derived group of the centralizer of Tc and (Tc )K 2 , where Tc := (ker c)0red . Thus, once again applying Proposition A.8.14(2), we pred

get an étale unipotent isogeny (Gc )K 2

pred

→ (GK 2 )c . This isogeny must be pred

central (as the source group is connected), so its kernel is trivial since (Gc )K 2 has trivial center (Theorem 9.4.3(2)). Applying R K 2 /k , we get a composite isomorphism pred

pred

Gc R K 2 /k ((Gc )K 2 ) R K 2 /k ((GK 2 )c ), and as in the non-multipliable case it follows from the deﬁnitions that composing pred this with the inclusion into R K 2 /k (GK 2 ) yields !G |Gc . Thus, !G restricts to an isomorphism between c-root groups for multipliable c as well. The preceding argument gives a lot of information about the image of !G . pred Consider an open Bruhat cell in R K 2 /k (GK 2 ) associated to a choice of maximal k-torus T in G and a positive system of roots + in (G, T ) = (GK 2 , TK 2 ). pred

pred

The absolutely pseudo-simple K 2 -group GK 2 of rank n with a non-reduced root system and K/K 2 as the ﬁeld of deﬁnition of its geometric radical is explicitly described by Theorem 9.4.3(1): it is Gα,n for any α ∈ K − K 2 . The structure of its open Bruhat cell relative to T and + is therefore provided by Theorem

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9.3.10(2), from which we see that the Cartan subgroup containing TK 2 is a direct pred product of the Cartan subgroups in the K 2 -groups (GK 2 )c for simple positive + c ∈ determined by T . Since Weil restriction naturally commutes with the formation of Cartan subgroups (Proposition A.5.15(3)), it follows from the isomorphism property of the restrictions !G |Gc for simple c ∈ + that the image of !G contains the pred Cartan subgroup of R K 2 /k (GK 2 ) containing T . Hence, the image of !G contains an open Bruhat cell, and so !G is surjective. We argued above that this surjectivity, coupled with the isomorphism property on root groups that has been established, implies that !G is an isomorphism. 2 We conclude with analogues of Proposition 7.3.3 and Corollary 7.3.4. Proposition 9.4.12 Let k be an imperfect ﬁeld of characteristic 2 such that [k : k 2 ] = 2. Let G be an absolutely pseudo-simple k-group with rank n 1 and a non-reduced root system. Let K/k be the ﬁeld of deﬁnition of the geometric radical of G, G := GKss Sp2n , and ξG : G → R K/k (G ) the natural surjective map. (1) The natural map G(k) → G (K) is an isomorphism of groups and H1 (k, G) vanishes. (2) If k is complete with respect to an absolute value then the natural map G(k) → G (K) is a topological isomorphism. (3) If k is a global function ﬁeld then the natural map G(Ak ) → G (AK ) on adelic points is a topological isomorphism. Proof. By Theorem 9.4.10(1) we get the group isomorphism claim in (1). Applying the result over ks , we get H1 (k, G) H1 (K, G ). But G Sp2n , whose degree-1 cohomology vanishes over any ﬁeld whatsoever (as nondegenerate symplectic spaces over any ﬁeld are determined up to isomorphism by their even rank). This settles (1). The completeness hypothesis in (2) ensures that the valuation topology on K is its topology as a ﬁnite-dimensional k-vector space. Thus, since the formation of topological groups of ﬁeld-valued and adelic points in (2) and (3) is naturally compatible with Weil restriction, the isomorpred phism !G : G R K 2 /k (GK 2 ) and Proposition 9.4.8 thereby reduce the proofs pred

of (2) and (3) to the case of GK 2 in place of G. That is, we can assume K = k 1/2 . In this case there is an explicit description of G, namely G Gα,n for any α ∈ K − k. Since ξG is surjective with unipotent kernel, for any maximal k-torus T in G the image ξG (T ) is a maximal k-torus in R K/k (G ). This image is the maximal k-torus in R K/k (T ) for a unique maximal K-torus T in G (Proposition

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The absolutely pseudo-simple groups in characteristic 2

A.5.15(2)), and explicitly T is the image of TK under the quotient map GK G corresponding to ξG via the universal property of Weil restriction. The natural isomorphism of X(T ) with X(T ) identiﬁes (G , T ) with the set of nonmultipliable roots in (G, T ) (Theorem 2.3.10), so there is a natural bijective correspondence between positive systems of roots in the two root systems. Note that when T is identiﬁed with a maximal k-torus in R K/k (G ), the root system (R K/k (G ), T ) is identiﬁed with (G , T ) (Example 2.3.2). For G = Gα,n there is a canonical isomorphism of G with Sp2n (depending on α, a k-split maximal k-torus T in G, and a positive system of roots + in (G, T )) so that for the corresponding positive system of roots + in (R K/k (G ), T ) = (G , T ) the restriction of ξG between open Bruhat cells for (G, T , + ) and (R K/k (G ), T , + ) can be described explicitly. To be precise, by the proof of Theorem 9.3.10, the map ξG between translated open Bruhat cells is identiﬁed with the map f : U w × B → R K/k (U wB) introduced in Theorem 9.3.5. This map is constructed as a direct product of several maps: the identity map on R K/k (D) (for the standard maximal diagonal K-torus D in Sp2n ), copies of the identity map on R K/k (Ga ) (for short roots of (Sp2n , D)), and copies of the map R K/k (Ga ) × Ga → R K/k (Ga ) deﬁned by (x, y) → x2 + yα. Thus, to establish (2) we just have to check that the bijective continuous map K × k → K deﬁned by (x, y) → x2 + yα has continuous inverse. Since K = k ⊕ kα and the squaring map K → k is a topological isomorphism (as we are in characteristic 2 and k is complete), the required continuity for (2) is clear. To prove assertion (3), we can use a straightforward variation on the method of proof as in the exotic case (i.e., Proposition 7.3.3(3), applied to ξG ). Note that it is not necessary to invoke the fact that Weil’s theory of birational group laws works over a Dedekind base (such as the ring of -integers of k). In view of Corollary 3.4.3(1) the main point that must be checked is that for a k-split maximal k-torus T in G and the corresponding K-split maximal K-torus T = TK ⊆ GKss = G , the induced map W (G, T )(k) → W (G , T )(K) is bijective. Since the equality X(T ) = X(T ) carries (G , T ) over to the set of non-multipliable roots in (G, T ), the Weyl groups of the two root systems are naturally isomorphic. By

9.4 Classiﬁcation of absolutely pseudo-simple groups

341

Proposition 3.2.7, the groups W (G, T )(k) and W (G , T )(K) map isomorphically onto the Weyl groups of the respective root systems. In other words, ξG induces an isomorphism W (G, T )(k) W (G , T )(K). 2 Corollary 9.4.13 Let (G, K/k, G ) be as in Proposition 9.4.12. (1) If T ⊆ G is a maximal K-torus then there exists a unique maximal k-torus T ⊆ G such that T ⊆ ξG−1 (R K/k (T )), and the map T → R K/k (T ) is an isomorphism onto the maximal k-torus of R K/k (T ). (2) The correspondence in (1) deﬁnes a bijection between sets of maximal tori, also characterized by the property T (k) ⊆ T (K) via the equality G(k) = G (K), and T is k-split if and only if T is K-split. (3) All maximal k-split k-tori in G are G(k)-conjugate. Assertion (3) is a special case of a conjugacy result in all smooth connected afﬁne groups over any ﬁeld (Theorem C.2.3), but it is more elementary to give a direct proof in this case (using the more familiar connected reductive case of the conjugacy result). Proof. Since k is inﬁnite and tori are unirational, the condition T (k) ⊆ T (K) is equivalent to the condition T ⊆ ξG−1 (R K/k (T )). Also, the maximal k-torus S in R K/k (T ) is k-split if and only if T is K-split, since the natural map SK → T is an isomorphism. Thus, (2) is a consequence of (1). To prove (1), since ξG : G → R K/k (G ) is surjective with unipotent kernel, the method of proof of Corollary 7.3.4 carries over essentially unchanged (replacing Proposition 7.3.3 with Proposition 9.4.12). The only novelty is the intervention of R K/k , so we use the good behavior of the formation of maximal tori with respect to Weil restriction (Proposition A.5.15(2)). For (3), ﬁrst observe that the connected reductive K-group G Sp2n is K-split, so its maximal K-split K-tori are all maximal as K-tori. Thus, if S is a maximal k-split k-torus in G then the K-split K-torus SK of G is contained in a K-split maximal K-torus T , so ξG (S) ⊆ R K/k (T ). The maximal k-torus T of G corresponding to T is therefore k-split, and it contains S. By maximality we must then have S = T , so S is maximal as a k-torus of G. In other words, the maximal k-split k-tori of G are precisely the maximal k-tori of G that are k-split. By (2), these naturally correspond to the K-split maximal K-tori of G . The latter are G (K)-conjugate to each other, so the deﬁnition of the bijective correspondence in (2) and the bijectivity of the map ξG : G(k) → G (K) imply the G(k)-conjugacy of maximal k-split k-tori of G. 2

10 General case

Since an absolutely simple and semisimple group of type Bn or Cn is either simply connected or adjoint but not both (and in characteristic 2 the two extremes are related by a purely inseparable central isogeny of degree 2), our technique for reducing the classiﬁcation problem to the case of absolutely pseudo-simple k-groups when char(k) = 2 will need to be modiﬁed if char(k) = 2. The notion of standard pseudo-reductive group has to be replaced with something that incorporates exotic constructions right from the beginning. The resulting “generalized standard” construction will classify all pseudo-reductive groups with a reduced root system (over ks ) in any characteristic, assuming [k : k 2 ] 2 when char(k) = 2. This only has novelty in characteristics 2 and 3, since in all other characteristics the generalized standard construction will literally be the standard construction.

10.1 Factors with non-reduced root system and the generalized standard construction Let k be a ﬁeld. The ﬁrst fact we wish to establish is that for any pseudo-reductive k-group G, if (Gks , Tks ) is non-reduced (so k is imperfect with characteristic 2) then its non-reduced irreducible components can be uniquely split off as the root system arising from a k-group direct factor of G, provided [k : k 2 ] = 2. We will often use without comment the elementary observation that if k has characteristic p > 0 and [k : k p ] = p then every ﬁnite extension of k satisﬁes the same degree condition. Deﬁnition 10.1.1 A pseudo-reductive k-group G is totally non-reduced if G = D (G) = 1 and every irreducible component of (Gks , Tks ) is non-reduced, where T is a maximal k-torus of G. 342

10.1 Factors with non-reduced root system

343

Since the formation of the root system of a pseudo-simple group over a separably closed ﬁeld is unaffected by a separable extension of the base ﬁeld to another separably closed ﬁeld, if k /k is a separable extension then a pseudoreductive k-group G is totally non-reduced if and only if Gk is totally nonreduced. By Proposition 3.2.10, G is totally non-reduced precisely when it is perfect and every pseudo-simple normal ks -subgroup of Gks has a non-reduced root system. Deﬁnition 10.1.2 A basic non-reduced pseudo-simple k-group is an absolutely pseudo-simple k-group G with a non-reduced root system such that the ﬁeld of deﬁnition K/k for the geometric radical of G is quadratic over k. For example, if k is imperfect with characteristic 2 and K/k is any purely inseparable quadratic extension then for any α ∈ K − k and n 1, the k-group Gα,n from Theorem 9.3.10 is a basic non-reduced pseudo-reductive k-group. If char(k) = 2 and [k : k 2 ] = 2 then up to k-isomorphism there is exactly one basic non-reduced pseudo-reductive k-group of each rank n 1 (Theorem 9.4.3(1)). If we allow the ﬁeld of deﬁnition of the geometric radical to have larger degree, or we relax absolute pseudo-simplicity to perfectness, we can make more examples of totally non-reduced pseudo-reductive k-groups: Example 10.1.3 Let k be an imperfect ﬁeld of characteristic 2, k a nonzero ﬁnite reduced k-algebra, and G a k -group such that each ﬁber over Spec k is absolutely pseudo-simple.Also assume, for ks := k ⊗k ks , that every ﬁber of Gk s over Spec ks has a non-reduced root system. Then we claim that G := R k /k (G ) is totally non-reduced, and that it is perfect if [k : k 2 ] = 2. We may assume k = ks and pass to factor ﬁelds of k so that k is a ﬁeld. For a maximal k -torus T in G , the maximal k-torus T in R k /k (T ) is a maximal k-torus in G (Proposition A.5.15(2)) and T Tk since k /k is purely inseparable. We then have (G, T ) = (G , T ) via the identiﬁcation of X(T ) and X(T ), by Example 2.3.2. Hence, (G, T ) is non-reduced and irreducible. To prove that G is perfect when [k : k 2 ] = 2 it sufﬁces to prove that the group G(k) = G (k ) is perfect. Since [k : k 2 ] = 2, by Theorem 9.4.10(2) we have G (k ) Sp2n (k ) for some ﬁnite extension k /k . But Sp2n (F) is perfect for any inﬁnite ﬁeld F, so we are done. The preceding class of examples is exhaustive when char(k) = 2 and [k : k 2 ] = 2: Proposition 10.1.4 Let k be an imperfect ﬁeld of characteristic 2 such that [k : k 2 ] = 2, and let G be a totally non-reduced pseudo-reductive k-group.

344

General case

(1) There exists a nonzero ﬁnite reduced k-algebra k and a k -group G such that the ﬁbers of G → Spec k are basic non-reduced pseudo-simple k-groups and G R k /k (G ). (2) If (G , k /k) is another such pair, then every k-group isomorphism R k /k (G ) R k /k (G ) is induced by a unique k-algebra isomorphism α : k k and a unique group isomorphism G G over Spec(α). In particular, the pair (G , k /k) in (1) is uniquely determined by G up to unique k-isomorphism. (3) If {ki } is the set of factor ﬁelds of k then every smooth connected normal ksubgroup of G is a product among the factor groups R ki /k (Gi ). In particular, G is pseudo-simple over k if and only if k is a ﬁeld, and it is absolutely pseudo-simple if and only if k /k is purely inseparable. Proof. By Galois descent, it sufﬁces to treat the case where k = ks . Let {Gi }i∈I be the set of pseudo-simple normal k-subgroups of G. By hypothesis G is totally non-reduced, so it follows from Proposition 3.2.10 that every Gi is totally nonreduced. But each Gi is (absolutely) pseudo-simple, so since [k : k 2 ] = 2 we may use the classiﬁcation in Theorem 9.4.10 to infer that ZGi = 1 and Gi R ki /k (Gi ) for a ﬁnite extension ki /k and a basic non-reduced pseudosimple ki -group Gi . The triviality of ZGi for all i implies that the surjective map Gi → G with central kernel has trivial kernel and hence is an isomorphism. Thus, letting k = ki and G = Gi yields the isomorphism G

Gi R k /k (G ).

This proves (1). To prove (2), consider a second pair (G , k /k), say with {kj }j∈J the factor ﬁelds of k and Gj the kj -ﬁber of G (which we assume to be a basic nonreduced pseudo-simple kj -group). By Example 9.4.9, R kj /k (Gj ) is absolutely pseudo-simple with a non-reduced root system. Consider any isomorphism R ki /k (Gi ) = R k /k (G ) R k /k (G ) = R kj /k (Gj ). f : j

i

The k-subgroups Gi R ki /k (Gi ) were constructed as the pseudo-simple normal k-subgroups of G, so by the proof of Corollary 8.1.4 every smooth connected normal k-subgroup of G is a product among the k-subgroups Gi .Applying this to the R kj /k (Gj ), we conclude that f identiﬁes each R kj /k (Gj ) with R k /k (Gi( j) ) i( j) for a unique i( j), with j → i( j) a bijection from J to I .

10.1 Factors with non-reduced root system

345

This reduces the proof of (2) to the case where k and k are both ﬁelds, in which case the problem is to show that necessarily k = k (as purely inseparable ﬁnite extensions of k) and that the isomorphism f is the Weil restriction of a unique isomorphism of groups G G over the common ground ﬁeld k = k . By Example 9.4.9, the k-group R k /k (G ) intrinsically determines k : the ﬁeld of deﬁnition over k for the geometric radical is k 1/2 . Hence, indeed k = k . It remains, for the proof of (2), to show that if G and H are two basic non-reduced pseudo-simple k -groups then the natural map Isomk (G , H ) → Isomk (R k /k (G ), R k /k (H )) is bijective. The rank of the k -group G is the same as the rank of the kgroup R k /k (G ), so both sides are empty unless G and H have the same rank. Assuming this equality of ranks, G and H are isomorphic since each is a basic non-reduced pseudo-simple k -group and [k : k 2 ] = 2 (Theorem 9.4.3(1)). Thus, we can rephrase the problem as showing that the Weil restriction functor R k /k carries the k -automorphism group of G isomorphically onto the k-automorphism group of R k /k (G ). But each has only inner automorphisms and trivial center (Theorem 9.4.3(2) for G , and Theorem 9.4.10(2) for R k /k (G )), so the map induced by the functor R k /k between automorphism groups is identiﬁed with the map G (k ) → R k /k (G )(k) assigning to any k -morphism g : Spec k → G the k-morphism ι

g : Spec k R k /k (Spec k ) → R k /k (G ) (where the map ι is the canonical map, which is an isomorphism). We have to check that the correspondence g → g is bijective, and we will show that it is the inverse of the natural bijection from R k /k (G )(k) to G (k ). In other words, we must check that the composite map gk

q

Spec k −→ R k /k (G )k −→ G is g (where q is the natural map). For general quasi-projective k -schemes X , the natural map qX : R k /k (X )k → X in functorial in X . Applying this functoriality to the k -map g : Spec k → G thereby reduces the problem to checking that the composite map ιk

qk

Spec k −→ R k /k (Spec k )k −→ Spec k

346

General case

is the identity map, where ι = jSpec k . This is the special case Y = Spec k of the general claim that for any quasi-projective k-scheme Y , the composite map

Yk

( jY )k

/ R k /k (Yk )k

q(Y

k )

/ Yk

(using q and j as in Proposition A.5.7) is the identity. This in turn is a special case of a general fact about adjoint functors, so (2) is proved. The asserted description of the smooth connected normal k-subgroups in (3) was established in the preceding proof of (2). 2 Proposition 10.1.5 Let k be an imperfect ﬁeld of characteristic 2 such that [k : k 2 ] = 2. If G is a totally non-reduced pseudo-reductive k-group then ZG = 1 and in the absolutely pseudo-simple case all k-automorphisms of G are inner. Moreover, for any commutative k-group scheme Z of ﬁnite type not containing any nontrivial smooth connected k-subgroup, every central extension 1→Z →E→G→1 over k is uniquely split over k. Proof. Once it is shown that the center is trivial, the assertion about inner automorphisms says exactly that every k-automorphism of G is conjugation by a unique g ∈ G(k). This uniqueness and the asserted uniqueness for the splitting claim allow us to use Galois descent to reduce to the case k = ks . Proposition 10.1.4 provides a decomposition G Gi with Gi = R ki /k (Gi ) for ﬁnite extensions ki /k and a basic non-reduced pseudo-simple ki -group Gi for each i. By Theorem 9.4.10(2) we have ZGi = 1 for all i and all automorphisms of Gi are inner. Thus, ZG = 1 and in the absolutely pseudo-simple case (i.e., only one i) every automorphism of G is inner. Since ki /k is purely inseparable, the splitting result for each Gi is Theorem 9.4.10(4). The splitting result for the product G = Gi can be proved by exactly the same method, using the compatibility of open Bruhat cells with respect to Weil restriction and direct products. 2 When char(k) = 2 and [k : k 2 ] = 2, we have the following decomposition of a pseudo-reductive group: Proposition 10.1.6 Let k be a ﬁeld of characteristic 2 such that [k : k 2 ] = 2. Let G be a pseudo-reductive k-group whose root system (over ks ) is non-reduced. There is a unique decomposition G = G1 × G2 where G2 is totally non-reduced and G1 has a reduced root system (over ks ).

10.1 Factors with non-reduced root system

347

Proof. By the uniqueness and Galois descent we may assume k = ks . Thus, if {Gi }i∈I is the non-empty set of pseudo-simple normal k-subgroups of G then by Proposition 3.2.10 some Gi has a non-reduced root system. Since absolutely pseudo-simple totally non-reduced pseudo-reductive k-groups have trivial center and only inner automorphisms over k, and Proposition 3.2.10 relates irreducible components of the root system to the root systems of the pseudo-simple normal k-subgroups (relative to compatible choices of maximal k-tori, as in Proposition 3.2.10), the proof of the splitting result in Theorem 8.2.10 applies verbatim (e.g., take G2 to be the k-subgroup generated by the pseudo-simple normal k-subgroups of G whose root system is non-reduced). The reducedness of the root system of G1 follows from the construction of G1 and Proposition 3.2.10. 2 In view of Propositions 10.1.6 and 10.1.4, to classify pseudo-reductive k-groups when char(k) = 2 and [k : k 2 ] = 2, we can focus our attention on the case where the root system (over ks ) is reduced. There is no analogue of the splitting result in Theorem 8.2.10 for exotic pseudo-simple normal ks subgroups of types Bn or Cn with n 2, due to the nontriviality of their center, so we need to revisit the deﬁnition of the standard construction. The idea is to use not only Weil restrictions of absolutely simple and simply connected groups, but also Weil restrictions of basic exotic pseudo-reductive groups. This will allow us to state and prove a general classiﬁcation result in characteristic 2 (assuming that [k : k 2 ] = 2) in a manner that does not require treating speciﬁc types (with reduced root systems) in a special way. In fact, the methods will be largely characteristic-free. Here is a generalization of the standard construction. Let k be an arbitrary ﬁeld, k a nonzero ﬁnite reduced k-algebra, and G a k -group whose ﬁbers are absolutely pseudo-simple. Assume moreover that each ﬁber is either simply connected semisimple or (if char(k) = 2, 3) basic exotic. Let G = R k /k (G ), and let T be a maximal k -torus in G , so the maximal k-torus T in R k /k (T ) is a maximal k-torus in G (and all maximal k-tori of G uniquely arise in this way; see Proposition A.5.15(2)). Let C = ZG (T ), so C := ZG (T ) = R k /k (C ) by Proposition A.5.15(1). By Theorem 2.4.1, Aut G ,C is represented by an afﬁne k-group of ﬁnite type; let ZG ,C denote its (commutative) maximal smooth connected k-subgroup. We can similarly deﬁne the functor Aut G ,C on k -schemes, and if {ki } is the set of factor ﬁelds of k then for any k -algebra R = Ri (with Spec Ri the ki -ﬁber of Spec R) we have Aut G ,C (R ) = Aut G ,C (Ri ) i i where (Gi , Ci ) is the ki -ﬁber of (G , C ). Thus, by treating the ﬁbers of G over Spec k separately we see that Aut G ,C is represented by the k -group Aut G ,C whose ki -ﬁber represents Aut G ,C . In particular, the maximal smooth ﬁberwise i

i

348

General case

connected k -subgroup ZG ,C in Aut G ,C is ZGi ,Ci over Spec k = Spec ki and R ki /k (Aut G ,C ). R k /k (Aut G ,C ) = i

i

Lemma 10.1.7 The natural map R k /k (Aut G ,C ) → Aut G ,C deﬁned by applying R k /k to automorphisms carries R k /k (ZG ,C ) isomorphically onto ZG ,C . Proof. Since R k /k (ZG ,C ) = R ki /k (ZGi ,Ci ) is smooth and connected, it lands in ZG ,C . To check that this is an isomorphism, we can assume k = ks (so ki /k is purely inseparable for all i). If all ﬁbers of G are standard then the result is Theorem 1.3.9, and if all ﬁbers of G are basic exotic then it is Proposition 8.2.6. The general case goes exactly as in the proof of Proposition 8.2.6, since each k-group Gi = R ki /k (Gi ) is absolutely pseudo-simple (use the criterion in Lemma 3.1.2, along with Proposition A.5.11 and Corollary A.7.11 in the standard case and Corollary 8.1.4 in the exotic case) with Ci = R ki /k (Ci ) a Cartan k-subgroup. 2 The preceding lemma provides an identiﬁcation of ZG ,C with R k /k (ZG ,C ). The conjugation action of C = ZG (T ) on G deﬁnes a homomorphism C → Aut G ,C , and this factors through a unique k-homomorphism C → ZG ,C . Consider any factorization φ

ψ

C −→ C −→ ZG ,C where C is a commutative pseudo-reductive k-group. It is not really necessary to mention ψ here: we could instead just say that C is equipped with an abstract left action on G restricting to the identity on C , and that φ is a k-homomorphism respecting the actions on G . However, it is more concrete and very useful in practice that we can encode the action in terms of the k-homomorphism ψ valued in ZG ,C . For example: Lemma 10.1.8 The natural map C → ZG0 ,C between smooth connected commutative afﬁne k-groups restricts to an isogeny between maximal k-tori. Proof. We may assume k = ks and then pass to factor ﬁelds of k to reduce to the case where k is a ﬁeld. Since ZG ,C = R k /k (ZG ,C ), the compatibility of maximal tori with respect to Weil restriction (Proposition A.5.15(2)) reduces us to considering (G , T , k ) instead of (G , T , k). The simply connected semisimple case is obvious (with ZG ,C = T /ZG ), and the basic exotic case follows from Corollary 8.2.7 (as we may compute by using Theorem 7.2.3(3) and a Levi k -subgroup of G containing T , which exists since k is separably closed). 2

10.1 Factors with non-reduced root system

349

Consider the semidirect product G C. The homomorphism α : C → G C deﬁned by z → (z −1 , φ(z)) is an embedding of C into G C as a central subgroup. We will view C as a subgroup of G C via α. Let G = (G C)/C . By Proposition 1.4.3, G is pseudo-reductive over k. If all ﬁbers of G are absolutely simple and simply connected then C = R k /k (T ) (Proposition A.5.15(3)), so this recovers the standard construction of non-commutative pseudo-reductive k-groups (in the precise form given in Theorem 4.1.1). More generally: Deﬁnition 10.1.9 Let k be a ﬁeld. A pseudo-reductive k-group G is generalized standard if it is commutative or if there exists a 4-tuple (G , k /k, T , C) consisting of a nonzero ﬁnite reduced k-algebra k , a k -group G whose ﬁbers are absolutely pseudo-simple and either simply connected semisimple or (if char(k) = 2, 3) basic exotic, a maximal k -torus T of G , a commutative pseudo-reductive k-group C, and a factorization diagram φ

ψ

C −→ C −→ ZG ,C = R k /k (ZG ,C )

(10.1.1)

with G = R k /k (G ), C = ZG (T ), and C = R k /k (C ) such that there is an isomorphism of k-groups G (G C)/C .

(10.1.2)

The 4-tuple (G , k /k, T , C) equipped with the factorization (10.1.1) and isomorphism (10.1.2) is a generalized standard presentation of G. It is clear that separable extension of the ground ﬁeld preserves the property of being generalized standard (though descent of this property through such a ground ﬁeld extension is not obvious; see Corollary 10.2.5 for separable algebraic extensions). In particular, exactly as in the standard case it follows that the root system (over ks ) for any generalized standard group is reduced, as this is true for the absolutely pseudo-simple groups that are either semisimple or basic exotic. Remark 10.1.10 In the diagram (10.1.1), the kernel ker φ is contained in the center R k /k (ZG ) of R k /k (G ) (see Proposition A.5.15(1) for this determination of the center). Indeed, we just have to check that the composite map ψ ◦ φ has kernel R k /k (ZG ). But this composite map is the canonical map (arising from the conjugation action of C on G ), so by left exactness of Weil restriction it is equivalent to prove that the conjugation action of C on G has functorial kernel ZG . By deﬁnition of the scheme-theoretic kernel, this is clear (since ZG is certainly contained in C ).

350

General case

Also, in the diagram (10.1.1), the ﬁnal term R k /k (ZG ,C ) can be replaced with its identity component R k /k (ZG0 ,C ). In fact, ZG ,C is always connected (but we never need this fact). In the standard case it follows from Theorem 1.3.9, and in the exotic case it requires making explicit the description provided in Corollary 8.2.7. As in the standard case, it is difﬁcult to directly characterize all central k-subgroups Z ⊆ R k /k (ZG ) such that R k /k (G )/Z is pseudo-reductive. Note that such pseudo-reductivity is equivalent to pseudo-reductivity of the Cartan k-subgroup C := R k /k (C )/Z, due to the isomorphism (R k /k (G ) C)/R k /k (C ) R k /k (G )/Z and the pseudo-reductivity of Cartan k-subgroups of pseudo-reductive k-groups. The pseudo-reductivity of such a quotient C is generally not easy to determine (given an arbitrary Z). Remark 10.1.11 In the setting of Deﬁnition 10.1.9, since the derived group of G C is G it follows that the derived group D (G) of the generalized standard pseudo-reductive group G is naturally isomorphic to G / ker φ. As G = R k /k (G ), and ker φ is contained in R k /k (ZG ), there is a natural surjective homomorphism D (G)k G /ZG . There is one property of central quotients of R k /k (G ) that is very useful for applications to classiﬁcation results: Proposition 10.1.12 Let (G , k /k) be as in Deﬁnition 10.1.9, Z ⊆ R k /k (ZG ) a k-subgroup scheme, and G := R k /k (G )/Z the smooth connected afﬁne central quotient over k. (1) For any separable extension of ﬁelds F/k, every F-automorphism f of GF uniquely lifts to an endomorphism f of R k /k (G )F , and f is an automorphism. Moreover, f restricts to the identity on a smooth connected F-subgroup of R k /k (G )F if f restricts to the identity on its image in GF . (2) If H is a smooth connected k-group equipped with a left action on G then this uniquely lifts to a left action on R k /k (G ). This lifted action restricts to the identity on a smooth connected k-subgroup of R k /k (G ) if H acts trivially on its image in G. Proof. In the standard case this is Proposition 5.1.7, and the proof in general is identical, apart from two modiﬁcations. First, we replace references to Example 5.1.4 in the standard case with references to Proposition 8.1.2 in the exotic

10.2 Classiﬁcation via generalized standard groups

351

case. Second, the ﬁbers of ZG may not be ﬁnite: they could be Weil restrictions of μ2 from an extension ﬁeld. Such group schemes receive no nontrivial homomorphisms from a smooth connected group (due to the universal property of Weil restriction), so we can argue exactly as in the proof of Proposition 5.1.7. 2

10.2 Classiﬁcation via generalized standard groups Our main theorem incorporating characteristic 2 is: Theorem 10.2.1 Let k be a ﬁeld, and assume [k : k 2 ] 2 if char(k) = 2. Let G be a pseudo-reductive k-group. (1) If Gks has a non-reduced root system then G has a unique decomposition G1 × G2 for which (G1 )ks has a reduced root system and G2 is totally non-reduced. (2) Assume Gks has a reduced root system, and let T be a maximal k-torus of G. Then G is generalized standard, and it is standard if and only if for all roots in a component of (Gks , Tks ) the root spaces in Lie(Gks ) are of the same dimension. Since non-reduced root systems only exist over imperfect ﬁelds of characteristic 2, the assertion in (1) is settled by Proposition 10.1.6. Thus, to prove Theorem 10.2.1 it remains to prove (2), for which we can assume D (G) = 1. Our proof of this special case will be modeled on the strategy from the standard case, namely to reduce to the case of perfect G with k = ks , and then to the (absolutely) pseudo-simple case. In this latter case we have essentially already classiﬁed all possibilities as generalized standard groups aside from exceptional cases of types Bn and Cn (n 2) for imperfect k of characteristic 2 when G ss k is adjoint rather than simply connected. We will need to do a bit more work to establish the generalized standard property in these remaining cases. Curiously, the reduction to the absolutely pseudo-simple case over k = ks will never use any hypotheses on [k : k 2 ] when char(k) = 2. Thus, in what follows we take k to be an arbitrary ﬁeld. As a ﬁrst step, we give intrinsic interpretations of some ingredients in the generalized standard construction, much like in the standard case. Proposition 10.2.2 Let G be a non-commutative generalized standard group over a ﬁeld k, arising from a 4-tuple (G , k /k, T , C) via a ﬁxed choice of isomorphism (10.1.2).

352

General case

(1) Let j : R k /k (G ) → G be the natural map, and let T be the maximal k-torus in R k /k (T ). The image of j is D (G), and if T is the maximal ktorus of G that is an almost direct product of the maximal central k-torus Z and j(T ) then C = ZG (T ). (2) The natural map T → j(T ) is an isogeny, and there is a bijective correspondence between the set of maximal k -tori S in G and the set of maximal k-tori in G via S → j(S ) · Z where S is the maximal k-torus in R k /k (S ). (3) Let S be a maximal k-torus in G, S the associated maximal k -torus in G , and S the maximal k-torus in R k /k (S ). The conjugation action of ZG (S) on D (G) = j(R k /k (G )) uniquely lifts to an action of ZG (S) on R k /k (G ) restricting to the identity on the maximal torus S . For the generalized standard pseudo-reductive k-group associated to (G , k /k, S , ZG (S)) and the resulting factorization diagram R k /k (ZG (S )) −→ ZG (S) −→ R k /k (ZG ,S ), the natural k-homomorphism γ : (R k /k (G ) ZG (S))/R k /k (ZG (S )) → G is an isomorphism. This result allows us to speak of a generalized standard presentation of a noncommutative pseudo-reductive k-group G being compatible with a ﬁxed choice of maximal k-torus T of G, and part (3) shows that if there is such a presentation compatible with one choice of T then there is one for any choice of T . Proof. In the standard case this is Proposition 4.1.4, and the proof carries over to the general case with the following minor changes. We have to use Proposition 8.1.2 to get perfectness of purely inseparable Weil restrictions of exotic pseudoreductive groups, and use Lemma 10.1.8 to see that the factorization diagram (10.1.1) canonically identiﬁes the maximal k-torus S in C up to k-isogeny with an almost direct product of the maximal k-torus in R k /k (T ) and the maximal k-torus in ker ψ. Since ker φ is central in R k /k (G ) (as in the standard case) and contains no nontrivial k-tori (Lemma 10.1.8), up to writing ZG (T ) in place of T and writing ZG ,T in place of T /ZG the earlier argument carries over unchanged. (We also need to make the same observation concerning the structure of ZG as we did at the end of the proof of Proposition 10.1.12.) 2 Before we can pass to the case k = ks , it is convenient to reduce to the consideration of the perfect pseudo-reductive k-group D (G):

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Proposition 10.2.3 A pseudo-reductive k-group G is generalized standard if and only if D (G) is generalized standard. If either admits a generalized standard presentation resting on the pair (G , k /k) then so does the other. Proof. The proof of Proposition 5.2.1 carries over with just minor notational and cross-reference changes, as in the proof of Proposition 10.2.2. (For example, we use Proposition 10.1.12(2) instead of Proposition 5.1.7(2).) 2 To pass to the case k = ks , we have to show that the pair (G , k /k) is canonically determined by G. We cannot appeal to the simply connected datum as in the standard case, but the following analogue of Proposition 5.2.2 carries over: Proposition 10.2.4 Let (G , k /k, T , C1 ) and (G , k /k, T , C2 ) be two 4tuples that give generalized standard presentations of a non-commutative pseudo-reductive k-group G, and choose both to be compatible with a ﬁxed maximal k-torus T in G. Under the resulting identiﬁcations of C1 and C2 with C := ZG (T ) via Proposition 10.2.2(1), the composite isomorphism (R k /k (G ) C)/R k /k (ZG (T )) G (R k /k (G ) C)/R k /k (ZG (T )) is induced by a unique pair consisting of a k-algebra isomorphism α : k k and an isomorphism of pairs (G , T ) (G , T ) over Spec α respecting the given factorization diagram (10.1.1) and its counterpart for (G , k /k, T , C). In particular, for each maximal k-torus T of G there is a unique generalized standard presentation of G compatible with T . Proof. In the standard case this is Proposition 5.2.2, and the proof works in general with the following minor changes. We use Proposition 10.1.12 instead of Proposition 5.1.7, and to prove that any k-group isomorphism f : R k /k (G ) R k /k (G ) arises from a unique k-algebra isomorphism α : k k and a group isomorphism φ : G G over Spec α, we ﬁrst have to check that the factor groups arising from the factor ﬁelds of k and k are precisely the pseudo-simple smooth connected normal k-subgroups. An elementary argument with commutator subgroups (as in the proof of Corollary 8.1.4) shows that if N1 , . . . , Nr are pseudo-simple k-groups then in Ni the smooth connected normal k-subgroups are the products of the Ni . The pseudosimplicity of the factors arising from the exotic ﬁbers of G and G follows from Proposition 8.1.2 and Corollary 8.1.4. For the standard (i.e., absolutely simple and simply connected) ﬁbers the perfectness follows from Corollary A.7.11, and so for the pseudo-simplicity we can use the criterion in Lemma 3.1.2 when

354

General case

k = ks . Considerations with Galois orbits of pseudo-simple factors of Gks then settles the pseudo-simplicity assertion in the standard case for general k. Now we may and do assume that k and k are ﬁelds. In this case, if G and G are both standard then the existence and uniqueness of (α, φ) is provided by Proposition 5.2.2. If both are exotic then the result instead is provided by Proposition 8.2.4. The other possibility, that one is standard and the other is exotic, cannot happen since Weil restriction preserves standardness but the Weil restriction (through a ground ﬁeld extension) of a basic exotic pseudo-reductive group is never standard (Remark 8.2.3). 2 Corollary 10.2.5 Let G be a non-commutative pseudo-reductive k-group. If Gks is generalized standard over ks , then so is G over k. Proof. In the standard case this is Corollary 5.2.3, and the proof works in general up to writing ZG (T ) instead of T in several places. 2 The following result has the same proof as its analogue (Corollary 4.1.6 and Proposition 5.3.1) in the standard case: Proposition 10.2.6 Let G be a non-commutative pseudo-reductive group over k = ks , and {Gi } the non-empty collection of pseudo-simple normal k-subgroups of G. (1) If each Gi is generalized standard then so is G. (2) Let T be a maximal k-torus of G and assume all Gi are generalized standard. Consider the generalized standard presentation (Gi , ki /k, Ti , Ci ) of G compatible with the maximal k-torus Ti = T ∩ Gi . The identiﬁcation of i X(Ti )Q = X(Ti )Q with X(T ∩ D (G))Q via Proposition 3.2.10 also identiﬁes the root systems (Gi , Ti ) with the irreducible components of (G, T ), and for each a ∈ (Gi , Ti ) the root space Lie(G)a is identiﬁed with Lie(Gi )a as k-vector spaces. The preceding results reduce the proof of Theorem 10.2.1(2) to a special case: k = ks and G is absolutely pseudo-simple with a reduced root system. It must be proved that G is generalized standard, and that it is standard precisely when the dimension criterion on root spaces as in Theorem 10.2.1(2) is satisﬁed.Applying Proposition 10.2.6(2), the dimension criterion follows because it holds in the simply connected semisimple case (with all root groups of dimension 1) and it fails in the basic exotic cases (with long root groups of dimension 1 and short root groups of strictly larger dimension). We may therefore focus our efforts on establishing the generalized standard property. Let T be a maximal k-torus of G, = (G, T ), and for a ∈ let Ga denote the rank-1 absolutely pseudo-simple subgroup generated by the root

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groups U±a . Each Ga has a reduced root system, and Theorem 6.1.1 implies that away from characteristic 2 it is standard; Proposition 9.1.10 leads to the same conclusion when char(k) = 2 (ﬁnally using that [k : k 2 ] = 2!). In particular, ξGa is surjective for all a ∈ . Theorem 6.3.4 now yields standardness in all higher rank cases except when G is exceptional in the sense of Deﬁnition 6.3.3. So we may now assume that k is imperfect with characteristic p ∈ {2, 3} and that G ss has type G2 when p = 3, and type Bn (n 2), Cn (n 2), or F4 k when p = 2. In all of these cases, if G ss is simply connected then G was k proved to be exotic (and hence generalized standard) in Theorem 8.2.1 (using the reducedness hypothesis on the root system in case G ss is simply connected k of type Cn ). Thus it only remains to address the cases with p = 2 and G ss of k adjoint type Bn or Cn for n 2 (so the root system for G is automatically reduced). Hence, we now assume char(k) = 2, [k : k 2 ] = 2, and that G ss is of k adjoint type Bn or Cn with n 2. Let K/k be the ﬁeld of deﬁnition of the geometric radical of G, and let Ka /k be the corresponding ﬁeld for Ga . We can assume that G is not standard, so K = k and by Proposition 6.3.2 we get a proper subﬁeld k ⊂ K containing kK 2 such that Ka = K for all short a ∈ and Ka = k for all long a ∈ . Let G = GKss and T = TK viewed as a maximal K-torus of G . For a ∈ , let Ga denote the connected absolutely simple subgroup of rank 1 generated by the root groups in G corresponding to the roots ±a. By hypothesis G is → G is of adjoint type Bn or Cn , so its simply connected central cover G a double cover, with kernel μ μ2 . For each long a ∈ , Ga is simply connected, hence it is isomorphic to SL2 and ξGa : Ga → R k/k (Ga ) is an isomorphism (Proposition 9.1.10). In particular, Ga is isomorphic to R k/k (SL2 ) for long a. Since G is perfect, the natural map iG : G → R K/k (G ) factors through the derived group of its target. This derived group is computed by Proposition 1.3.4, so we obtain a canonical k-homomorphism )/R K/k (μ). ξG : G → D (R K/k (G )) = R K/k (G A key point is to prove: ), where G ⊆ Proposition 10.2.7 The k-group ξG (G) is the image of R k/k (G R K/k (G ) is a basic exotic pseudo-reductive k-group. Proof. Let L be a Levi k-subgroup of G containing T . Then, as in Step 3 of the proof of Theorem 8.2.1, we can see that ξG (L) is a Levi subgroup of both ξG (G) and R K/k (G ), and ξG |L : L → ξG (L) is an isomorphism. We rename

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General case

ξG (L) as L. Then TK ⊂ LK recovers the maximal K-torus T of G , and hence (L, T ) is a k-descent of (G , T ). Let π : L → L be the simply connected central cover of L, and let T ) which maps onto T (resp. T ). (resp. T ) be the maximal torus of L (resp. G , The pair ( L, T ) is a k-descent of the pair (G T ), so ( Lk , Tk ) is a k-descent of (G , T ). Moreover, Lk is a Levi k-subgroup of R K/k (G ) via the inclusion ), so it gives rise to a basic exotic pseudo-reductive Lk → R K/k ( LK ) = R K/k (G is the sub )) containing k-group G ⊂ R K/k (LK ) (= R K/k (G Tk . Explicitly, G group generated by the long root groups of Lk (⊂ R K/k ( LK )) relative to Tk and the Weil restriction from K to k of the short root groups of LK = G relative to are its root groups with respect to TK = T , and these k-subgroups of G Tk . The k-group R k/k (G ) is perfect, by Proposition 8.1.2, and any perfect pseudo-reductive k-group is generated by the root groups with respect to a ) in ﬁxed maximal k-torus (Lemma 3.1.5). Therefore, the image of R k/k (G R K/k (LK ) = R K/k (G ) is the subgroup generated by the Weil restriction from k to k of the long root groups of Lk (⊂ R K/k (LK )) relative to Tk and the Weil restriction from K to k of the short root groups of LK = G relative to TK = T . We claim that these Weil restrictions are precisely the root groups of ξG (G) with respect to T . Proposition 9.1.10 and (6.3.1) imply that ξG (Gc ) = ξGc (Gc ) = D (R K/k ((Lc )K )) for every short root c. Proposition 1.3.4 describes this derived group. If c is a long root then it follows from the discussion preceding this proposition that ξG (Gc ) (= ξGc (Gc )) is a subgroup of R K/k ((Lc )K ) (⊂ R K/k (LK )) which contains Lc and is isomorphic to R k/k ((Lc )k ), so by an observation in Step 4 of the proof of Theorem 8.2.1, ξG (Gc ) equals R k/k ((Lc )k ) inside R K/k ((Lc )K ). Since the perfect pseudo-reductive k-group ξG (G) is generated by its root groups ξG (Uc ) (Remark 2.3.6 and Lemma 3.1.5), we conclude that ) coincides with ξG (G). the image of R k/k (G 2 Continuing with the proof of Theorem 10.2.1(2), now we will use Proposition 10.2.7 to deduce the generalized standard property for G. Let T be the maximal K-torus of G for which T := T /μ is TK viewed as a maximal K , torus of G . The reducedness of implies that = (G , T ) = (G T ) via the embedding X(T ) → X( T ). Choose c ∈ . The proof of Proposition 10.2.7 implies that ξG carries Gc onto D (R Kc /k (Gc )) via ξGc for a Kc -structure Gc = (Gc )ss Kc on the K-subgroup Gc of G generated by the root groups U±c . By Proposition 9.1.10, the k-group G is standard, so by Remark 5.3.6 the map ξGc : Gc → D (R Kc /k (Gc )) is a central quotient that carries the root groups for (Gc , T ∩ Gc ) over k isomorphically onto the Weil restrictions (from Kc down to k) of the root groups for (Gc , (T ∩ Gc )Kc ). Therefore, ξG carries the c-root group of (G, T ) isomorphically onto a copy of R Kc /k (Ga ) inside

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of the c-root group R K/k (Uc ) of (R K/k (G ), T ) (with Uc the cK -root group of (G , T )). The smooth connected afﬁne k-group ξG (G) is perfect, and it is also pseudoreductive (Proposition 4.2.5). Consider the short exact sequence ξG

1 → ker ξG −→ G −→ ξG (G) → 1. The preceding analysis of root groups implies that the action of T on the Lie algebra of ker ξG is trivial. It then follows (by Lemma 5.3.2 and Corollary A.8.11) that ker ξG is central in G. As G is pseudo-simple, ker ξG does not contain any nontrivial smooth connected k-subgroup. be as in Proposition 10.2.7. We can form a pullback central extension Let G 1

1

/ ker ξG

/ E

/ R k/k (G )

/ ker ξG

/ G

/ ξG (G)

/ 1

π

ξG

(10.2.1) / 1

The top line in (10.2.1) is uniquely split over k (by Proposition 8.1.2), so we obtain a factorization ξ

j G ) → G ξG (G) R k/k (G

of the surjective π. The kernel of π is central (as it is contained in the central )), and ker ξG is unipotent. Hence, ker j is central subgroup R K/k (μ) of R K/k (G and j carries maximal tori onto maximal tori, so upon choosing compatible maximal tori it also carries root groups into root groups with trivial kernel. But π and ξG each restrict to isomorphisms between root groups, so it follows that j does too. In particular, since G is generated by its root groups relative to a maximal k-torus (due to perfectness of G; see Lemma 3.1.5), we conclude that j is surjective. ). In other words, the pseudo-reductive G is a central quotient of R k/k (G Such a central quotient presentation can be promoted to a generalized standard presentation, as was explained immediately following Remark 10.1.10. This completes the proof of Theorem 10.2.1(2).

11 Applications

This chapter gives some applications of Theorem 5.1.1 over general ﬁelds. The reader is referred to [Con2] for arithmetic applications of Theorem 5.1.1 (and Proposition 7.3.3) to general ﬁniteness theorems for all afﬁne group schemes of ﬁnite type over global function ﬁelds (over ﬁnite ﬁelds).

11.1 Maximal tori in pseudo-reductive groups Our concrete description of pseudo-reductive groups enables us to analyze the structure of such groups by means of maximal tori, as well as to analyze problems related to the structure of maximal tori. We begin with a toric criterion for a pseudo-reductive group to be reductive (away from characteristic 2). Theorem 11.1.1 Let k be a ﬁeld with char(k) = 2. A pseudo-reductive k-group G is reductive if and only if some (equivalently, every) Cartan k-subgroup is a torus. We do not know a direct proof of Theorem 11.1.1 using Tits’ structure theory for pseudo-reductive groups as in §§2.1–3.5. An interesting application of this result is given in Theorem C.1.9 (which is never used elsewhere in this monograph). Proof. It is a well-known fact that maximal tori in connected reductive groups over any ﬁeld are their own centralizers [Bo2, 13.17, Cor. 2(c)], which is to say that Cartan subgroups are tori in such groups. For the converse, we will use that char(k) = 2. Let T be a maximal k-torus of G and C = ZG (T ) the associated Cartan subgroup. Assume that C = T ; we will prove that G is reductive. We may

358

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replace k with ks so that k is separably closed. Let = (G, T ) be the root system of G with respect to T . For a ∈ , let Ga be the subgroup generated by the root groups U±a . This is an absolutely pseudo-simple k-subgroup of rank 1 normalized by T (see Proposition 3.4.1 for an alternative description of Ga ), and Sa := T ∩ Ga is the 1-dimensional maximal torus of Ga contained in T . The codimension-1 subtorus Ta = (ker a)0red in T centralizes Ga , so T is an almost direct product of Sa and Ta . In particular, the Cartan k-subgroup ZGa (Sa ) is equal to Ga ∩ ZG (T ) = Ga ∩ C = Ga ∩ T = Sa . We now compute ZGa (Sa ) in another way, using that char(k) = 2. By Theorem 6.1.1, Ga is isomorphic to R K/k (G ), where G is SL2 or PGL2 and K/k is a ﬁnite extension of ﬁelds. From this description over k = ks , the Cartan k-subgroup ZGa (Sa ) is R K/k (GL1 ). But we saw that ZGa (Sa ) = Sa GL1 , so comparison of dimensions forces K = k. Hence, Ga is reductive, so the root groups U±a are 1-dimensional. Thus, the equality = (G red , Tk ) (as k

char(k) = 2) implies that the smooth quotient mapping Gk G red restricts to k an isomorphism between Lie algebras (as we can see on separate Tk -weight spaces, using 1-dimensionality for the nontrivial weights and the equality C = T for the trivial weight). Hence, this quotient map is étale, so Gk is reductive and therefore so is G. 2 Example 11.1.2 The reductivity criterion in Theorem 11.1.1 is false over every imperfect ﬁeld k with characteristic 2, even for standard pseudo-reductive groups. Indeed, a counterexample is given by the non-reductive standard pseudo-reductive k-group G = R k /k (SL2 )/R k /k (μ2 ) for any purely inseparable quadratic extension k /k with char(k) = 2: a Cartan k-subgroup is given by C := R k /k (GL1 )/R k /k (μ2 ) for the diagonal k -torus GL1 ⊆ SL2 , yet the natural inclusion GL1 /μ2 ⊆ C over k is an equality for dimension reasons since [k : k] = 2. For an exotic pseudo-reductive k-group G = R K/k (G ) over an imperfect ﬁeld k of characteristic p ∈ {2, 3} (with G → Spec K having ﬁbers that are basic exotic pseudo-reductive groups), the following result parameterizes the maximal k-tori in G in terms of the data (G , K /K, G) used to construct G . The difﬁculty in the proof is due to the problem of lifting tori through very special isogenies. Proposition 11.1.3 With notation as above, there is a natural bijection between the set of maximal k-tori T in the k-group R K/k (G ) ⊆ R K /k (G ) and the set of maximal K -tori T ⊂ G whose image T ⊂ G under the very special

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Applications

isogeny π : G → G = G K is deﬁned over K relative to the Levi K-subgroup G ⊂ R K /K (G ). The bijection is deﬁned by the condition that T is the maximal k-torus in R K /k (T ), and T is the image of TK → G . The analogous such bijection holds between the set of maximal k-split tori in R K/k (G ) and the set of maximal K -split tori S in G whose image S ⊂ G under π is deﬁned over K relative to the Levi K-subgroup G ⊂ RK /K (G ). When k = ks , R K /k (G ) contains maximal k-tori which are not contained in the proper k-subgroup R K/k (G ). Thus, it is not possible to eliminate the constraint on T related to the K-structure G. Proof. The assertion for maximal split tori is immediate from the assertion for maximal tori, so we may assume that k is separably closed. By passing to factor ﬁelds we can also arrange that K is a ﬁeld; it is purely inseparable over k. Let T be a maximal k-torus in R K/k (G ). By Proposition A.5.15(2), T is the maximal k-torus in R K/k (T ) for a unique maximal K-torus T of G . Let G be a Levi K-subgroup of G containing T (Theorem 3.4.6). The image of TK under the maximal pseudo-reductive quotient map R K/k (G )K G is T . Also, the image T of TK under the maximal reductive quotient map GK G is a maximal K -torus in G . Hence, T is the image of TK under the maximal reductive quotient map R K/k (G )K G . It follows that T is a maximal k-torus of R K /k (T ). Since the very special isogeny G → G is a K-descent of the very special isogeny G → G , the image T of T in G descends to the image T of T in G. In particular, this shows that the maximal K -tori T ⊂ G arising from maximal k-tori T ⊂ R K/k (G ) must satisfy the asserted compatibility requirement with respect to the K-structure G on G . The only remaining problem is to show that every maximal K -torus T in G for which T = T K for a maximal K-torus T in G does arise from a maximal k-torus T of R K/k (G ). Since K and K are separably closed, the K -split K torus T uniquely descends to a K-split K-torus T , and the isogeny T → T between K -split K -tori uniquely descends to an isogeny T → T between K-split K-tori. Thus, we get a commutative diagram T

/ R K /K (T )

T

/ R (T ) K /K

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in which the bottom and right sides are induced by the bottom and right sides in the cartesian diagram G

/ R K /K (G )

G

/ R (G ) K /K

Thus, we get a K-homomorphism h : T → G whose composition with G → G is the K-isogeny T → T (so h(T ) is a maximal K-torus in G ) and whose composition with G → R K /K (G ) is the inclusion of T into R K /K (TK ) = R K /K (T ) (so ker h = 1). We conclude that T is a maximal K-torus in G . Since the inclusion G → R K /K (G ) corresponds to a K -group map GK → G that is a K -descent of the maximal geometric reductive quotient (e.g., it is surjective) this latter map carries TK onto T . Thus, if T is the maximal k-torus in R K/k (T ) ⊂ R K/k (G ) then TK → G has image T . 2 At the end of [Ti3, II] Tits raised the question, for an imperfect ﬁeld k, of characterizing those non-reductive absolutely pseudo-simple k-groups G for which the map iG in (1.6.1) is an isomorphism. He proved this isomorphism property whenever G ss is unique in its central isogeny class (i.e., it is simk ply connected and adjoint), which is to say of type E8 , F4 , or G2 , subject to some restrictions on the characteristic (avoiding characteristic 2 for type F4 , avoiding characteristic 3 for type G2 ). We can now prove a vast generalization in any characteristic, subject to a degree restriction in characteristic 2, as follows. Let k be an imperfect ﬁeld with characteristic p > 0, and let G be an absolutely pseudo-simple k-group that is not reductive. Let the nontrivial ﬁnite purely inseparable extension K/k be the ﬁeld of deﬁnition over k of the k-subgroup Ru (Gk ) = R(Gk ) in Gk , and let G = GK /R(GK ) be the K-descent of the maximal semisimple quotient of Gk . We seek necessary and sufﬁcient conditions that characterize when the canonical morphism iG : G → R K/k (G ) is an isomorphism. Note that this question does not apply to the non-reductive pseudo-reductive k-groups of the form R k /k (PGLp ) with k /k a nontrivial purely inseparable ﬁnite extension, since such k-groups are not even perfect (Example 1.3.5). The source of this phenomenon is that PGLp has fundamental group whose order is divisible by p. It turns out that if p > 3 then for any G as above, iG is an isomorphism if and only if the fundamental group of G ss has order not divisible by p. More generally: k

362

Applications

Proposition 11.1.4 With notation as above, if iG is an isomorphism then Gks has a reduced root system and G ss has fundamental group with order not divisible k by p. Assume conversely that Gks has a reduced root system and G ss has funk damental group with order not divisible by p, and if p = 2 then assume [k : k 2 ] = 2. (1) The map iG is an isomorphism except possibly when p ∈ {2, 3} and G ss is k simply connected with an edge of multiplicity p in its Dynkin diagram. (2) Assume p ∈ {2, 3} and that G ss is simply connected with an edge of mulk tiplicity p in its Dynkin diagram. The map iG is not an isomorphism onto R K/k (G ) precisely when it is an isomorphism onto R k/k (G ), with k K a proper subﬁeld over k containing K p and G ⊆ R K/k (G ) a basic exotic pseudo-reductive k-group associated to the simply connected K-group G and a choice of Levi k-subgroup of R K/k (G ), where G → G is the very special K-isogeny. Proof. If iG is an isomorphism then Gks has a reduced root system (Example 2.3.2). Thus, for the rest of the argument we may and do assume that this root system is reduced. G be the simply connected central cover of G , and let μ denote Let G the kernel of this covering map. Since G is absolutely pseudo-simple over k, the canonical map ξG as in (5.3.3) makes sense. By Proposition 1.3.4, the target )/R K/k (μ) of ξG is the derived group of R K/k (G ), and by deﬁnition R K/k (G the composition of ξG with the canonical inclusion )/R K/k (μ) → R K/k (G ) R K/k (G onto the derived group is iG . Hence, if iG is an isomorphism then this inclusion is an equality. By dimension and connectedness considerations, equality happens if and only if dim R K/k (μ) = 0. The ﬁnite extension K/k is nontrivial and purely inseparable, so R K/k (μ) has dimension > 0 whenever μ has nontrivial p-part (as we can check by working over ks and replacing K with a subﬁeld of degree p over k). Hence, iG can be an isomorphism only if μ has order not divisible by p. The Cartier dual of μk is the fundamental group of G , and absolutely simple groups of types E8 , F4 , k and G2 are both simply connected and adjoint whereas groups of types Bn and Cn in characteristic p = 2 have fundamental group of order not divisible by p precisely in the simply connected case. Thus, for the remainder of the argument we may restrict attention to the case where G ss has fundamental group with order not divisible by p. In particular, k

11.1 Maximal tori in pseudo-reductive groups

363

the ﬁnite multiplicative K-group μ is étale, so )/R K/k (μ) = R K/k (G /μ) = R K/k (G ). R K/k (G Hence, now iG is identiﬁed with ξG . Since Gks has a reduced root system and [k : k 2 ] = 2 when p = 2, by Theorem 6.1.1, Theorem 8.2.1, and Proposition 9.1.10 it follows from Theorem 6.3.4 that G is standard when G is not exotic. Thus, the ﬁnal claim in Theorem 5.3.8 ensures that the map iG = ξG is an isomorphism in such cases. In the exotic case, Theorem 8.2.1 gives that iG = ξG is an isomorphism onto the Weil restriction of a basic exotic pseudo-reductive group in the desired manner. 2 Remark 11.1.5 It is natural to ask when Weil restrictions of the sort arising in Proposition 11.1.4 actually satisfy the initial hypotheses on the k-group G there. (That is, when are such Weil restrictions non-reductive and absolutely pseudosimple?) This corresponds to a kind of converse to Proposition 11.1.4. To make it precise, ﬁx an imperfect ﬁeld k with characteristic p > 0 and consider a kgroup G := R K/k (G ) with K/k a nontrivial purely inseparable ﬁnite extension and G a connected semisimple K-group that is absolutely simple. The k-group G is pseudo-reductive over k but not reductive (by Example 1.6.1), and Gks has a reduced root system (Example 2.3.2). By Theorem 1.6.2(2), G ss G ⊗K k k and the ﬁeld of deﬁnition of R(Gk ) = Ru (Gk ) ⊆ Gk over k is equal to K. We claim that if the fundamental group of G has order not divisible by p then G necessarily is absolutely pseudo-simple over k. To see this, note that the → G is a ﬁnite étale cover, so the natural simply connected central cover G k-homomorphism ) → R K/k (G ) = G R K/k (G ) is a ﬁnite étale cover (Corollary A.5.4(2), Proposition A.5.13). Since R K/k (G is its own derived group (Corollary A.7.11), it follows that G is perfect and hence G is absolutely pseudo-simple over k by Lemma 3.1.2. In the non-standard direction, suppose p ∈ {2, 3} and consider a purely inseparable ﬁnite extension k/k, a basic exotic pseudo-reductive k-group G , and the k-group G := R k/k (G ). Once again, Gks has a reduced root system (Example 2.3.2). We claim that G is always absolutely pseudo-simple over k: this follows from Proposition 8.1.2 and Proposition 8.1.5 (due to the absolute pseudo-simplicity criterion in Lemma 3.1.2). Note that by Proposition 8.1.2, such k-groups G are not reductive.

364

Applications

11.2 Pseudo-semisimplicity It is natural to seek a theory of pseudo-semisimplicity, as a special case of pseudo-reductivity. Intuition derived from experience with connected semisimple groups can be misleading, and even ﬁnding an appropriate deﬁnition leads to some surprises. Consider a connected reductive group G over a ﬁeld k. The derived group D (G) is semisimple, and in particular D (G) is its own derived group. If Z denotes the maximal central k-torus in G (i.e., maximal central connected reductive k-subgroup of G) then the natural map D (G) × Z → G is an isogeny. For pseudo-reductive G it is generally not true that G is generated by its derived group and its maximal central pseudo-reductive k-subgroup. Here are some counterexamples that are standard pseudo-reductive groups: Example 11.2.1 For a ﬁeld k, consider a pseudo-reductive k-group G containing no nontrivial central k-tori. We claim that the k-radical Rk (G) is necessarily trivial (exactly as in the connected reductive case), but that over any imperfect ﬁeld k such G exist that are standard and have the additional property G = D (G) (which never happens in the connected reductive case). In particular, such G are not generated by D (G) and the maximal central pseudo-reductive k-subgroup. To prove Rk (G) = 1, ﬁrst observe that since Rk (G) is solvable and pseudoreductive (by normality in G), it is commutative. In particular, its maximal k-torus S is unique and thus is normal in G. But the automorphism functor of a k-torus is represented by an étale k-group, so the action on S by the connected G must be trivial. That is, the k-torus S is central in G, so by hypothesis S = 1. This forces Rk (G) = Ru,k (G) = 1. To make such G for which G = D (G), consider a non-commutative standard pseudo-reductive k-group G equipped with a standard presentation G (R k /k (G ) C)/R k /k (T ). The structure of maximal k-tori in G (as in Proposition 10.2.2(2)) implies that triviality of the maximal central k-torus of G is equivalent to the map φ : R k /k (T ) → C restricting to an isogeny between maximal k-tori. Moreover, D (G) is the image of R k /k (G ) in G, so G/D (G) = coker φ. Hence, we seek standard presentations (G , k /k, T , C) such that φ is not surjective but restricts to an isogeny between maximal k-tori. For example, consider C = R k /k (T /ZG ) and the canonical φ: the Weil restriction of the quotient map T → T /ZG . This is an isogeny between maximal k-tori, so we just need to arrange that φ is not surjective. For example, if char(k) = p > 0 and k /k is a nontrivial purely inseparable ﬁnite extension

11.2 Pseudo-semisimplicity

365

then we can take G = SLp (so coker φ is identiﬁed with the nontrivial cokernel of p-multiplication on R k /k (GL1 )). There are (at least) two ways to characterize when a connected reductive k-group G is semisimple: the triviality of the k-radical Rk (G) and the perfectness of G. In the pseudo-reductive case the smooth connected solvable normal k-subgroup Rk (G) is always commutative (by Proposition 1.2.3), but Example 11.2.1 shows that requiring Rk (G) = 1 does not generally imply that G = D (G) when k is imperfect. Thus, we are led to make the following deﬁnition. Deﬁnition 11.2.2 Let k be a ﬁeld. A k-group G is pseudo-semisimple if it is perfect and pseudo-reductive over k. By Proposition 3.4.9, a pseudo-semisimple k-group G is reductive if and only if all root groups of Gks are 1-dimensional. (There is no analogue in the general pseudo-reductive case since commutative pseudo-reductive groups do not admit root groups.) As with the case of semisimplicity for smooth afﬁne connected groups, any nontrivial pseudo-semisimple group is non-solvable. Likewise, pseudo-semisimplicity is insensitive to separable extension of the base ﬁeld. For any pseudo-semisimple k-group G we have Rk (G) = 1 because the perfectness of G forces the radical R(Gk ) to be unipotent (i.e., G red is k semisimple) and hence Rk (G) = 1 by Lemma 1.2.1. For generalized standard pseudo-reductive groups over an arbitrary ﬁeld k, there is a concrete characterization of pseudo-semisimplicity in terms of generalized standard presentations, as follows. By Deﬁnition 3.1.1, any pseudo-simple k-group is pseudo-semisimple, so if G is a non-commutative pseudo-reductive k-group admitting a generalized standard presentation (G , k /k, T , C), then by Proposition 10.2.2(1) the k-group G is pseudosemisimple if and only if the given map φ : R k /k (T ) → C is surjective. (For the “only if” implication, use that G maps onto the commutative coker φ.) By Proposition 10.1.4, if k is imperfect of characteristic 2 and [k : k 2 ] = 2 then all totally non-reduced pseudo-reductive k-groups are perfect. Since a product of k-groups is pseudo-semisimple if and only if each factor is pseudosemisimple, we have therefore described the pseudo-semisimple groups within the classiﬁcation of pseudo-reductive groups in Theorem 5.1.1. Remark 11.2.3 We may restate Proposition 1.2.6 as follows: if G is a pseudoreductive group over a ﬁeld k then D (G) is pseudo-semisimple.

366

Applications

We now use Theorem 5.1.1 to make some of our earlier work with pseudosemisimple groups more explicit. Proposition 3.1.8 describes the pseudosemisimple normal k-groups N of any non-commutative pseudo-reductive k-group G as follows: if {Ni } is the collection of such minimal nontrivial N then each N is generated by a collection of some of the Ni . It is natural to ask for an explicit description of each Ni (and hence of every N ) when G is given in an explicit form as in our structure theorem (Theorem 5.1.1) for pseudo-reductive groups. The following discussion addresses this matter. Let G be a non-commutative pseudo-reductive group over a ﬁeld k, and assume [k : k 2 ] = 2 if k is imperfect with characteristic 2. In view of the splitting off of the non-reduced part of the root system in characteristic 2, and the general structure of totally non-reduced pseudo-reductive groups in such cases (see Propositions 10.1.4 and 10.1.6), any Ni that is totally non-reduced uniquely splits off as a direct factor of any N that meets it nontrivially. Thus, we may and do now assume that Gks has a reduced root system. By Theorem 5.1.1, G is generalized standard. For a generalized standard group G over any ﬁeld k whatsoever (i.e., no restriction on [k : k 2 ] when char(k) = 2), we shall determine the minimal such nontrivial N in terms of a generalized standard presentation of G, and then directly compute N and G/N in all cases. In particular, we will gain some insight into the phenomenon already seen in Example 1.6.4 whereby G/N can fail to be pseudo-reductive even when G and N are pseudo-semisimple. Consider a generalized standard presentation G = (R k /k (G ) C)/R k /k (ZG (T )) using a suitable k-group map φ : R k /k (ZG (T )) → C. Here, C is a Cartan k-subgroup of G. Consider the decomposition k = i∈I ki into a product of ﬁelds and the associated decompositions G =

Gi , T =

Ti

into a disjoint union of ﬁbers over Spec k . For each subset J ⊆ I deﬁne kJ =

j∈J

kj , GJ =

j∈J

Gj , TJ =

Tj .

j∈J

The Weil restriction R kJ /k (ZGJ (TJ )) is a direct factor of R k /k (ZG (T )); let CJ denote its image in C under φ. Note that CJ is pseudo-reductive since it is a smooth connected k-subgroup of the commutative pseudo-reductive k-group C.

11.2 Pseudo-semisimplicity

367

We therefore get a generalized standard pseudo-reductive normal k-subgroup GJ := (R kJ /k (GJ ) CJ )/R kJ /k (ZGJ (TJ )) ⊆ G that is pseudo-semisimple (due to the deﬁnition of CJ ). Clearly the G{i} pairwise commute and the G{ j} for j ∈ J generate GJ . Proposition 11.2.4 With notation and hypotheses as above, every perfect smooth connected normal k-subgroup N of G has the form N = GJ for a unique subset J ⊆ I . Proof. By Proposition 3.1.8, it sufﬁces to prove that the G{i} are the minimal nontrivial perfect smooth connected normal k-subgroups N of G (in which case J consists of exactly those i such that G{i} ⊆ N ). If we can prove this minimality property in the case of a separably closed base ﬁeld then it follows in general by Galois descent considerations. Hence, to prove N = GJ for some J we now may and do assume k = ks . By Proposition 3.1.6, it sufﬁces to show that as we vary J , the image GJ of (GJ )k in G red varies without repetition through the perfect smooth normal k

connected subgroups of G red . It is equivalent to verify the analogous assertion k

using G ss in place of G red , since a connected reductive k-group and its maximal k k semisimple quotient admit an evident natural correspondence between their sets of perfect smooth connected normal subgroups. For the same reason we can even pass to the maximal adjoint semisimple quotient G ss /ZGss . This adjoint k k group is the direct product of its simple factors, and (by using the generalized standard presentation of G) the simple factors are clearly the maximal adjoint semisimple quotients of the (Gi )k . 2 For N = GJ , the quotient G/N = G/GJ is k-isomorphic to the k-group (R kI −J /k (GI −J ) (C/CJ ))/R kI −J /k (ZGI −J (TI−J )),

(11.2.1)

so this is pseudo-reductive if C/CJ is pseudo-reductive. More importantly, the converse is true: if G/GJ is pseudo-reductive then its k-subgroup C/CJ must be pseudo-reductive. The reason for this is that C/CJ is the image of the Cartan ksubgroup C ⊆ G, so it is a Cartan k-subgroup of G/GJ , and Cartan k-subgroups of pseudo-reductive k-groups are pseudo-reductive (Proposition 1.2.4). Consider the special case that G is pseudo-semisimple (i.e., G is perfect), which is to say that the natural map φ : R k /k (ZG (T )) → C is surjective. Let Z := ker φ and deﬁne the projection map πI −J : R k /k (ZG (T )) → R kI −J /k (ZGI −J (TI−J )).

368

Applications

We are given that the Cartan k-subgroup R k /k (ZG (T ))/Z in G is pseudoreductive over k, and we have just seen that the quotient G/GJ is pseudoreductive if and only if its Cartan k-subgroup R kI −J /k (ZGI −J (TI−J ))/πI −J (Z)

(11.2.2)

is pseudo-reductive. It can happen for suitable triples (G , k /k, T ) and k-subgroups Z ⊆ R k /k (ZG (T )) contained in R k /k (ZG ) that (11.2.2) is not pseudo-reductive over k for some non-empty J even though R k /k (ZG (T ))/Z is pseudo-reductive over k! This is the essential content of the standard examples given in Example 1.6.4 (corresponding to the case I = {1, 2}, k1 = k2 , G1 = G2 , T1 = T2 , and J = {1}).

11.3 Unirationality It is a well-known and very useful fact [Bo2, 18.2(ii)] that a connected reductive group over a ﬁeld is unirational, and in particular has a Zariski-dense locus of rational points when the ground ﬁeld is inﬁnite. This ultimately rests on the unirationality of tori over any ﬁeld, a result which has no analogue for general commutative pseudo-reductive groups: Example 11.3.1 Over every imperfect ﬁeld k there are commutative pseudoreductive k-groups that are not unirational over k, and in case k is a rational function ﬁeld k0 (v) with char(k0 ) > 0 we can always ﬁnd such examples whose locus of k-rational points is not Zariski-dense. To make such examples, let k be imperfect with char(k) = p > 0 and choose a p-power q = pr > 1 and any t ∈ k not in k p . (The case q > p will be of most interest for p = 2.) Consider the smooth k-group U = {yq = x − tx p } in Ga × Ga . This becomes isomorphic to Ga over k = k(t ) with t = t 1/q . Indeed, if we deﬁne x0 = x and xj = y p

r−j

+ t

pr−j

xj−1 r−j

for 1 j r then by induction on j we ﬁnd y p = xj − t p xj since Uk is reduced. Thus, for z = xr we have y = z − t z p ∈ k [z]. r−j r−j Since xj−1 = t −p (xj − y p ) for 1 j r, by descending induction on j we also have x ∈ k [z]. It follows that Uk Ga as k -groups via the coordinate z. By [Ru, Cor. 2.3.1], for an extension K/k we have UK Ga as K-groups if and only if K contains k over k. r−j

p

11.3 Unirationality

369

We claim that if q > 2 then the smooth geometrically connected k-scheme U of dimension 1 is not unirational over k. Suppose there were a dominant k-morphism P → U with P open in an afﬁne space over k. By choosing P of minimal dimension we see via consideration of afﬁne k-lines meeting P that P must be 1-dimensional, so k(U ) is k-rational by Lüroth’s theorem over k. But by [KMT, 6.9.2] such k-rationality can only happen if p = 2 and U {w2 = v − cv 2 } as k-schemes for some non-square c in k, in which √ √ case UK AK1 over K = k( c) via the coordinate z = w + c · v (z 2 = v, √ z − c · z 2 = w). Since Ga is the unique group scheme structure on the afﬁne line over a ﬁeld with identity at the origin, in such a situation UK Ga as K-groups and hence K contains k over k, so q = [k : k] [K : k] = 2. Before we use such U to make non-unirational pseudo-reductive groups over k, we brieﬂy digress to consider a special case in which U (k) is not Zariski-dense in U . Example 11.3.2 Suppose k = k0 (t) for a subﬁeld k0 over which t is transcendental, and q > 2. Then U (k) = {(0, 0)} if p = 2 and U (k) = {(0, 0), (1/t, 0)} if p = 2, so U (k) is not even Zariski-dense in U . Indeed, we can assume k0 is algebraically closed, and if (x0 , y0 ) ∈ U (k) then t-differentiation gives p x0 = x0 := dx0 /dt. Thus, if x0 = 0 then (x0 , y0 ) = (0, 0), so we can assume x0 = 0. Hence, ea := ord a (x0 ) is not divisible by p for some a ∈ k0 , so the p equality ord a (x0 ) = ord a (x0 ) gives ea − 1 = pea . Thus, p = 2 and ea = −1, so x0 = h20 /f0 for some separable monic non-constant f0 ∈ k0 [t] and some h0 ∈ k0 (t)× with divisor on Spec k0 [t] disjoint from div( f0 ). The identity q

y0 = x0 − tx02 =

h20 ( f0 − th20 ) f02

q

then gives that y0 has a pole of order 2 at the zeros of f0 away from t = 0. Since q > 2 this forces f0 = t, so h0 = 1 any hence (x0 , y0 ) = (1/t, 0). Continuing with Example 11.3.1, consider a k-group C representing a class ξ ∈ Ext 1k (U , GL1 ) (where we compute the Ext-group in the category of commutative afﬁne k-group schemes of ﬁnite type). Obviously such a C is smooth and connected. If q > 2 then such a C is not unirational over k since its quotient U is not (and likewise if k = k0 (t) with t transcendental over a subﬁeld k0 then the set C(k) is not Zariski-dense in C when q > 2). The induced map h : Ru,k (C) → U has kernel Ru,k (C) ∩ GL1 that is both unipotent and multiplicative type, hence trivial (Example A.1.7). Thus, if Ru,k (C) = 1 then h is an isomorphism for dimension reasons (forcing ξ = 0). Hence, C is pseudo-reductive if ξ = 0, so to give examples of non-unirational

370

Applications

commutative pseudo-reductive k-groups it sufﬁces to prove Ext 1k (U , GL1 ) = 0 for all q > 1 (allowing q = 2). This will work out nicely for p = 2, but we will have to choose t ∈ k carefully if p > 2. Let U = {y p = x − tx p }, so the map (x, y) → (x, yq/p ) deﬁnes an exact sequence of k-groups 0 → αq/p → U → U → 0. The pullback map Ext 1k (U , GL1 ) → Ext 1k (U , GL1 ) is injective because Homk (αq/p , GL1 ) = 1. Thus, it sufﬁces to prove Ext 1k (U , GL1 ) = 0. According to [KMT, 6.13.3] (whose proof is explicit and works even if p = 2 and k is not separably closed, contrary to what is suggested by the statement there), this Ext-group is isomorphic (functorially in k/Fp (t)) to the abelian group of (p − 1)-tuples (c0 , . . . , cp−2 ) ∈ k p−1 such that cp−2 =

p

cj t j .

(11.3.1)

0jp−2

We seek a nonzero solution to this equation in k for some choice of t. (Note that if p = 3 then this equation is c1 = c03 + tc13 , which we have seen in Example 11.3.2 has no nonzero solution if k = k0 (t) with t transcendental over a subﬁeld k0 . Hence, in general we will not be able to use an arbitrary t ∈ k.) Set cj = 0 for all j < p − 2, so (11.3.1) with cp−2 = 0 collapses to p−1 cp−2 = 1/t p−2 . (For p = 2 the original equation is c0 = c02 , so there is exactly one nonzero solution for each t. By [SGA3, XVII, 6.4(a)], this distinguished nontrivial extension corresponds to the k-group R k(√t)/k (GL1 ).) Thus, if at the start we choose v ∈ k × that is not in k p and deﬁne t = v p−1 then t is not in k p and we can set cp−2 = 1/v p−2 ∈ k × . To make examples in which the locus of k-points is not Zariski-dense, consider k = k0 (v) a rational function ﬁeld and choose t = v p−1 . It sufﬁces to show that if q > 2 then U (k) is {(0, 0), (ζ /v, 0)|ζ p−1 = 1}. For p = 2 this is the settled case t = v from Example 11.3.2, so we can assume p > 2. For (x0 , y0 ) ∈ U (k) differentiating the equation q

p

y0 = x0 − v p−1 x0

gives x0 = −v p−2 x0 , so if x0 = 0 then (x0 , y0 ) = (0, 0). p

11.3 Unirationality

371

We may assume k0 is algebraically closed and x0 = 0, so there exists some a ∈ k0 such that ea := ord a (x0 ) is not divisible by p. If a = 0 then ea − 1 = ord a (x0 ) = ord a (−v p−2 x0 ) = pea , p

which is impossible since p > 2. Hence, necessarily a = 0 and e0 − 1 = p p − 2 + pe0 , so e0 = −1. That is, x0 = x1 /v where x1 ∈ k0 (v)× has neither a zero nor a pole at 0. Then p

q

p

y0 = x0 − v p−1 x0 =

p(p−1)

x1 (1 − x1 v

)

p−1

=

(x1 (1 − x1 v

)) p

in K(v), so y0 = 0 and x1 = ζ ∈ μp−1 (k0 ). Hence, x0 = ζ /v. We can push Example 11.3.1 a bit further: Example 11.3.3 Let k be a (possibly archimedean) local ﬁeld. It is a wellknown fact [Ser3, III, §4.3] that H1 (k, G) is ﬁnite for any connected reductive k-group G. It is therefore natural to wonder if such ﬁniteness holds for pseudoreductive groups. This is false in the commutative case over every local function ﬁeld k. Indeed, choose an isomorphism k F((v)) for a ﬁnite ﬁeld F with characteristic p > 0, and let U ⊆ Ga × Ga be the k-group {yq = x − v p−1 x p } with a p-power q = pr > 2. In Example 11.3.1 we constructed a non-split commutative extension C of U by GL1 over k and showed that C is pseudo-reductive (and non-unirational) over k. We claim that H1 (k, C) is inﬁnite. Since U is p-torsion, the long exact Galois cohomology sequence associated to the extension structure on C provides an exact sequence 1 → H1 (k, C) → H1 (k, U ) → Br(k)[p]. But Br(k) Q/Z, so it sufﬁces to prove that H1 (k, U ) is inﬁnite. The exact sequence 1 → U → Ga × Ga → Ga → 1 (with quotient map deﬁned by (x, y) → yq − (x − v p−1 x p )) yields an isomorphism H1 (k, U ) k/{yq − (x − v p−1 x p ) | x, y ∈ k}, so we have to show that this quotient of k is inﬁnite. Consideration of poleorders gives this when p > 2 since expressions of the form yq − (x − v p−1 x p )

372

Applications

with negative order only have orders that are 0 or −1 mod p. When p = 2 we have 4|q (since q > 2 by hypothesis), so a simple variant of this argument works by considering elements of k with negative order that is 2 mod 4. By Example 11.3.1, a unirationality result for pseudo-reductive groups must avoid the commutative non-toric case. Likewise, by Example 11.3.3, a ﬁniteness result for H1 (k, G) for pseudo-reductive G over local function ﬁelds k must avoid the commutative non-toric case. By Remark 11.2.3, it is natural to consider the pseudo-semisimple case for both problems. Since pseudo-semisimple groups are perfect, their unirationality is a special case of Proposition A.2.11. For pseudo-semisimple groups G over local function ﬁelds k the ﬁniteness of H1 (k, G) requires arguments that are speciﬁc to the arithmetic case, so we refer the reader to [Con2] for the details on this result.

11.4 Structure of root groups and pseudo-parabolic subgroups We saw in Proposition 2.2.9 that a pseudo-parabolic k-subgroup of a connected reductive k-group G is the same thing as a parabolic k-subgroup of G, and hence for any such subgroup P, G/P is proper over k . If G is a general smooth connected afﬁne k-group then a natural generalization is to inquire about the structure of G/P for pseudo-parabolic k-subgroups P of G. There is no loss of generality in restricting attention to the case of pseudo-reductive G, due to Proposition 2.2.10, and this is the case of most interest to us anyway. Now let G be a pseudo-reductive group over a ﬁeld k, and let P be a pseudoparabolic k-subgroup. In general G/P can fail to be proper, as we saw explicitly in Example 2.1.3. It seems difﬁcult to understand this failure of properness directly from the deﬁnitions. As an application of the theory of root groups in pseudo-reductive groups over a separably closed ﬁeld, we saw in Proposition 3.4.9 that if G = D (G) then G fails to be reductive precisely when Gks admits a root group with dimension larger than 1 and that in such cases G/P fails to be k-proper whenever P does not contain any nontrivial perfect smooth connected normal k-subgroups of G (i.e., P is not “too big”). In this section we shall use our main structure theorem (Theorem 5.1.1) to give a more explicit link from root groups and pseudo-parabolic k-subgroups in G to root groups and parabolic k-subgroups in connected reductive groups over ﬁnite extensions of k, assuming [k : k 2 ] 2 when char(k) = 2. This will clarify the structure of coset spaces G/P for pseudo-reductive G and pseudoparabolic subgroups P over an arbitrary ﬁeld k (assuming [k : k 2 ] 2 when char(k) = 2).

11.4 Structure of root groups and pseudo-parabolic subgroups

373

We begin with a result that provides a useful substitute in the pseudo-reductive case for the 1-dimensionality of root groups in the connected reductive case. Proposition 11.4.1 Let G be a pseudo-split pseudo-reductive group over a ﬁeld k, and T a split maximal k-torus in G. If char(k) = 2 then assume [k : k 2 ] 2. For each non-multipliable a ∈ (G, T ) there is a T -equivariant isomorphism Ua R ka /k (Ga ) for a purely inseparable ﬁnite extension ﬁeld ka /k, with Tka acting on Ga through aka and the quotient map (Ua )ka (R ka /k (Ga ))ka Ga that is a ka -descent of the quotient map from (Ua )kp onto the a-root . group of Gkred p If char(k) = 2 then the ZG (T )-action on Ua is irreducible in the sense that it preserves no proper nontrivial smooth k-subgroup of Ua . We remind the reader that Ua is normalized by ZG (T ) due to the intrinsic characterization of Ua in terms of the T -action on G; see Remark 2.3.5 (with U(a) = Ua when a is not divisible). Also, the avoidance of characteristic 2 for the irreducibility claim is essential. For example, if K/k is a purely inseparable quadratic extension in characteristic 2 then for G = R K/k (SL2 ) the action of ZG (T ) on Ua is the action of R K/k (GL1 ) on R K/k (Ga ) through scaling by squaring. But the squaring map on R K/k (GL1 ) is valued in the k-torus GL1 , and R K/k (Ga ) Ga2 is not irreducible for its usual GL1 -action. Proof. By Theorem 2.3.10, since a is not multipliable it lies in the subset (G red , Tk ). The structure of the k-subgroup Ga generated by Ua and k

U−a is given in Proposition 3.4.1, and T is generated by (ker a)0red and the 1-dimensional torus Ga ∩ T , so replacing G with Ga does not affect Ua and brings us to the case where G is absolutely pseudo-simple of rank 1 with a reduced root system {±a} (since we assumed at the outset that a is not multipliable). Thus, we may apply Theorem 6.1.1 away from characteristic 2 and Proposition 9.1.10 in characteristic 2 to conclude that G R K/k (SL2 )/Z for a purely inseparable ﬁnite extension K/k and a k-subgroup Z ⊆ R K/k (μ2 ) such that R K/k (GL1 )/Z is pseudo-reductive. Passage to central quotients does not affect the formation of root groups (Proposition 2.3.15), and any quotient map carries Cartan subgroups onto Cartan subgroups (Proposition A.2.8), so it sufﬁces to treat the case G = R K/k (SL2 ). This establishes the desired description of root groups, and reduces the irreducibility of the ZG (T )-action on Ua (for char(k) = 2) to the irreducibility of the action of R K/k (GL1 ) on R K/k (Ga ) through (t, x) → t 2 x. The squaring map on R K/k (GL1 ) is surjective (hence faithfully ﬂat) when char(k) = 2, so it sufﬁces to consider the action (t, x) → tx instead. Suppose H ⊆ R K/k (Ga ) is a smooth k-subgroup stable under the action of R K/k (GL1 ).

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Then, Lie(H ) is an R K/k (GL1 )-stable k-linear subspace of K. Equivalently, it is a K × -stable k-linear subspace of K, so it is either 0 or K. If Lie(H ) = 0 then H is étale, and so it cannot be stable under the action of R K/k (GL1 ) unless it is trivial. If Lie(H ) = K then the inclusion H ⊆ R K/k (Ga ) is an equality because H is smooth and R K/k (Ga ) is smooth and connected. This completes the veriﬁcation of ZG (T )-irreducibility for root groups. 2 Remark 11.4.2 The preceding method of proof also makes root groups explicit in terms of the parameters of a generalized standard presentation, assuming [k : k 2 ] = 2 in characteristic 2, as follows. Consider a generalized standard presentation G (R k /k (G ) C)/R k /k (ZG (T )) compatible with a k-split maximal k-torus T in G, so T is k -split. The behavior of root groups under central quotients (see Proposition 2.3.15) identiﬁes each root group for (G, T ) with a root group for (R ki /k (Gi ), Ti ) for a unique i, where {ki } is the set of factor ﬁelds of k and (Gi , Ti ) is the ki -ﬁber of (G, T ). Moreover, the root systems (Gi , Ti ) are identiﬁed with the irreducible components of (G, T ). The corresponding root groups are R ki /k (Ua i ) for ai ∈ (Gi , Ti ), where Ua i ranges through the root groups of (Gi , Ti ). When Gi is semisimple, Ua i is ki -isomorphic to Ga . In the basic exotic case, so k is imperfect of characteristic p ∈ {2, 3}, we have Ua i Ga over ki for long ai , whereas Ua i R ki /ki (Ga ) for a nontrivial ﬁnite extension ki /ki satisfying ki ⊆ (ki )1/p when ai is short. In the case of totally non-reduced pseudo-reductive groups with [k : k 2 ] = 2 we can again describe the root groups explicitly, and it comes out rather similarly to the exotic case. This is most easily seen by inspection of the open Bruhat cell for such k-groups, and we leave it to the interested reader. (For multipliable roots, the root group is not the Weil restriction of Ga from a single extension ﬁeld, but rather from a product of two extension ﬁelds.) Now we turn our attention to the structure of pseduo-parabolic k-subgroups P of G and the coset spaces G/P. In contrast with the preceding considerations for root groups, we no longer assume G to be pseudo-split. We ﬁx a maximal k-torus T and seek to describe the pseudo-parabolic k-subgroups P of G containing T in concrete terms so that we can understand the structure of G/P, assuming [k : k 2 ] 2 when char(k) = 2. Recall from Lemma 1.2.5(iii) that there is a bijective correspondence between the sets of maximal k-tori in G and D (G) via T → T ∩ D (G) and S → Z · S with Z the maximal central k-torus in G (and Z · S an almost direct product). We claim

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that an analogous procedure which works for pseudo-parabolic k-subgroups is P → P ∩ D (G) and Q → NG (Q), with the natural map D (G)/Q → G/P an isomorphism. In one direction, choose a pseudo-parabolic k-subgroup P of G and write P = PG (λ). For a maximal k-torus T of P containing λ(GL1 ), by Remark 2.1.7 we may replace λ with λn for a sufﬁciently divisible n > 0 depending on the degree of the isogeny Z × S → T (S = T ∩ D (G)) so that we obtain the alternative description P = PG (μ) for some μ : GL1 → S. Hence, Q := P ∩ D (G) = PD (G) (μ) is indeed a pseudo-parabolic k-subgroup of D (G). Since ZG (T ) maps onto G/D (G) and clearly ZG (T ) ⊆ ZG (λ) ⊆ PG (λ) = P, the natural map of coset spaces D (G)/Q → G/P is an isomorphism. It is clear that P normalizes Q, and the resulting containment P ⊆ NG (Q) forces NG (Q) to be a pseudo-parabolic k-subgroup of G (Proposition 3.5.8). Thus, to prove that P = NG (Q) it sufﬁces to show that the k-groups Pk and NG (Q)k = NGk (Qk ) have the same image in G = G red (Proposition 3.5.4). Letting T denote the k (isomorphic) image of Tk in G , the image of ZG (T )k in G is T . But ZG (T ) normalizes Q and the resulting k-group ZG (T ) · Q coincides with P, so the image P of Pk in G is T · Q , where Q is the parabolic image of Qk in D (G ) = D (G)red . Hence, P = Z · Q where Z is the maximal central torus k in G , so NG (Q ) ⊆ NG (P ) = P . But NG (Q ) contains the image of NGk (Qk ), so we get the desired equality. In the reverse direction, if Q is a pseudo-parabolic k-subgroup of D (G) then we claim that P := NG (Q) is a pseudo-parabolic k-subgroup of G and Q = P ∩ D (G). Writing Q = PD (G) (μ) for a 1-parameter k-subgroup μ of D (G), since PG (μ) ∩ D (G) = Q we have PG (μ) ⊆ P. Hence, P is pseudoparabolic in G (Proposition 3.5.8), and if T := Z · S for a maximal k-torus S in Q containing μ(GL1 ) then ZG (T ) normalizes Q and the resulting k-group ZG (T ) · Q is contained in PG (μ). But it is clear that the image P of Pk in G coincides with the image of (ZG (T )Q)k , so the containments ZG (T )Q ⊆ PG (μ) ⊆ P are equalities (Proposition 3.5.4). The preceding proves that P → P ∩ D (G) and Q → NG (Q) are inverse bijective correspondences between the set of pseudo-parabolic k-subgroups P of G and the set of pseudo-parabolic k-subgroups Q of D (G). Moreover, it shows (working over ks ) that if T is a maximal k-torus of G and S = T ∩ D (G) is the corresponding maximal k-torus of D (G) then P contains T if and only if Q contains S. Thus, for the purpose of concretely describing P and G/P for all P containing T we lose no generality by replacing G with D (G); i.e., we can assume G is perfect (and G = 1). We will address two cases: the nontrivial perfect generalized standard case over an arbitrary ground ﬁeld k, and the totally non-reduced case when

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char(k) = 2 with [k : k 2 ] = 2. By our main result (Theorem 5.1.1), these two cases cover all nontrivial perfect pseudo-reductive groups over any ﬁeld k, provided we require [k : k 2 ] 2 when char(k) = 2. (We have used the elementary fact that a pseudo-parabolic k-subgroup in a direct product G1 × G2 is precisely P1 × P2 for pseudo-parabolic k-subgroups Pi ⊆ Gi .) Proposition 11.4.3 Let G be a nontrivial perfect generalized standard pseudoreductive group over a ﬁeld k, and T a maximal k-torus in G. Consider the generalized standard presentation G (R k /k (G ) C)/R k /k (ZG (T )) compatible with T , where T is the unique maximal k -torus in G such that T contains the maximal k-torus of j(R k /k (T )), where j : R k /k (G ) → G is the canonical map. The pseudo-parabolic k-subgroups P of G containing T are in natural bijection with the ﬁberwise pseudo-parabolic k -subgroups P ⊆ G containing T via the recipes P = (R k /k (P ) C)/R k /k (ZG (T )),

(11.4.1)

P /ZG = im(Pk → G /ZG ), using the natural quotient map Gk G /ZG from Remark 10.1.11. In particular, G/P = R k /k (G /P ). This result reduces the problem of describing P and G/P to the case of (the ﬁbers of) G . For the standard ﬁbers of G → Spec k we get parabolic subgroups containing a ﬁxed maximal torus, and these are parameterized in terms of root systems in the pseudo-split case. Since such standard ﬁbers are descents of simple factors of the simply connected central cover of D (G red ) = D (G)ss , k k Proposition 11.4.3 thereby provides a concrete procedure reversing the one given in Proposition 3.5.4, at least in the standard case. Note also that the construction of P from P via the second procedure in (11.4.1) deﬁnes a bijection between the set of pseudo-parabolic k-subgroups of G and the set of ﬁberwise pseudo-parabolic k -subgroups of G without mentioning T and T ; we will revisit this in Remark 11.4.4. Proof. According to Proposition 3.5.2, pseudo-parabolicity of a subgroup is insensitive to separable extension of the ground ﬁeld. Thus, since the formation of the constructions in (11.4.1) is compatible with separable extension on k, by Galois descent we may assume that k is separably closed. In particular, the characterization of possibilities for P in terms of root spaces in Proposition 3.5.1 is available.

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If T is the maximal k-torus in R k /k (T ) then T = j(T ). But PG (λ) = PG (λn ) for any n 1 (Remark 2.1.7), so by replacing λ with a suitable power we may assume that it arises from a 1-parameter k-subgroup of T . In view of the compatibility of PG (λ) with respect to general quotient maps (Corollary 2.1.9), our problem is reduced to the special case G = R k /k (G ) (since the expression for P /ZG in (11.4.1) follows from Proposition 3.5.4). Moreover, the bijectivity in Proposition 3.5.4 reduces the problem in this special case to verifying that each R k /k (P ) actually is a pseudo-parabolic k-subgroup of G (as it certainly contains T ). Let {ki } be the set of factor ﬁelds of k and (Gi , Ti ) the ﬁber of (G, T ) over ki . A ﬁberwise pseudo-parabolic k -subgroup P in G containing T is a k subgroup Pi such that Pi is a pseudo-parabolic ki -subgroup of Gi containing Ti . That is, P = PG (λ ) for a 1-parameter k -subgroup λ : GL1 → G such that T ⊆ P . We may choose λ to be valued in T , by Proposition 2.2.11. In other words, P represents the functor assigning to each k -algebra R the subgroup PG (λ )(R ) = {g ∈ G (R ) | lim λ (t)gλ (t)−1 exists in G (R )}. t→0

This limit process is deﬁned in the evident manner (using R -maps AR1 → GR ). For any k -homomorphism λ : GL1 → T over k , we deﬁne the 1-parameter k-subgroup R k /k (λ )

φ

λ : GL1 → R k /k (GL1 ) −→ R k /k (T ) −→ C → G. This lands inside of the maximal k-torus in C, which is T , so PG (λ) is a pseudoparabolic k-subgroup of G containing T . It remains to check that PG (λ) = R k /k (PG (λ )),

(11.4.2)

so the right side really is pseudo-parabolic in G. Since both sides of (11.4.2) are smooth k-subgroups of G, to establish an inclusion in either direction in (11.4.2) it sufﬁces to work with k-points because k = ks . (In what follows we could work with R-points for any k-algebra R. We restrict to the sufﬁcient case R = k only to keep the notation simple.) The group of k-points on the right side of (11.4.2) is P (k ), and we ﬁrst check that such points lie in PG (λ)(k). That is, for g ∈ P (k ) we claim that the k-map GL1 → G deﬁned by t → λ(t)g λ(t)−1

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extends to a k-map Ak1 → G. Since g ∈ P (k ) = PG (λ )(k ), the k -map (GL1 )k → G deﬁned by t → λ (t )g λ (t )−1 extends to a k -map Ak1 → G . Applying R k /k to this latter map yields a composite map j

Ak1 → R k /k (Ak1 ) → R k /k (G ) → G whose restriction to GL1 is t → λ(t)g λ(t)−1 because the diagram R k /k (T ) R

k /k

(G )

φ

j

/ C / G

commutes. Thus, we have built the required k-map extension to Ak1 , so P (k ) ⊆ P(k). This completes the proof that the right side of (11.4.2) is contained in the left side (as k-subgroups of G). To establish the reverse inclusion in (11.4.2), by smoothness and connectedness it sufﬁces to check equality of Lie algebras inside of g = g where g = Lie(G) and g = Lie(G ) = gi (with gi = Lie(Gi )). The Lie algebra of PG (λ) is the span gλ0 of the weight spaces ga for a in the set λ0 ∪ {0} = {a ∈ (G, T ) ∪ {0} | a, λ 0}. Likewise, PG (λ ) has Lie algebra over k = ki that is the span gλ 0 of the weight spaces (gi )ai for all i ∈ I and ai ∈ (Gi , Ti ) ∪ {0} such that ai , λ 0. ?

Hence, we want to prove gλ0 = gλ 0 inside of g. (Note that the right side makes sense since λ is valued in T .) Due to the inclusion already proved for (11.4.2), the problem is to prove that the left side is contained in the right side. In other words, for each a ∈ (G, T ) ∪ {0} such that a, λ 0 we claim that ga is contained in gλ 0 . In case a = 0 we have ga = Lie(ZG (T )) = Lie(ZG (T )) = g0 since ZG (T ) = R k /k (ZG (T )). Hence, it sufﬁces to prove that if a ∈ λ0 then ga is inside of gλ 0 . This follows from the identiﬁcation (G, T ) = (Gi , Ti ) from Corollary 4.1.6. More precisely, by that corollary a is identiﬁed with an element ai ∈ (Gi , Ti ) for a unique i ∈ I , and ga = (gi )ai inside of g = g . Since a, λ = ai , λ , we are done. 2

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Remark 11.4.4 The proof of Proposition 11.4.3 used rather little about the properties of G . In fact, by setting C = R k /k (ZG (T )) the argument works verbatim to classify the pseudo-parabolic k-subgroups in G = R k /k (G ) for any ﬁeld k, nonzero ﬁnite reduced k-algebra k , and smooth afﬁne k -group G with pseudo-reductive ﬁbers (so G is pseudo-reductive over k). For example, by Proposition 10.1.4 this applies with G a totally non-reduced pseudo-reductive group over any imperfect ﬁeld k of characteristic 2 such that [k : k 2 ] = 2. To be precise, we have a bijection between the set of pseudo-parabolic ksubgroups P ⊆ G and the set of pseudo-parabolic k-subgroups P ⊆ G , deﬁned via P = R k /k (P ) and P = im(Pk → G ). (The pseudo-reductivity hypothesis on the ﬁbers of G can be relaxed to connectedness, since reduction to the pseudo-reductive case is elementary.) It also follows from the explicit formulas relating P and P that if T is a maximal k-torus of G and T is the corresponding maximal k -torus of G then T ⊆ P if and only if T ⊆ P . It remains to describe pseudo-parabolic subgroups containing a given maximal k-torus in a basic exotic pseudo-reductive k-group (with k imperfect of characteristic p ∈ {2, 3}) and in a basic non-reduced pseudo-simple group over an imperfect ﬁeld k of characteristic 2 (assuming [k : k 2 ] = 2). The basic non-reduced case is easier (in both statement and proof), so we begin with that case. Proposition 11.4.5 Let k be an imperfect ﬁeld of characteristic 2 such that [k : k 2 ] = 2, and let K = k 1/2 . Choose α ∈ K − k and n 1, and let G = Gα,n be the associated basic non-reduced pseudo-simple k-group. Let G = GKss Sp2n . Under the non-central quotient map ξG : G → R K/k (G ), there is a unique bijection between the set of pseudo-parabolic k-subgroups P of G and the set of parabolic K-subgroups P of G via either of the properties ξG (P) = R K/k (P ),

P = im(PK → G ).

Moreover, P(k) = G(k) ∩ P (K) via ξG : G(k) G (K), and P(k) is Zariskidense in P. By Corollary 3.5.11, if T is a maximal k-torus of G and T is the corresponding maximal K-torus of G as in Corollary 9.4.13 then T ⊆ P if and only if T ⊆ P . Proof. By Proposition 3.5.4 and Corollary 3.5.11, for each P the image P of PK in G is a pseudo-parabolic K-subgroup that determines P uniquely. It is a parabolic K-subgroup, since G is reductive (Proposition 2.2.9). Since every

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maximal K-torus in G arises in a similar manner from a unique maximal ktorus in G (Corollary 9.4.13), it likewise follows from Proposition 3.5.4 that every parabolic K-subgroup of G has the form P for some choice of P. It now remains to prove that any choice of P and the corresponding parabolic K-subgroup P := im(PK → G ) we have P(k) = G(k) ∩ P (K), ξG (P) = R K/k (P ), and P(k) is Zariski-dense in P. Since P = PG (λ) for some k-homomorphism λ : GL1 → G, we see that ZG (λ) contains a maximal ksplit k-torus of G (choose one containing λ(GL1 )). But these are all maximal as k-tori in G and are G(k)-conjugate, by Proposition 9.4.7(1) and Corollary 9.4.13, so by applying a suitable G(k)-conjugation it sufﬁces to treat the case of P containing a single ﬁxed choice of k-split maximal k-torus of G. We will work with T corresponding to D0 as in the construction of Gα,n G. The parabolic set of roots (P, T ) determines where P meets an open Bruhat cell for (G, T ) (relative to a choice of positive system of roots contained in (P, T )), and (P , T ) is the set of non-multipliable roots in (P, T ). Explicitly, P ∩ is a direct product of ZG (T ) and some root groups for (G, T ). But each root group is either R K/k (Ga ) or R K/k (Ga ) × Ga , and ZG (T ) = R K/k (GL1 )n , and by inspection the k-points are Zariski-dense in each of these k-groups. Thus, P(k) is Zariski-dense in P. For each non-divisible root a ∈ (G, T ) and the corresponding nonmultipliable root a ∈ (G , T ), the explicit description of ξG on root groups gives that ξG (Ua ) = R K/k (Ua ). Likewise, ξG restricts to an isomorphism between Cartan k-subgroups, so ξG (P) is open in R K/k (P ) (as they meet an open Bruhat cell in the same locus), and hence equality holds, due to connectedness considerations. To prove that the containment P(k) ⊆ G(k) ∩ P (K) is an equality, we work with the Zariski closure Q in G of the subgroup G(k) ∩ P (K). This contains P since P(k) is Zariski-dense in P, so Q is pseudo-parabolic (Proposition 3.5.8). But then PK ⊆ QK , and (by construction of Q) a Zariski-dense locus in QK maps into P in G = GKss . Hence, Q = P , so the containment P ⊆ Q is an equality. Thus, G(k) ∩ P (K) ⊆ Q(k) = P(k), as required. 2 Remark 11.4.6 Since G as in Proposition 11.4.5 is absolutely pseudo-simple and not reductive, it follows from Proposition 3.4.9(2) that G/P is non-proper for any pseudo-parabolic k-subgroup P of G distinct from G. This can be seen rather easily in the present situation: the natural map G/P → R K/k (G )/R K/k (P ) = R K/k (G /P )

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is surjective with ﬁber over 1 isomorphic to (ker ξG )/(P ∩ ker ξG ), and the explicit description of ker ξG in Theorem 9.3.10(4) implies (by root group considerations) that P ∩ ker ξG has strictly smaller dimension than ker ξG when P = G. Since a positive-dimensional coset space for a unipotent group scheme is never proper, we get a direct proof of the non-properness of G/P when P = G. Now we address the description of pseudo-parabolic k-subgroups in the basic exotic case over a ﬁeld k with char(k) = p ∈ {2, 3}. Let (G , k /k, G) be the corresponding triple from the basic exotic construction, where kk p ⊆ k ⊆ k with Spec k → Spec k having all ﬁber-degrees > 1, π : G → G = G k the (ﬁberwise) very special isogeny, and G a Levi k-subgroup of R k /k (G ) such −1 that G = R k /k (π ) (G) is smooth. Then G is a basic exotic pseudo-reductive k-group and it canonically determines (G , k /k, G), by Proposition 8.1.5. Proposition 11.4.7 Let G be the basic exotic pseudo-reductive k-group corresponding to a triple (G , k /k, G). (1) There is a natural bijection between the set of pseudo-parabolic ksubgroups P ⊆ G and the set of parabolic k -subgroups P ⊆ G whose image P under the very special isogeny π : G → G = G k descends to a k-subgroup P in G. Explicitly, P = G ∩ R k /k (P ), P = im(Pk → G ), and the coset space G/P is given by the ﬁber product G/P = G/P ×R

k /k (G

/P )

R k /k (G /P ).

(11.4.3)

In particular, the natural map G/P → R k /k (G /P ) is a closed immersion. (2) Let T ⊆ G be a maximal k-torus, and T := im(Tk → G ) the image maximal k -torus in G . Then P contains T if and only if P contains T , and every pseudo-parabolic k -subgroup of G containing T arises as such a P . This proposition provides a concrete procedure reversing the one in Proposition 3.5.4 in the exotic case. Proof. We can pass to factor ﬁelds of k to reduce to the case where k is a ﬁeld. We may also assume k = ks , as in the proof of Proposition 11.4.3. Step 1. Choose a pseudo-parabolic k-subgroup P in G, and let T be a maximal k-torus in P (equivalently, a maximal k-torus of G contained in P), so P = PG (λ) for some 1-parameter k-subgroup λ : GL1 → T ⊆ G. The k -subgroup

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Pk ⊆ Gk is PGk (λk ). The natural map q : Gk → G is surjective (in fact, it is a k -descent of the maximal reductive quotient of Gk , by Theorem 7.2.3(3)). Thus, the image of Pk → G is P := PG (λ ), where λ = q ◦ λk . This is a pseudo-parabolic k -subgroup of the connected reductive k -group G , so it is a parabolic k -subgroup (Proposition 2.2.9). It is clear by construction that P contains the maximal k -torus T := im(Tk → G ) in G , and that P ⊆ G ∩ R k /k (P ). Step 2. We next show that each parabolic k -subgroup P of G containing T arises from a pseudo-parabolic k-subgroup P in G containing T as in Step 1, and that P = G ∩ R k /k (P ). This will settle the second claim in assertion (2). It will also imply the ﬁrst claim in (2) provided that we prove P = G ∩ R k /k (P ) for any pseudo-parabolic k-subgroup P in G, since the maximal ktorus in R k /k (T ) is T (as k /k is purely inseparable) and so when T ⊆ P we have T ⊆ G ∩ R k /k (T ) ⊆ G ∩ R k /k (P ) = P. To construct P containing T such that im(Pk → G ) equals the chosen P containing T , we will argue by exhaustion of possibilities without directly studying any speciﬁc P . That is, we make no initial choice of P and will instead directly prove that as we vary P through the set of pseudoparabolic k-subgroups of G containing T , the corresponding k -subgroups P in G vary without repetition through the set of pseudo-parabolic k -subgroups containing T . To this end, ﬁrst observe that by Proposition 7.2.7(2) and Proposition 7.2.3(3), the maximal reductive quotient map q : Gk G carries Lie(G)k onto Lie(G ) and identiﬁes (G, T ) with (G , T ) inside of X(T ) = X(Tk ) = X(T ). Now choose any pseudo-parabolic k-subgroup P in G containing T and choose λ ∈ X∗ (T ) such that P = PG (λ) as in Step 1. Also deﬁne P in terms of P as in Step 1. Under the identiﬁcation (G, T ) = (G , T ), the set (P, T ) = (G, T )λ0 of nontrivial T -weights on Lie(P) is carried over to the set (G , T )λ 0 of nontrivial T -weights on Lie(P ). In a pseudoreductive group over a separably closed ﬁeld F, the set of pseudo-parabolic F-subgroups containing a given maximal F-torus is in natural bijection with the set of parabolic subsets of the root system (Proposition 3.5.1). Thus, as we vary through all possibilities for P containing T in G, the resulting pseudoparabolic k -subgroups P in G that we get do indeed vary without repetition through all parabolic k -subgroups of G containing T . To prove that the inclusion P ⊆ G ∩ R k /k (P ) inside of G is an equality for any pseudo-parabolic k-subgroup P of G (without any a priori reference

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to containment of speciﬁc maximal tori), we may make an auxiliary choice of maximal k-torus T in P and λ ∈ X∗ (T ) such that P = PG (λ). Consider the 1-parameter k-subgroup

[λ ] : GL1

/ R k /k (GL1 )

R k /k (λk )

/ R k /k (T )

/ R k /k (G ).

By Remark 11.4.4, R k /k (P ) = PRk /k (G ) ([λ ]). Functoriality considerations with Weil restriction imply that [λ ] factors as the composition of λ : GL1 → T ⊆ G and the inclusion of G into R k /k (G ). Hence, compatibility with closed subgroup inclusions gives that P = PG (λ) = G ∩ R k /k (P ). In particular, this completes the unconditional proof of (2), where it is understood (since (1) has not yet been fully proved) that P is deﬁned to be im(Pk → G ). Step 3. Now we begin the most difﬁcult task, which is to prove the sufﬁciency of the criterion in assertion (1) for a parabolic k -subgroup of G to arise from a pseudo-parabolic k-subgroup of G (so then (2) will be fully proved). More precisely, if Q is a parabolic k -subgroup of G such that the image Q ⊆ G = G k is deﬁned over k then we shall prove that Q arises from a pseudo-parabolic k-subgroup Q in the basic exotic pseudo-reductive k-group G. (The source of the difﬁculty is that it is not evident a priori that Q contains a maximal k torus T whose image T in Q arises from a k-torus T in G. Thus, we cannot easily apply Proposition 11.1.3 to obtain a suitable maximal k-torus T in G for which (G, T ) provides a mechanism to construct Q from the set of roots (Q , T ) ⊆ (G , T ).) Pick a maximal k-torus T in G, and let T = im(Tk → G ) be the corresponding maximal k -torus in G . Choose g ∈ G (k ) such that P := g Q g −1 is a parabolic k -subgroup of G containing T . (To ﬁnd such a g , we may ﬁrst pick a maximal k -torus S in Q , so S is maximal in G , and then choose g such that g S g −1 = T .) Thus, by (2), g Q g −1 = im(Pk → G ) for a (unique) pseudo-parabolic k-subgroup P of G that moreover contains T . Note that g is well deﬁned up to left multiplication by NG (P )(k ) = P (k ). To ﬁnd Q, it sufﬁces to prove that g ∈ P (k )G(k). Indeed, assuming g ∈ P (k )G(k), we can modify g by left P (k )-multiplication so that g ∈ G(k). Thus, g −1 Pg is a pseudo-parabolic k-subgroup of G whose k -ﬁber in Gk maps onto Q . We can therefore take Q = g −1 Pg .

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We now take up the task of proving g ∈ P (k )G(k). Let g ∈ G (k ) be the image of g under the very special k -isogeny π : G → G , so the −1 parabolic k -subgroup g −1 P g = Q in G is carried onto g P g = Q in G = G k . By hypothesis, Q = Qk for a k-subgroup Q in G. Note that Q is a parabolic k-subgroup of G, since parabolicity in smooth connected afﬁne groups is insensitive to ground ﬁeld extension. Moreover, by construction P = P k for the parabolic k-subgroup P ⊆ G that is the image of P ⊆ G under the quotient map f : G G. The parabolic k-subgroups P and Q in the connected reductive k-group G become conjugate (by a rational point) over the ﬁeld k . Hence, there exists g ∈ G(k) such that g −1 Pg = Q (by [Bo2, 20.9(iii)]). We conclude that g g −1 ∈ NG (P )(k ) = P (k ), so g ∈ P (k )g. Since we want to prove g ∈ P (k )G(k), we are reduced to proving that if an element g ∈ G (k ) is carried into P (k )G(k) under the map

R k /k (π ) : R k /k (G ) → R k /k (G ) on k-points then g ∈ P (k )G(k). Step 4. The k -group P is the image of Pk under the maximal reductive quotient map Gk G . Our problem is to prove that the preimage under G (k ) → G (k ) = G(k ) of the subset P (k )G(k) is P (k )G(k). This problem is expressed in terms of (G , k /k, G) (which is used to deﬁne the basic exotic pseudo-reductive k-group G) and the pseudo-parabolic k-subgroup P ⊆ G (from which the parabolic k -subgroup P ⊆ G arises). By applying inversion throughout, we may switch to working with cosets g P (k ) and g P (k ) rather than P (k )g and P (k )g . We will now study the map

G (k )/P (k ) → G (k )/P (k )

(induced by π : G → G ) between the coset spaces modulo k -points of parabolics and prove that the preimage of the P (k )-coset of a point in G(k) is always represented by a point in G(k) (with G := f −1 (G) for f = R k /k (π )). To do this, we will work with the root system := (G, T ) = (G , T ) and a ﬁxed positive system of roots + contained in the parabolic subset := (P, T ) = (P , T ). Observe that is the union of + and the set of negative roots that are Z0 -linear combinations of a unique set J of simple positive roots. Let W denote the Weyl group of the root system . We deﬁne WJ to be the subgroup generated by the reﬂections in the roots belonging to J (which

11.4 Structure of root groups and pseudo-parabolic subgroups

385

is also naturally identiﬁed with the Weyl group of the maximal reductive quo tients of the parabolic subgroups P and P , relative to the maximal tori T and T of G and G respectively). By the proof of [Bo2, 21.24], each coset in W /WJ has a unique representative with shortest length; we let W J denote the set of such representatives. By (the proof of) [Bo2, 21.29], there is a disjoint union decomposition G (k ) = Uw (k ) · nw · P (k ) w∈W J

where nw ∈ NG (T )(k) represents w and each of the product sets is a direct product. (Here, Uw denotes the k -subgroup of G directly spanned, in any order, by the root groups associated to the roots a ∈ + such that w−1 (a) is negative.) We also have compatible decompositions G (k ) = U w (k ) · nw · P (k ), G(k)P (k ) = U w · nw · P (k ) w∈W J

w∈W J

of G (k ) and G(k)P (k ) (⊂ G (k )), with

U w = π (Uw ), U w = G(k) ∩ U w (k ), and nw = π (nw ). These compatible coset decompositions imply that under the natural map G (k ) = Uw (k ) · nw · P (k ) −→ U w (k ) · nw · P (k ) = G (k ), w∈W J

w∈W J

the preimage of G(k)P (k ) consists of classes in G (k )/P (k ) whose unique representative u nw (w ∈ W J , u ∈ Uw (k )) has the property that it is carried by the map Uw (k ) → U w (k ) into an element of U w (⊂ G(k)). In particular, π (u ) ∈ G(k), so by the deﬁnition of G = f −1 (G) it follows that u ∈ G(k) inside of G (k ). Since the point nw was chosen inside G(k), we have proved that u nw ∈ G(k) inside of G (k ). Step 5. With assertion (2) and the necessary and sufﬁcient criterion in (1) now proved, to ﬁnish the proof of (1) we have to establish the ﬁber product description of G/P in terms of G/P, G /P , and G /P , and deduce the closed immersion claim. Since we have reduced to the case K = k with K renamed as k, G is the exotic pseudo-reductive k-group arising as G = f −1 (G) = G ×R

k /k (G

)

R k /k (G ).

386

Applications

The recipe P = G ∩ R k /k (P ) likewise can be rewritten as P = P ×R

k /k (P

)

R k /k (P ).

Since Weil restriction is compatible with the formation of smooth group quo tients, and the maps G → R k /k (G ) and P → R k /k (P ) are subgroup inclusions, it follows that G/P is given by the ﬁber product as in (11.4.3); keep in mind that we are now in the case K = k. The natural map h : G/P → R k /k (G /P ) is thereby identiﬁed with a base change of the natural map h : G/P → R k /k (G /P ), so to show that h is a closed immersion it sufﬁces to prove that h is. But G /P = (G/P)k , so h is an instance of the general map jX : X → R k /k (Xk ) for quasi-projective k-schemes X . This map is always a closed immersion, by Proposition A.5.7. 2

Example 11.4.8 We now show that the constraint on P in relation to G in Proposition 11.4.7 cannot be dropped (though part (2) of the proposition implies that this constraint is satisﬁed when P contains a maximal k -torus T arising from a maximal k-torus T in G). Let k be an imperfect ﬁeld with char(k) ∈ {2, 3} and G a basic exotic pseudoreductive k-group associated to a triple (G , k /k, G) with k a ﬁeld. Assume k = ks , pick a Levi k-subgroup L in G , and choose a Borel k-subgroup B in L and a maximal k-torus T in B. Since B is pseudo-parabolic in L, there is a 1-parameter k-subgroup λ : GL1 → T such that B = PL (λ). Deﬁne B = PG (λ). By Proposition 2.3.10, the groups L, G , and G have the same root system (with respect to compatible maximal tori), so the characterization of minimality in terms of the root system in the pseudo-split case (Proposition 3.3.15(1)) implies that B is a minimal pseudo-parabolic k-subgroup of G and B := im(Bk → G ) is a Borel k -subgroup of G with maximal k -torus T := im(Tk → G ). Let B be the image Borel subgroup of B in the very special quotient G , and let B be the common image of B and B under the quotient map G G, so B is a Borel k-subgroup of G descending B . Choose g ∈ G (k ) not in G (k)B (k ). Such a g exists, as can be seen by working in an open Bruhat cell of G = Lk relative to (B , T ) (which arises by scalar extension from the open Bruhat cell of L relative to (B, T )). Consider P = g B g −1 . This is a minimal pseudo-parabolic k -subgroup of G , and we claim that it cannot arise from a pseudo-parabolic k-subgroup P in G . Suppose P arises from some such P, which is to say that P = im(Pk → G ). Consideration of the root systems (relative to a maximal k-torus in P and the associated maximal k -torus in P ) shows that P must be a minimal

11.4 Structure of root groups and pseudo-parabolic subgroups

387

pseudo-parabolic k-subgroup of G . Thus, by rational conjugacy of minimal pseudo-parabolic subgroups over a separably closed ﬁeld (Proposition 3.3.17), P = gBg −1 for some g ∈ G (k). Passing to images under the maximal reductive quotient map q : Gk G , we get P = q(g)B q(g)−1 in G . But P = g B g −1 by deﬁnition, so q(g)−1 g ∈ NG (B )(k ) = B (k ). Thus, g ∈ G (k)B (k ), contrary to how g was chosen.

PA RT IV Appendices

Appendix A Background in linear algebraic groups

We assume the reader is familiar with the general theory of linear algebraic groups over a ﬁeld, as well as with the special features of the connected reductive case over a ﬁeld. More speciﬁcally, the book of Borel [Bo2] covers much of this background. However, we need to use non-smooth groups, and some deﬁnitions of common notions (such as unipotence, quotient, etc.) in [Bo2] in the smooth case need to be modiﬁed to work more generally. To avoid any confusion, in §A.1 we review a number of important deﬁnitions from the scheme-theoretic point of view that is necessary in our work; in general we follow [SGA3]. As an additional convenience to the reader, in §A.2 we record some facts from [Bo2] that we frequently apply, incorporating generalizations without smoothness hypotheses (since we need such generality). In §A.3 we review the relative Frobenius morphism and discuss some results for non-afﬁne groups (mainly for justifying a few examples); non-afﬁne groups can be ignored for the development of the theory of pseudo-reductive groups. In §A.4 we record versions of the Existence, Isomorphism, and Isogeny theorems for split connected reductive groups over an arbitrary ﬁeld (since the literature on this topic aside from [SGA3] appears to always assume the ground ﬁeld is algebraically closed). We also show how to deduce these results from the more well-known case of an algebraically closed ground ﬁeld. The case of an arbitrary separably closed ground ﬁeld arises often in our work. Within the theory of linear algebraic groups, Weil restriction through a ﬁnite separable ﬁeld extension is an important and widely used operation. Less familiar, but ubiquitous in our work, is the case of an inseparable ﬁnite extension of ﬁelds. In §A.5 we review the Weil restriction functor in a general setting, and an interesting example concerning Levi subgroups (and a variant on Weil restriction due to Greenberg) is the focus of attention in §A.6. In §A.7 we address the interaction of Weil restriction and Lie algebras. Finally, in §A.8 we study group schemes M of multiplicative type and their actions on group schemes G of ﬁnite type over a ﬁeld. This is a familiar situation 391

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when M is a torus and G is smooth and afﬁne, but we need to allow for the possibility that M or G is not smooth.

A.1 Review of deﬁnitions The additive group is denoted Ga and the multiplicative group is denoted GL1 , always with the base ring understood from context. A maximal k-torus T in a group scheme G locally of ﬁnite type over a ﬁeld k is a k-torus not strictly contained in another k-torus in G; see Corollary A.2.6 for the (well-known) invariance of this property with respect to any ground ﬁeld extension when G is smooth and afﬁne, and Lemma C.4.4 for the case of general smooth G (which we do not need). Deﬁnition A.1.1 A vector group over a ﬁeld k is a smooth commutative k-group scheme V that admits an isomorphism to Gan for some n 0. The GL1 -scaling action arising from such an isomorphism is a linear structure on V . Observe that the GL1 -action on V arising from a linear structure induces the canonical k × -action on Lie(V ) (e.g., if char(k) = p > 0 then the composition of such a GL1 -action on V with the p-power map on GL1 does not arise from a linear structure on V when V = 0). Example A.1.2 If W is a ﬁnite-dimensional k-vector space then the associated vector group W represents the functor R R ⊗k W on k-algebras and its formation commutes with any extension of the ground ﬁeld. Explicitly, W = Spec(Sym(W ∗ )) and it has an evident linear structure relative to which W (k) is naturally identiﬁed with W as a k-vector space. When linear structures are speciﬁed on a pair of vector groups, a homomorphism respecting them is called linear. Over a ﬁeld of characteristic 0 there is a unique linear structure and all homomorphisms are linear. Over a ﬁeld with characteristic p > 0 the linear structure is not unique in dimension larger than 1 (e.g., a.(x, y) := (ax+(a−a p )y p , ay) is a linear structure on Ga2 , obtained from the usual one via the non-linear k-group automorphism (x, y) → (x + y p , y) of Ga2 ). A smooth connected solvable group G over a ﬁeld k is k-split if it admits a composition series over k whose successive quotients are k-isomorphic to GL1 or Ga . Any quotient of a k-split smooth connected solvable k-group is k-split [Bo2, 15.4(i)]. A general discussion of k-split groups is in [Bo2, 15.1ff.]. The notion of unipotence for afﬁne groups of ﬁnite type over a ﬁeld is of most interest to us in the familiar smooth case, where it can be deﬁned in terms of Jordan decomposition of geometric points. It is sometimes convenient (though not essential) to allow the non-smooth case, so we now record the deﬁnition in general and address alternative formulations.

A.1 Review of deﬁnitions

393

Deﬁnition A.1.3 A group scheme U over a ﬁeld k is unipotent if U is afﬁne of ﬁnite type and Uk admits a ﬁnite composition series over k with each successive quotient isomorphic to a k-subgroup of Ga . The deﬁnition of unipotence given in [SGA3, XVII, 1.3] does not assume U to be afﬁne, but it is equivalent to Deﬁnition A.1.3 by [SGA3, XVII, 2.1]. Proposition A.1.4 A group scheme U over a ﬁeld k is unipotent if and only if U is k-isomorphic to a closed k-subgroup of the standard upper triangular unipotent subgroup of some GLn . If U is smooth, connected, and unipotent then it is k-split if k is perfect, and in general if such a U is k-split then so is any quotient of U . Proof. The equivalence with an upper-triangular presentation over k is [SGA3, XVII, 3.5(i),(v)]. The k-split property in the smooth connected case over a perfect ﬁeld k is [Bo2, 15.5(ii)], and its preservation under passage to a quotient over any k is [Bo2, 15.4(i)]. 2 Deﬁnition A.1.5 A group scheme M over a ﬁeld k is of multiplicative type if Mk is isomorphic to a closed k-subgroup scheme of a k-torus. Its character group is deﬁned to be X(M ) = Homk (Mk , GL1 ) equipped with its natural action by Gal(ks /k) = Aut(k/k). In [SGA3, IX, 1.1] a more general deﬁnition over k is given, allowing kgroup schemes that may not be of ﬁnite type. The distinction is that we require the geometric character group to be ﬁnitely generated as a Z-module. Example A.1.6 If M = GL1 then X(M ) = Z with trivial Galois action, and if M = μn then X(M ) is the Z-module Z/nZ equipped with trivial Galois action. Example A.1.7 A k-group scheme that is both unipotent and of multiplicative type is trivial. Indeed, we may assume k is algebraically closed, so it remains to show that a nontrivial k-subgroup scheme M of GL1 cannot be a k-subgroup of Ga . Every nontrivial k-subgroup scheme of GL1 contains μ for some prime , so we just have to show that μ has no nontrivial k-homomorphism to Ga . The case char(k) = is clear, so we may assume = p = char(k) > 0. A k-homomorphism from μp to Ga is an element f ∈ k[t]/(t p − 1) such that f (xy) = f (x)+f (y) in k[x, y]/(x p −1, y p −1). Direct comparison of monomial terms on both sides forces f = 0. The following deﬁnitions for group schemes over a ﬁeld k are schemetheoretic versions of some familiar notions from the smooth case as in [Bo2]. Deﬁnition A.1.8 The kernel of a k-homomorphism f : G → G between kgroup schemes is f −1 (1); it is denoted ker f . We call G (or really (G, f )) a k-subgroup of G when ker f = 1, and a closed k-subgroup when f is a closed

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immersion. A k-subgroup G in G is normal if G(S) is a normal subgroup of G (S) for all k-schemes S (or equivalently, all afﬁne k-schemes S). The kernel is the unique closed subscheme of G representing the functor on k-schemes S ker(G(S) → G (S)). The reason it is closed is that every group scheme over a ﬁeld is separated (as the diagonal of a group scheme is a base change of the identity section, and any rational point in a scheme over a ﬁeld is closed). In Proposition A.2.1 we will see that k-subgroups are always closed when working with k-groups of ﬁnite type. Deﬁnition A.1.9 Let G be a scheme over a ﬁeld k. Let X be a locally ﬁnite type k-scheme equipped with a left G-action, and let V be a geometrically reduced (hence generically smooth) closed subscheme of X . If V is separated then the scheme-theoretic centralizer ZG (V ) of V in G is the unique closed k-subgroup of G such that, for each k-algebra A, the group ZG (V )(A) is the subgroup of g ∈ G(A) such that the g-action on XA is trivial on VA . If V is ﬁnite type (e.g., X is ﬁnite type) then the scheme-theoretic normalizer NG (V ) of V in G is the unique closed subscheme of G such that, for each kalgebra A, NG (V )(A) is the set of g in G(A) such that the g-action on XA carries VA back into itself. A k-subgroup H of G normalizes V if H ⊆ NG (V ). The functor NG (V ) does in fact act on V via automorphisms since its A-points act via monomorphisms VA → VA and any monic endomorphism of a ﬁnitely presented scheme is an automorphism [EGA, IV4 , 17.9.6]. Thus, NG (V ) really is a subgroup functor of G. To see that ZG (V ) and NG (V ) exist as closed subschemes of G, since V (ks ) is Zariski-dense in Vks (due to geometric reducedness) it follows from considerations with relative schematic density (cf. [EGA, IV3 , 11.10.6]) that ZG (V ) (resp. NG (V )) is represented by the Gal(ks /k)-descent of the schemetheoretic intersection of the centralizer schemes in Gks of the points v ∈ V (ks ) (resp. intersection of the preimage of Vks under the maps Gks → Xks deﬁned by g → g.v for v ∈ V (ks )). To make sense of such (generally inﬁnite) intersections we need to check that the centralizers and preimages being used are closed subschemes. For the case of NG (V ) it follows from the fact that Vks is closed in Xks . For the case of ZG (V ), we use that the diagonal of V is closed in V × V (by the separatedness hypothesis) and that V × V is closed in X × X . It is an elementary consequence of the deﬁnitions that the formation of ZG (V ) and NG (V ) commutes with any extension of the ground ﬁeld, but they need not be smooth, even when G and V are smooth. For example, if char(k) = p > 0 and G = SLp then ZG (G) = μp .

A.1 Review of deﬁnitions

395

Deﬁnition A.1.10 The scheme-theoretic center ZG of a smooth k-group scheme G is ZG (G); i.e., for any k-algebra A, ZG (A) is the group of g ∈ G(A) whose conjugation action on GA is trivial. A k-subgroup Z of a k-group scheme G is central if the conjugation action of Z of G is trivial. Note that we only deﬁne the notion of scheme-theoretic center for smooth G. There is another group-theoretic construction which we need without smoothness restrictions: the formation of coset spaces, and especially quotients by normal subgroup schemes. In [Bo2], quotient maps between smooth afﬁne groups over a ﬁeld are generally required to be separable (equivalently, smooth). This restriction is due to the avoidance of non-smooth groups in [Bo2]. In our work it is crucial to go beyond this classical setting and use a more general theory of quotients as developed by Grothendieck: Deﬁnition A.1.11 If G is a group scheme locally of ﬁnite type over a ﬁeld k and H is a closed k-subgroup scheme of G then the quotient scheme G/H is taken in the sense of Grothendieck [SGA3, VIA , Thm. 3.2(iv)]. This is a locally ﬁnite type k-scheme equipped with faithfully ﬂat map of k-schemes G → G/H that is initial for being invariant under right translation by H on G (and G/H is separated). If H is normal in G then G/H admits a unique k-group structure making G → G/H a k-homomorphism, and its kernel is H . This is proved via descent theory. Example A.1.12 Let f : G G be a surjective homomorphism between smooth afﬁne groups over a ﬁeld k, and let H = ker f . We claim that the induced map G/H → G is always an isomorphism. In other words, f has the universal property of a quotient map modulo H regardless of the structure of H (e.g., this works even when H is not smooth). The reason is that f has all ﬁbers equidimensional of dimension dim G − dim G, so the regularity of G and G implies that f is automatically ﬂat [Mat, 23.1] and hence the quotient property for f follows from descent theory and the evident isomorphism H × G G ×G G. We often use this quotient property for such f when H is non-smooth. (The same argument shows in general that a surjective homomorphism between arbitrary afﬁne groups of ﬁnite type over k is a quotient map if and only if it is ﬂat.) The quotient G/H is smooth when G is smooth (as it inherits geometric reducedness from G, and then homogeneity via the G-action propagates the generic smoothness globally). If G is smooth and afﬁne and H is smooth then there is another notion of (smooth separated) quotient G/H deﬁned in [Bo2, II, §6], and these quotients coincide because they have the same universal mapping property in the category of smooth separated k-schemes of ﬁnite type.

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Proposition A.1.13 Let k be a ﬁeld, G a smooth connected k-group, and N a smooth normal k-subgroup of G. If the center ZN is of multiplicative type then ZN is central in G. In particular, if G is reductive then ZN ⊆ ZG . Proof. We may and do assume k = k. The normality of N implies that ZN is carried into itself under G(k)-conjugations. Thus, by smoothness of G, it follows that ZN is normal in G. The conjugation action of G on ZN corresponds to a homomorphism of group functors f : G → Aut(ZN ). Since ZN is of multiplicative type, Aut(ZN ) is represented by an étale k-scheme. Hence, connectedness of G forces f to be trivial. This implies that ZN is central in G. 2 Let k be a ﬁeld with char(k) = p > 0, and let Fk : k → k be the ppower map. For a k-group G of ﬁnite type, the relative Frobenius morphism FG/k : G → G (p) := G ⊗k, Fk k is very useful. Loosely speaking, it is deﬁned by the p-power map in local coordinates over k. A systematic discussion of this map, including a proof that it is an isogeny of degree pdim G when G is smooth, is given in §A.3. (This map is never ﬂat in the non-smooth case.) In general the k-map FG/k is a k-homomorphism and its formation commutes with any scalar extension on k. For smooth G we call FG/k the Frobenius isogeny; the case of non-smooth G does arise in our work, such as inﬁnitesimal G. The map FG/k is made explicit in general in Example A.3.4 (with n = 1 there). When char(k) = p > 0, there is yet another kind of quotient for smooth afﬁne k-groups G deﬁned in [Bo2, V, §17], the concept of quotient G/h for any Lie subalgebra h ⊆ Lie(G) that is stable under the adjoint action of G and the p-operation on Lie(G); see §A.7 for a systematic discussion of Lie algebras for group schemes and the p-operation on them in characteristic p > 0. In Proposition A.7.14 we review the sense in which any such h uniquely “exponentiates” to an inﬁnitesimal normal closed k-subgroup H ⊆ G such that the relative Frobenius morphism FH /k vanishes and Lie(H ) = h inside of Lie(G). In Example A.7.16 we explain why the quotient G/h coincides with the quotient G/H . If G is a smooth afﬁne group over a ﬁeld k and H and H are smooth closed k-subgroups, at least one of which is connected, then (H , H ) denotes the commutator k-subgroup [Bo2, I, §2.3]. This is the unique smooth closed k-subgroup of G such that for any algebraically closed extension K/k the subgroup (H , H )(K) in G(K) is generated by the commutators hh h−1 h −1 for h ∈ H (K) and h ∈ H (K). Beware that if H and H are disconnected then no such k-subgroup may exist (e.g., G = SL2 over Q and suitable ﬁnite H and H in G(Q)). Deﬁnition A.1.14 The derived group D (G) of a smooth group G of ﬁnite type over a ﬁeld k is the unique smooth closed k-subgroup such that (D (G))(K) is the commutator subgroup of G(K) for any algebraically closed extension K/k.

A.1 Review of deﬁnitions

397

Note that the derived group exists without connectedness hypotheses on G; see [Bo2, I, 2.4] for the afﬁne case (which is all we need) and [SGA3, VIB , 7.2(vii), 7.10] for the general case. The formation of D (G) commutes with any extension of the base ﬁeld, and the quotient map G → G/D (G) is initial among all k-homomorphisms from G to a commutative k-group scheme (see Lemma 5.3.4 for a generalization). We never consider derived groups for non-smooth groups. A smooth k-group G of ﬁnite type is called perfect if G = D (G). Deﬁnition A.1.15 For a smooth afﬁne group G over a ﬁeld k, the geometric unipotent radical (resp. geometric radical) of G is the maximal smooth connected normal closed k-subgroup of Gk that is unipotent (resp. solvable). It is denoted Ru (Gk ) (resp. R(Gk )). We say G is reductive (resp. semisimple) if its geometric unipotent radical (resp. geometric radical) is trivial. A smooth afﬁne k-group G is reductive (resp. semisimple) if and only if the identity component G 0 is reductive (resp. semisimple), so if G is reductive then the scheme-theoretic centralizer in G of a semisimple element of Lie(G) is reductive [Bo2, 13.19]. Such centralizers can be disconnected even when G is connected, but we only use their identity component (e.g., PropositionA.8.14(2) and its proof). The reader may safely follow the convention in [SGA3, XIX, 2.7] according to which reductive k-groups are connected. Deﬁnition A.1.16 Let G be a smooth connected afﬁne group over a ﬁeld k. The maximal geometric reductive (resp. semisimple) quotient of Gk is the quotient G red := Gk /Ru (Gk ) (resp. G ss := Gk /R(Gk )) of Gk ; it uniquely dominates k k all other reductive (resp. semisimple) quotients of Gk . Since we are most interested in the case char(k) = 0 the possible existence of non-central ﬁnite normal subgroup schemes in G (e.g., the kernel of the Frobenius isogeny of G) forces us to distinguish between general isogenies and central ones. The quotient G ss /ZGss of G ss modulo its ﬁnite multiplicative k k k scheme-theoretic center is called the maximal (geometric) adjoint semisimple quotient of Gk . Asimple semisimple (resp. simple adjoint semisimple) quotient of G is called maximal if it is not dominated by any others except through isomorphisms. Every simple semisimple (resp. simple adjoint semisimple) quotient of Gk is uniquely dominated by a maximal one. These notions of maximal quotient are sometimes considered as quotients of G rather than of Gk , but only when the k-subgroup Ru (Gk ) ⊆ Gk is deﬁned over k. Remark A.1.17 Note that the maximal adjoint semisimple quotient of a connected reductive group G is G/ZG , and up to unique isomorphism this is the

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only adjoint semisimple quotient of the form G/Z with Z ⊆ G a central subgroup scheme. (For example, over any ﬁeld of nonzero characteristic the central isogeny SLn → PGLn for n > 1 is the maximal adjoint semisimple quotient of SLn , and its composite with the Frobenius isogeny of PGLn is an adjoint semisimple quotient of SLn modulo a non-central ﬁnite subgroup.) Deﬁnition A.1.18 A Levi k-subgroup of a smooth connected afﬁne group G over a ﬁeld k is a smooth connected k-subgroup L ⊆ G such that the natural map Lk → Gk /Ru (Gk ) is an isomorphism (equivalently, Ru (Gk ) Lk → Gk is an isomorphism). We use a similar deﬁnition when k is a nonzero ﬁnite product of ﬁelds and all ﬁbers of G → Spec k are connected (treating the ﬁbers over Spec k separately). We do not consider the notion of Levi subgroup in an afﬁne k-group in the absence of the smoothness and connectedness requirements. This agrees with Borel’s convention in [Bo2, 11.22]. Levi subgroups need not exist in nonzero characteristic, even over an algebraically closed ﬁeld; see §A.6.

A.2 Some results from the general theory If a homomorphism between smooth afﬁne groups over a ﬁeld has trivial kernel in the scheme-theoretic sense then it is injective on geometric points and on Lie algebras, so it is a closed immersion. The smoothness of the groups is not necessary: Proposition A.2.1 Let k be an artin local ring, G a k-group scheme of ﬁnite type, and G a k-group scheme locally of ﬁnite type. If f : G → G is a k-homomorphism and ker f = 1 then f is a closed immersion. Proof. This is [SGA3, VIB , 1.4.2].

2

Example A.2.2 Let G be a group scheme of ﬁnite type over an artin local ring k such that G admits a faithful linear representation in the sense that there is a subgroup functor inclusion G → GLn . This is automatically a closed immersion, by Proposition A.2.1 (so G is afﬁne). For a ﬁeld k and afﬁne k-scheme V equipped with a left action by a k-group scheme G, a ﬁnite-dimensional k-subspace W ⊆ k[V ] is G-stable if, for any k-algebra R, the R-submodule R ⊗k W ⊆ R[V ] is G(R)-stable. In particular, we get a map of group functors G → GL(W ), so by Yoneda’s Lemma it is a k-group scheme homomorphism (i.e., the G-action on W must be “algebraic”). Proposition A.2.3 Let k be a ﬁeld and G an afﬁne k-group scheme of ﬁnite type. Consider an afﬁne k-scheme V endowed with a left action by G over k. The coordinate ring k[V ] is the directed union of G-stable ﬁnite-dimensional

A.2 Some results from the general theory

399

k-linear subspaces. In particular, taking V = G with its natural left action, G is a closed k-subgroup of GLn for some n 1. Proof. The construction of the directed union is proved in [Bo2, 1.9–1.10] assuming that G and V are smooth. The role of smoothness is to ensure the sufﬁciency of computations using ﬁeld-valued points. By using points valued in any k-algebra, the argument works without smoothness. It remains to construct a closed k-subgroup inclusion into some GLn . By taking V = G with G-action by left translation, there is a G-stable k-linear subspace W ⊆ k[G] of ﬁnite dimension that contains a ﬁnite generating set of k[G] as a k-algebra. We claim that for any k-algebra R, the G(R)-action on R ⊗k W is faithful. Since this action is induced by the G(R)-action on the R-algebra R[G], and R ⊗k W generates R[G] as an R-algebra, it sufﬁces to show that G(R) acts faithfully on R[G]. But G is afﬁne, so any g ∈ G(R) is determined by the evaluations f (g) = (g −1 .f )(1) for all f ∈ R[G]. This implies faithfulness of the action, so we get a monomorphism of group functors G → GL(W ). By Example A.2.2, this is a closed immersion. 2 Proposition A.2.4 Let k be a ﬁeld and G an afﬁne k-group scheme of ﬁnite type. Let H be a closed k-subgroup scheme. There is a nonzero ﬁnite-dimensional k-vector space V , a closed k-subgroup inclusion ρ : G → GL(V ), and a line L in V such that H is the scheme-theoretic G-stabilizer of L viewed as a point in the projective space of lines in V . Proof. This is proved in [Bo2, 5.1] when H and G are smooth, using computations with ﬁeld-valued points. The same method of proof (ignoring the considerations with Lie algebras and convolutions there) works without smoothness by using points valued in k-algebras. Another reference, with a scheme-theoretic viewpoint, is [DG, II, §2, 3.5]. 2 There are two very useful results concerning torus centralizers that are proved in [Bo2] and which we state here for ease of reference. Proposition A.2.5 Let G be a smooth connected afﬁne group over a ﬁeld k, and S a k-torus in G. The scheme-theoretic centralizer ZG (S) is smooth and connected, with Lie algebra Lie(ZG (S)) equal to the S-ﬁxed part of Lie(G). Proof. The smoothness and determination of the Lie algebra are [Bo2, Cor. 9.2], and the connectedness is [Bo2, 11.12]. 2 Corollary A.2.6 Let G be a smooth afﬁne group over a ﬁeld k, T a k-torus in G, and K/k an extension ﬁeld. The k-torus T is maximal as such in G if and only if TK is maximal as a K-torus in GK . Likewise, if Z is the maximal central k-torus in G then ZK is the maximal central K-torus in GK . For the centrality, note that we do not assume G is connected.

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Proof. To prove that TK is maximal in GK when T is maximal in G, by Proposition A.2.5 we can replace G with ZG (T ) so that T is central in G. Passing to G/T lets us assume T = 1, so G has no nontrivial k-tori. By [Bo2, 18.2(i)] there is a maximal k-torus in Gk that is deﬁned over k, so Gk has no nontrivial k-tori and hence is unipotent [Bo2, 11.5(2)]. Thus, G is unipotent, so GK has no nontrivial K-tori. Now consider the assertion concerning maximal central tori. The case K = ks is a consequence of Galois descent, so we may assume k = ks and K = Ks . We have to show that if a central K-torus S in GK contains ZK then S = ZK . Let T be a maximal k-torus of G containing Z, so TK · S is a K-torus in GK yet TK is maximal in GK . Thus, S is an intermediate torus between TK and ZK that is central. But T is k-split, as k = ks , so every K-subtorus of TK descends to a k-subtorus of T . Thus, the k-descent of S is a central k-torus in G containing Z, so it equals Z by maximality of Z. That is, S = ZK . 2 Corollary A.2.7 Let G be a smooth afﬁne group over a ﬁeld k, and T a maximal k-torus in G. For any smooth connected k-subgroup N in G that is normalized by T , the scheme-theoretic intersection T ∩ N is a maximal k-torus in N ; in particular, it is smooth and connected. Note that this corollary is applicable whenever N is normal in G. Proof. By Corollary A.2.6, we can assume k = k (so all maximal k-tori in G are G(k)-conjugate). The afﬁne k-group H = N T is smooth and connected with ZH (T ) = (N ∩ ZG (T )) T , so N ∩ ZG (T ) is smooth and connected. Hence, to prove that T ∩ N is a maximal k-torus in N we can replace G with ZG (T ) and replace N with N ∩ ZG (T ) so that T is central in G. Thus, T contains all k-tori in G, so S := (T ∩ N )0red is the unique maximal k-torus in N . The quotient N /S is therefore unipotent, so every k-subgroup scheme of N of multiplicative type must be contained in S. But T ∩ N is such a k-subgroup, so the containment S ⊆ T ∩ N is an equality. 2 Proposition A.2.8 Let f : G G be a surjective homomorphism between smooth connected afﬁne groups over a ﬁeld k. Let S be a k-torus in G , and S = f (S ) the image k-torus in G. If S is maximal in G then S is maximal in G, and in general f (ZG (S )) = ZG (S). Proof. The compatibility with maximal tori is [Bo2, Prop. 11.14], and the compatibility with torus centralizers is [Bo2, Cor. 2, 11.14]. 2 Next, we wish to record a special case of a fundamental result of Grothendieck [SGA3, XII, 7.1(b)] concerning conjugacy of maximal tori for smooth group schemes over any base scheme. We focus on the case of a separably closed base ﬁeld. (In Theorem C.2.3 we prove a ﬁner result over an arbitrary ﬁeld, but that generality will never be used in our work.) First we establish a lemma.

A.2 Some results from the general theory

401

Lemma A.2.9 Let G be a smooth afﬁne group over a ﬁeld k, and S a k-torus in G. Then W (G, S) := NG (S)/ZG (S) is ﬁnite étale and NG (S) is smooth. Proof. By Proposition A.2.5 (applied to G 0 ), ZG (S) is smooth. Thus, smoothness of NG (S) will follow once we prove that W (G, S) is étale. Since W (G, S) is ﬁnite type, we just have to prove that it is étale. The automorphism functor Aut(S) of the k-torus S is an étale k-group (locally of ﬁnite type), and the conjugation action of NG (S) on S deﬁnes a k-group homomorphism W (G, S) → Aut(S) with trivial kernel. Thus, by Proposition A.2.1, W (G, S) is a closed k-subgroup of Aut(S). A closed subscheme of an étale scheme over a ﬁeld is étale, so W (G, S) is étale. 2 Proposition A.2.10 (Grothendieck) Let G be a smooth afﬁne group over a separably closed ﬁeld k. Any two maximal k-tori in G are G(k)-conjugate. Proof. We may and do assume that G is connected. Let T and T be two maximal k-tori in G. Consider the functor X on k-algebras deﬁned as follows: X (A) is the group of elements g ∈ G(A) such that gTA g −1 = TA in GA . We claim that X is represented by a smooth non-empty closed subscheme of G, so X (k) is non-empty since k is separably closed. To see that X is represented by a closed subscheme of G, for each n 1 that is not divisible by char(k) let Xn denote the subfunctor of G with Xn (A) consisting of those g ∈ G(A) such that gT [n]A g −1 = T [n]A inside of GA . Since T [n] and T [n] are ﬁnite étale subschemes of G, they are constant and hence each Xn is represented by a closed subscheme of G. By Zariski-density of the collection of n-torsion subgroups in a k-torus for n not divisible by char(k), we have X = n Xn as subfunctors of G, so X is represented by a closed subscheme. To prove that X is smooth and non-empty we may assume that k is algebraically closed. Then X is a translate of the normalizer NG (T ) if X is non-empty, and NG (T ) is smooth by Lemma A.2.9. Hence, it sufﬁces to prove conjugacy when k is algebraically closed. This case is well known [Bo2, Cor. 11.3]. 2 Proposition A.2.11 Let G be a smooth connected afﬁne group over an inﬁnite ﬁeld k, and let Gt be the smooth connected k-subgroup generated by the k-tori of G. The k-group Gt is unirational and normal in G, with G/Gt unipotent (so G = Gt if G is perfect). The formation of Gt commutes with any extension of the ground ﬁeld. In particular, (Gt )k contains all tori of Gk . For ﬁnite k we can deﬁne Gt by Galois descent from ks , so the proposition is true even for ﬁnite k except possibly for the condition that Gt is generated by all k-tori of G. (In fact, it can be shown that Gt is generated by k-tori when k is ﬁnite.)

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Proof. Since Gt is generated by k-tori, by [Bo2, 2.2] there is a ﬁnite set of k-tori T1 , . . . , Tn in G such that the multiplication map T1 × · · · × Tn → Gt is surjective. Hence, the unirationality of Gt follows from the unirationality of k-tori [Bo2, 8.13(2)]. Grant for a moment that the formation of Gt commutes with extension of the ground ﬁeld, so its normality in G follows from working over k. The smooth connected quotient G/Gt then makes sense as a k-group, and it has no nontrivial k-tori (due to the deﬁnition of Gt and Proposition A.2.8), so it is unipotent and hence solvable. A smooth connected unipotent k-group is trivial if it is perfect, so G = Gt when G is perfect. It remains to prove that the formation of Gt commutes with extension of the ground ﬁeld. Let Tor G be the “variety of maximal tori” of G. Concretely this is G/NG (T0 ) for a maximal k-torus T0 in G, and there is a ﬁberwise maximal closed Tor G -torus T ⊆ G × Tor G with the property that for any ﬁeld L/k the ﬁbers Tx ⊆ GL for x ∈ Tor G (L) vary without repetition through precisely the set of maximal L-tori in GL . To construct T , we form the twisted embedding T0 × (G/ZG (T0 )) → G × (G/ZG (T0 )) deﬁned via (t, g) → (gtg −1 , g) and pass to the quotient by the free right action of the ﬁnite étale k-group NG (T0 )/ZG (T0 ) by right translation on G/ZG (T0 ) and by conjugation on T0 . (The ﬁbers over L-valued points for any L/k are as expected, since Proposition A.2.10 applies over Ls and then Galois descent does the rest.) The pair (Tor G , T ) satisﬁes a stronger universal property among maximal tori in G parameterized by arbitrary k-schemes [SGA3, XII, Cor. 5.4], but we do not need this fact. What matters for our purposes is the important theorem of Chevalley– Grothendieck that Tor G is a unirational (even rational) variety over k [SGA3, XIV, Thm. 6.1]. In particular, Tor G (k) is Zariski-dense in Tor G since k is inﬁnite. Consider the k-map f : T → G deﬁned by restriction to T of the projection map G × Tor G → G. We shall prove that f lands in Gt , so indeed the formation of Gt commutes with any extension of the ground ﬁeld. Consider the closed subset Z = f −1 (Gt ) in T . To prove Z = T it sufﬁces to show that Z contains the generic torus (i.e., the ﬁber of T over the generic point of Tor G ). Let π : Z → Tor G be the natural map, so π(Z) contains the Zariski-dense subset Tor G (k) and hence (by constructibility) contains a dense open subset U of Tor G . By semicontinuity of ﬁber dimension, the ﬁbers of π have a common dimension over a dense open subset of U . But over U (k) the ﬁbers of π coincide with the ﬁbers of T → Tor G , all of which have the same dimension. Hence, Z contains the generic torus, so Z = T . 2

A.3 Frobenius morphisms and non-afﬁne groups We now turn to a pair of more technical topics. A useful but somewhat specialized construction in the study of algebraic groups in nonzero characteristic is

A.3 Frobenius morphisms and non-afﬁne groups

403

the notion of a relative Frobenius morphism. We shall generally need this in the smooth case (in which case it is an isogeny), but it will also be used for groups that may be non-smooth of any dimension. Hence, as a convenient reference for the reader, we discuss Frobenius morphisms in general for schemes and address special properties in the case of group schemes. We also record some useful general structural results for (possibly non-afﬁne) group schemes of ﬁnite type over a ﬁeld, as is needed to justify some later examples. For a ﬁxed prime p, let S be an Fp -scheme. The absolute Frobenius morphism FS : S → S is given on afﬁne open subsets by the p-power map on the coordinate ring. Equivalently, FS is the identity on topological spaces and the p-power map on OS . For any S-scheme X and n 1, the map FXn : X → X is generally not an S-map (aside from special cases, such as when S is the spectrum of a ﬁnite extension of Fp of degree dividing n). To promote this to an S-map, we are led to consider the S-scheme X (p ) := S ×FSn ,S X n

for any n 0. Loosely speaking, X (p ) is given by the same equations over S that deﬁne X except that the coefﬁcients in the equations are replaced with their pn -powers. The case of most interest to us will be when S = Spec k for a ﬁeld k of characteristic p and X is an afﬁne k-group of ﬁnite type (generally smooth, but not always). The formation of FS is functorial in S since all maps of Fp -algebras are functorial with respect to the p-power map. This functoriality has some useful n consequences. First, the formation of X (p ) is compatible with base change on S in the sense that for any map of schemes S → S and the associated S -scheme X = X ×S S , the natural S -map n

X

(pn )

→ X (p ) ×S S n

(A.3.1)

is an isomorphism. (This justiﬁes suppressing the mention of S in the notation n X (p ) .) Second, the compatibility of FXn and FSn yields the S-map FX /S,n : X → X (p ) = S ×FSn ,S X n

(A.3.2)

whose composite with the projection pr 2 : X (p ) → X is FXn . Loosely speaking, the S-map FX /S,n is given by the pn -power power on local coordinates over S. Via (A.3.1), it is straightforward to check that the formation of FX /S,n is compatible with base change on S. For example, if X = ASd is an afﬁne space over S then via base change from Spec Fp we may identify FX /S,n with the endomorphism of ASd deﬁned by the pn -power map on the standard coordinates. A third application of the functoriality of absolute Frobenius morphisms is that the formation of FX /S,n is functorial in X over S. n

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When n = 1 we write FX /S to denote FX /S,1 , so for n > 1 we have FX /S,n = FX (pn−1 ) /S ◦ · · · FX (p) /S ◦ FX /S . For example, if S = Spec k for an extension ﬁeld k/Fp then when [k : Fp ] is ﬁnite and divides n we have FX /S,n = FXn but otherwise FX /S,n and FXn generally have different targets. Proposition A.3.1 Let Z be a scheme that is smooth of pure relative dimension n d over an Fp -scheme S. The map FZ/S,n : Z → Z (p ) is a ﬁnite locally free nd surjection with degree p . This proposition will be of interest to us when S = Spec k for a ﬁeld k/Fp . Proof. The result is clear by inspection in the special case that Z is an afﬁne space over S. To handle the general case we wish to work locally for the étale topology on Z to reduce to the case of an afﬁne space. Thus, we now consider how FZ/S,n interacts with étale maps over S. Rather generally, for any étale map f : X → Y of S-schemes, consider the commutative diagram of S-maps f

FX /S,n

X

/ Y

X (p

n)

n f (p )

FY /S,n

/ Y (pn )

in which the horizontal maps are étale. We claim that this diagram is cartesian; that is, the natural map h : X → X (p ) ×Y (pn ) Y n

is an isomorphism. By inspection of the deﬁnition of relative Frobenius morphisms, the vertical maps in the diagram are ﬁnite, surjective, and radiciel (i.e., injective on topological spaces and purely inseparable on residue ﬁeld extensions). Hence, the map n h over X (p ) is ﬁnite, surjective, and radiciel. But h is also a Y -map between étale Y -schemes, so h is étale. Since étale radiciel maps are open immersions [EGA, IV4 , 17.9.1], it follows that h is an open immersion, so by surjectivity it is an isomorphism. Applying the above cartesian property to the case of a Zariski-open inclusion n −1 U → Z we see that FZ/S,n (U (p ) ) = U , so we may work Zariski-locally on Z. Hence, by smoothness, we can arrange (by [EGA, IV4 , 17.11.4]) that Z is étale over an afﬁne space over S. Then the above cartesian property identiﬁes FZ/S,n

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with a base change of the n-fold relative Frobenius morphism of an afﬁne space, so we are reduced to the settled case of an afﬁne space. 2 Example A.3.2 Let k be a ﬁeld of characteristic p > 0 and let G be a connected reductive k-group that is k-split. Hence, by the Existence and Isomorphism Theorems (which we review in §A.4 over any ﬁeld) there exists a split connected reductive Fp -group G0 such that G (G0 )k , and G0 is unique up to (generally non-unique) isomorphism. Under this identiﬁcation, FG/k,n is carried to (pn ) (pn ) F(G0 )k ,n = (FG0 )nk via the natural identiﬁcation of (G0 )k = (G0 )k with (G0 )k = G. However, G0 is not functorial in G, so unless G is deﬁned to be (G0 )k (in particular, an Fp -structure on G is speciﬁed, such as the natural one for G = GLN over k) it is best not to view FG/k,n as an endomorphism of G except when [k : Fp ] is ﬁnite and divides n. The functoriality of the absolute Frobenius morphism implies, via the comn patibility of the formation of X (p ) with respect to ﬁber products in X over S, that F(X ×S Y )/S,n = FX /S,n × FY /S,n for any S-schemes X and Y . Hence, if X is n an S-group then X (p ) is an S-group and FX /S,n is an S-group homomorphism. Deﬁnition A.3.3 Let X → S be a map of Fp -schemes. For any n 0, the S-map FX /S,n in (A.3.2) is the n-fold relative Frobenius morphism of X over S, and for n = 1 it is the relative Frobenius morphism of X over S. If S = Spec k for a ﬁeld k we write FX /k,n and FX /k , and when G is a smooth k-group we call FG/k : G → G (p) the Frobenius isogeny of G over k. In view of the preceding discussion, for any k-group G the map FG/k,n is a k-group homomorphism whose formation commutes with any extension of scalars on k, and it is functorial in G. In the smooth case it is an isogeny of degree pn dim G . It is not generally ﬂat in the non-smooth case (e.g., FG/k vanishes for G = αp and G = μp ). Example A.3.4 Let G be an afﬁne group scheme of ﬁnite type over a ﬁeld n k/Fp . Choose a k-subgroup inclusion j : G → GLN , so G (p ) is likewise n identiﬁed with a k-subgroup of GLN via j (p ) . In this way, FG/k,n is induced by n ) of GLN that is deﬁned by the pn -power map on the endomorphism (FGL N /Fp k matrix entries. In some general considerations with group schemes of ﬁnite type over a ﬁeld it is not necessary to restrict attention to the afﬁne case, and for such purposes the following general result is useful. Proposition A.3.5 Any group scheme G of ﬁnite type over a ﬁeld k is quasiprojective. Proof. The case of smooth G is originally due to Chow [Chow], and a modern proof is given in [Con1, Cor. 1.2] as an application of descent theory and

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Theorem A.3.7 below (which does not rely on the result we are presently proving). To handle the non-smooth case we may assume char(k) = p > 0. For n n 0, consider the n-fold relative Frobenius morphism FG/k,n : G → G (p ) . The kernel is a ﬁnite (inﬁnitesimal) normal k-subgroup scheme of G, so the quotient map G → G/ ker(FG/k,n ) is a ﬁnite map. Hence, it sufﬁces to prove quasi-projectivity of G/ ker(FG/k,n ) for some n 0. By [SGA3, VIIA , 8.3], if n is sufﬁciently large then this quotient of G is smooth, so we are done. 2 Remark A.3.6 Although quasi-projectivity is a sufﬁcient criterion for the existence of certain kinds of quotients, in view of the use of quotients G/H in the proof of Proposition A.3.5 (even in the smooth case) we note that the construction of such quotients in [SGA3, VIA , Thm. 3.2(iv)] does not use quasi-projectivity of G. To conclude this section, we state an important structure theorem of Chevalley over perfect ﬁelds and a less widely known variation on it over any ﬁeld. Theorem A.3.7 (Chevalley) Let G be a smooth connected group over a perfect ﬁeld k. There is a unique short exact sequence of smooth connected k-groups 1→H →G→A→1 with A an abelian variety and H afﬁne. Proof. This is proved in [Chev]; see [Con1] for a modern exposition.

2

Theorem A.3.7 is not applicable over imperfect ﬁelds if we insist on working with smooth H . Here are counterexamples, resting on our discussion of Weil restriction in §A.5. Example A.3.8 Let k be an imperfect ﬁeld, and k /k a nontrivial purely inseparable ﬁnite extension. For any nonzero abelian variety A over k , the Weil restriction G = R k /k (A ) makes sense as a k-scheme since A is quasi-projective over k . By Proposition A.5.11, G is a smooth connected commutative k-group. There is no nontrivial smooth connected afﬁne k-subgroup H in G. Indeed the inclusion map j : H → G corresponds to a k -group map Hk → A that must be trivial (since Hk is a smooth connected afﬁne k -group and A is an abelian variety over k ). Hence, j is also trivial, so H = 1 as claimed. It follows that if Theorem A.3.7 were applicable to G then G would have to be an abelian variety. However, G is not an abelian variety. One reason is that the natural map Gk → A has nontrivial smooth connected unipotent kernel, by Proposition A.5.11; another reason is that R k /k applied to a smooth proper k -scheme of positive dimension is never proper (Example A.5.6). Thus, G is a counterexample to the conclusion of Theorem A.3.7 over k when the perfectness hypothesis on k is dropped. In [BLR, 9.2/1] there is given a version of Theorem A.3.7 over general ﬁelds by allowing the “afﬁne part” to be non-smooth. For Galois cohomological and

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other purposes it is useful to avoid working with non-smooth groups. Thus, the following variant on Theorem A.3.7 (in which, surprisingly, the afﬁne part goes on the right instead of on the left) is extremely useful. This result applies over arbitrary ﬁelds and only uses smooth groups, so it deserves to be better-known. Theorem A.3.9 (Anti-Chevalley Theorem) Let k be a ﬁeld, and G a smooth connected k-group. Deﬁne G aff = Spec(O(G)) equipped with its natural k-group scheme structure. This afﬁne k-group is smooth (in particular, of ﬁnite type) and the natural map G → G aff ﬁts into a short exact sequence of smooth connected k-groups 1 → Z → G → G aff → 1 with O(Z) = k. The k-group Z is central in G and if char(k) > 0 then Z is semi-abelian. Proof. This is [DG, III, §3, 8.2, 8.3] apart from the fact that Z is semi-abelian when char(k) > 0. It remains to show that if char(k) = p > 0 then any smooth connected commutative k-group H satisfying O(H ) = k is semi-abelian (i.e., an extension of an abelian variety by a torus). This is proved in [Bri, Prop. 2.2] and [SS]; for the convenience of the reader, we now give another proof. It sufﬁces to check the semi-abelian property over an algebraic closure of the base ﬁeld, so we may assume that k is algebraically closed. Hence, Theorem A.3.7 is applicable to H , and we have to show that the maximal smooth connected afﬁne subgroup of H is a torus. Since H is commutative, it contains a unique maximal torus T . By Theorem A.3.7, H /T is an extension of an abelian variety A by a smooth connected unipotent group U . The k-group U is killed by pN for sufﬁciently large N , so pN -multiplication on the commutative H gives an isogeny-section s for the quotient H /T A. Hence, (H /T )/s(A) is a unipotent quotient of H . But O(H ) = k, so H has no non-constant maps to an afﬁne k-scheme. Thus, (H /T )/s(A) = 1. This forces U = 1 for dimension reasons. 2

A.4 Split reductive groups: Existence, Isomorphism, and Isogeny Theorems A connected reductive group G over a ﬁeld k is called split if it contains a k-split maximal k-torus T . We will call such (G, T ) a split reductive pair over k. If (G, T ) is a split reductive pair over k then every root group for Tk in Gk is deﬁned over k, and so every Borel subgroup of Gk containing Tk is deﬁned over k [Bo2, 18.7].

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Example A.4.1 Let G be a k-split connected semisimple group of rank 1, say with T a k-split maximal k-torus (of dimension 1). We claim that (G, T ) is k-isomorphic to either SL2 or PGL2 equipped with its diagonal k-torus. Since SL2 has μ2 as its only normal subgroup scheme of order 2, it is equivalent to construct an isogeny SL2 → G of degree dividing 2 such that the diagonal k-torus is carried onto T . Over an algebraically closed ﬁeld, this is [Spr, 7.2.4]. In general we have the result for (Gk , Tk ), so (G, T ) = {±a}. To handle the case of a general ground ﬁeld, we claim that the proof in [Spr, 7.2.4] can be applied in such generality. This rests on several observations. First, for N = NG (T ) there exists n ∈ N (k) representing the nontrivial class in N /T = Z/2Z. This follows from the vanishing of H1 (k, T ) (with T GL1 ); note that n−1 Ua n = U−a . Consider the map U−a × T × Ua → G deﬁned by multiplication. We claim that this is an isomorphism onto G − n−1 B, where B = T × Ua is PG (λ) for any λ ∈ X∗ (T ) such that a, λ > 0. It sufﬁces to check this property over k as well, in which case the veriﬁcation is explained in the proof of [Spr, 7.2.4]. Finally, the k-groups U±a are smooth connected unipotent of dimension 1 (as may be checked over k), and by Theorem B.4.3 each is k-split due to the nontrivial action by T . Thus, we can choose a k-group isomorphism u : Ga Ua . The proof of [Spr, 7.2.4] therefore applies over k to yield a k-rational homomorphism from SL2 to G. By Galois descent from ks (over which there is a Zariski-dense locus of rational points), this rational homomorphism uniquely extends to a k-group homomorphism, and the proof of [Spr, 7.2.4] shows that this is an isogeny with degree dividing 2 that carries the diagonal torus onto T , as desired. (Note that these properties can be proved by working over k.) Every G becomes split over a ﬁnite separable extension of the ground ﬁeld (since each maximal torus over the ground ﬁeld is split by such an extension). In particular, if k = ks then every connected reductive k-group is k-split. The split reductive pairs over a ﬁeld are classiﬁed in terms of their root data (which are reduced). This is made precise by the Existence and Isomorphism Theorems for split connected reductive groups over a ﬁeld (which we will need over a general separably closed ﬁeld), and there is a stronger formulation that records information about isogenies. The literature on these topics (aside from [SGA3, XXV]) generally treats only an algebraically closed ﬁeld, so below we explain how to deduce the Existense, Isomorphism, and Isogeny Theorems over a general ﬁeld from the case of an algebraically closed ﬁeld. Before we address the main theorems in this direction, we review the appropriate notion of isogeny for root data.

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Deﬁnition A.4.2 Let p 1 be a prime number or 1. A p-morphism of reduced root data ∨

∨

(X , R , X , R ) → (X , R, X ∨ , R∨ ) is a triple ( f , ι, q) with f : X → X an injective Z-linear map having ﬁnite cokernel (i.e., fQ is an isomorphism), ι : a → a a bijection from R to R (the other direction!), and q : R → {pn }n0 a map of sets such that f (a ) = q(a)a, f ∨ (a∨ ) = q(a)a

∨

for all a ∈ R. A composition of p-morphisms is deﬁned by composition in f and ι and multiplication in q. If q(a) = 1 for all a then ( f , ι, q) is called central. If p > 1, f (X ) = pX , and q(a) = p for all a ∈ R then ( f , ι, q) is called a Frobenius isogeny. Remark A.4.3 The injectivity of f implies that (−a) is proportional to a , so (−a) = −a since ι is injective and the root data are assumed to be reduced. It then also follows that q(−a) = q(a) in such cases. The terminology in Deﬁnition A.4.2 follows [Spr]; in [SGA3, XXI, 6.8.1] the notion of p-morphism does not require fQ to be an isomorphism. (This extra generality corresponds to considering maps (G, T ) → (G , T ) that may only be an isogeny between derived groups.) Example A.4.4 Consider split reductive pairs (G, T ) and (G , T ) over a ﬁeld k with characteristic exponent p 1, and let (X , R, X ∨ , R∨ ) = R(G, T ) and (X , R , X ∨ , R ∨ ) = R(G , T ) be the corresponding root data. Let φ : G → G be a k-isogeny such that φ(T ) = T . We now explain how to naturally associate to φ a p-morphism ( f , ι, q) between the corresponding root data. The map f := X(φ) : X → X is injective with ﬁnite cokernel since φ : T → T is an isogeny of k-tori. For each a ∈ R let Ua be the corresponding root group in G, and likewise let Ua be the root group in G associated to any a ∈ R . Root groups are uniquely characterized as nontrivial smooth connected unipotent k-subgroups normalized by the split maximal torus and on which a codimension-1 torus acts trivially, so for every a ∈ R there is a unique a ∈ R such that φ(Ua ) = Ua . Since φ is an isogeny, it follows that the map ι : R → R deﬁned by a → a is injective, so it is bijective since the number of roots is determined by the dimension of the group and the dimension of the maximal tori. The isogeny φ must respect the unique codimension-1 subtori Ta = (ker a)0red and Ta = (ker a )0red that act trivially on the respective root groups Ua and Ua ,

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which is to say φ(Ta ) = Ta . In particular, (−a) = −a since T−a = Ta and are the only two root groups on which T acts trivially. By using conjugaU±a a tion against suitable Weyl elements, the isogenies Ua → Ua and U−a → U−a have the same degree, which must be a p-power q(a) = q(−a). Upon choosing k-group isomorphisms xa : Ga Ua and xa : Ga Ua , we have φ(xa (t)) = xa (ca t q(a) ) for a unique ca ∈ k × . Thus, f (a ) = q(a)a for all a. Let Ga be the semisimple rank-1 group generated by Ua and U−a , and deﬁne Ga similarly in terms of Ua and U−a = U(−a) . The map f induces an isogeny Ga → Ga which restricts to an isogeny T ∩ Ga → T ∩ Ga between their maximal tori. In view of the construction of coroots, it follows that f ∨ (a∨ ) := φ ◦ a∨ is a rational multiple of a ∨ . We claim that the multiplier is q(a): ∨

f ∨ (a∨ ) = q(a)a .

(A.4.1)

Indeed, writing f ∨ (a∨ ) = ca ∨ with c ∈ Q, we have ∨

2c = ca , a = a , f ∨ (a∨ ) = f (a ), a∨ = q(a)a, a∨ = 2q(a). As a prelude for understanding the proofs of the Isomorphism and Isogeny Theorems, we now make a good choice of normalization of the parameterizations xa and xa so that we can relate ca and c−a . Since we are working with split reductive pairs, so Example A.4.1 applies to the pairs (Ga , T ∩ Ga ) and (Ga , T ∩ Ga ), we can choose the root group parameterizations xa and xa so that the elements na := xa (1)x−a (−1)xa (1) ∈ G(k), na := xa (1)x−a (−1)xa (1) ∈ G (k)

(A.4.2)

respectively lie in NG (T )(k) and NG (T )(k) and represent the reﬂections ra and ra in W (G, T )(k) and W (G , T )(k) respectively. By [Spr, 8.1.4(iv)] (whose proof is valid for a split reductive pair over any ground ﬁeld, due to Example A.4.1, and amounts to calculations with SL2 ), these requirements determine the collections {xa } and {xa } up to multiplication on Ga by unique constants ua , ua ∈ k × satisfying ua u−a = 1 and ua u−a = 1 for all a and a . Thus, if q(a) = 1 for all a then ca c−a = 1 for all a. We shall now prove that ca c−a = 1 for all a ∈ R, regardless of the value of each q(a). To prove ca c−a = 1 for all a, we ﬁrst recall that by direct calculations with SL2 (as in [Spr, 8.1.4] and its proof), we have xa (t)x−a (−t −1 )xa (t) = a∨ (t)na ∈ NG (T )

(A.4.3)

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for all geometric points t of GL1 . Applying φ then yields φ(xa (t))φ(x−a (−t −1 ))φ(xa (t)) ∈ NG (T ) with φ(xa (t)) = xa (ca t q(a) ) a non-vanishing geometric point of Ua . In particular, xa (ca t q(a) )φ(x−a (−t −1 ))xa (ca t q(a) ) ∈ NG (T )

(A.4.4)

with φ(x−a (−t −1 )) ∈ U(−a) = U−a a nontrivial element. But we also have

xa (ca t q(a) )x−a (−(ca t q(a) )−1 )xa (ca t q(a) ) ∈ NG (T ).

(A.4.5)

The properties (A.4.4) and (A.4.5) with middle terms nontrivial in U−a force the middle terms to be equal [Spr, 8.1.4(iii)]. That is, φ(x−a (−t −1 )) = x−a (−(ca t q(a) )−1 ). But there is another formula for this element:

φ(x−a (−t −1 )) = x(−a) (c−a (−t −1 )q(−a) ) = x−a (−c−a (t q(a) )−1 ), since (−1)q(a) = −1 in k. Comparing the two formulas for φ(x−a (−t −1 )), we get c−a = ca−1 as desired. We have associated to the isogeny φ : (G, T ) → (G , T ) a p-morphism f (φ) = ( f , ι, q). Before we state the Existence and Isomorphism Theorems over a general base ﬁeld, it is convenient to record a preliminary lemma that will reduce certain difﬁculties with roots to the case of simple positive roots. Lemma A.4.5 Let G be a split connected reductive group over a ﬁeld k, and T a k-split maximal k-torus. For each a ∈ (G, T ), let Ua be the corresponding root group and let Ga be the corresponding k-subgroup of type A1 generated by Ua and U−a . For any choice of positive system of roots in (G, T ), the k-subgroups Ga for simple positive a generate D (G). Proof. We may and do assume that k is algebraically closed (since the structure theory may be more familiar to the reader in that case). Since all Ga are contained in D (G), we may replace G with D (G) and replace T with the k-split subtorus T ∩ D (G) to reduce to the case when G is semisimple. Let H be the smooth connected closed k-subgroup of G generated by the k-subgroups Ga . We have T ⊆ H since T is generated by the tori a∨ (GL1 ) for simple positive a (as the corresponding coroots a∨ generate a lattice of ﬁnite index in X∗ (T ), due to the semisimplicity of G).

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In view of the structure of an open Bruhat cell of (G, T ), it sufﬁces to prove that Gb ⊆ H for all b. Since every root is carried to a simple positive root by some element of W (G, T ), it sufﬁces for H to contain a set of representatives for W (G, T ). The Weyl group W (G, T ) is generated by the reﬂections ra for simple positive a, so it is enough to ﬁnd a representative in H for each such a. For every root a, the subgroup Ga contains a representative na for ra . Thus, na ∈ Ga ⊆ H for simple positive a. 2 Theorem A.4.6 (Existence and Isomorphism Theorems) Let k be a ﬁeld. A split reductive pair (G, T ) over k is determined up to isomorphism by its reduced root datum, and the isomorphism is unique up to the natural action αt of a unique t ∈ (T /ZG )(k). Conversely, every reduced root datum arises from a split reductive pair (G, T ) over k. Proof. We ﬁrst address the uniqueness of t ∈ (T /ZG )(k). That is, if the t-action on G is trivial then we claim t = 1. It sufﬁces to check this when k = k, in which case t lifts to T (k), so the result is clear. Turning to the other assertions, the case of algebraically closed k is proved in [Spr, 9.6.2, 10.1.1]. To establish the result over a general ﬁeld k, ﬁrst consider the Existence Theorem. Every separably closed ﬁeld contains an algebraic closure of its prime ﬁeld, so the Existence Theorem over separably closed ﬁelds is clear. This will be sufﬁcient for our needs: the Existence Theorem and its consequences are used in this monograph only over a separably closed ﬁeld. For the reader who is interested in the Existence Theorem directly over an arbitrary ﬁeld, we just make some general remarks and give a reference for a complete proof. Consideration with central isogenies reduces the problem to the case of irreducible and reduced root data that are “simply connected” in the sense that R∨ generates X ∨ . This reduces the problem to the construction of absolutely simple and simply connected split groups over every ﬁeld. For the classical types one can write down the groups explicitly, and in the other cases it gets messy to argue by hand. For a uniform approach, a lot of work is required to overcome subtleties with “structure constants”. We refer the reader to [SGA3, XXV, §§2–3] for a proof of the Existence Theorem over Z (and in particular, over prime ﬁelds). We now give a complete proof of the Isomorphism Theorem over a general ﬁeld k as a consequence of the result over k. This will be an application of descent theory. Choose split reductive pairs (G, T ) and (G , T ) over k and an isomorphism ∨

∨

f : (X , R , X , R ) (X , R, X ∨ , R∨ ) between their root data. We seek a k-isomorphism φ : (G, T ) (G , T ), unique up to (T /ZG )(k)-conjugation, such that f = f (φ).

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Let us ﬁrst address uniqueness. In view of the result over K := k, we have to show that if t ∈ (T /ZG )(K) is such that the t -action αt on GK is deﬁned over k then t ∈ (T /ZG )(k). Since ZG is the scheme-theoretic center of G , T /ZG is a subfunctor of the automorphism functor Aut(G ) of G . That is, for any k-algebra R, an element of (T /ZG )(R) is uniquely determined by its action on GR . Hence, the two pullback maps (T /ZG )(K) ⇒ (T /ZG )(K ⊗k K) carry t to the same point since the analogue holds for the pullback maps ) Aut(GK ) ⇒ Aut(GK⊗ kK

applied to the automorphism αt (which we assume is deﬁned over k). By descent theory, it follows that t ∈ (T /ZG )(k), as desired. For the existence of the k-isomorphism φ inducing f , if we can settle the problem over ks then the uniqueness result implies that the obstruction to further descent down to k is a class in H1 (ks /k, T /ZG ). But T /ZG is a k-split torus, so the obstruction vanishes. Thus, it sufﬁces to consider the case of a separably closed base ﬁeld. We will only use that k is inﬁnite. By the result over K we get a K-isomorphism φ such that f = f (φ). We seek t ∈ (T /ZG )(K) such that αt ◦ φ is deﬁned over k. The isomorphism TK TK induced by φ must descend to a k-isomorphism T T since the k-tori are k-split. For any a ∈ (G, T ), let Ta = (ker a)0red . The root group (Ua )K is the unique nontrivial smooth connected unipotent K-subgroup of GK that is normalized by TK , centralized by (Ta )K , and has a single weight w on its Lie algebra which moreover satisﬁes w, a∨ > 0. We have a similar description for (Ua )K in terms of TK , where f (a ) = a. Since f ∨ (a∨ ) = a ∨ , it follows that φ carries (Ua )K isomorphically onto (Ua )K . Now choose normalized k-isomorphisms xa : Ga Ua and xa : Ga Ua for all a ∈ R and a ∈ R in accordance with the requirements as in (A.4.2). The parameterizations xa and xa convert φ : (Ua )K (Ua )K into multiplication by some unit ca ∈ K × . That is, (xa )K (ca t) = φ((xa )K (t)) for t ∈ Ga . Since f ∨ (a∨ ) = a ∨ , it follows from the uniqueness argument in Example A.4.4 (resting on [Spr, 8.1.4(iv)]) that c−a ca = 1 for all a. Fixing a positive system of roots + in R = (G , T ), the simple positive roots form a basis of the character group of T /ZG . It follows that there is a unique t ∈ (T /ZG )(K) such that a (t ) = ca−1 for all simple positive a . By replacing φ with αt ◦ φ we get to the case when ca = 1 for all simple positive a, so likewise c−a = 1 for such a. We have reached the situation in which φ ◦ (xa )K = (xa )K for all simple positive a and simple negative a. Letting Ga = Ua , U−a be the type-A1 subgroup associated to a root a ∈ R, the K-group isomorphism φ : (Ga )K

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(Ga )K is deﬁned over k between open Bruhat cells of Ga and Ga , so by Zariskidensity of these open Bruhat cells (and descent theory) the restriction of φ to (Ga )K is deﬁned over k for all simple positive a. By Lemma A.4.5, the k-subgroups Ga for simple positive a generate D (G). Thus, the groups Ga (k) for such a generate a subgroup ⊆ D (G)(k) that is Zariski-dense (as k is inﬁnite and the Ga are unirational). The map φ must carry into D (G )(k), so by descent theory and Zariski-density considerations it follows that φ|D (G)K descends to a k-homomorphism D (G) → G . If Z ⊆ T is the maximal central k-torus then we have also seen that φ|ZK descends to a k-homomorphism Z → T → G . Let π : Z × D (G) → G denote the canonical central isogeny over k. We have shown that φ ◦ πK is deﬁned over k. Since π is faithfully ﬂat, it follows by descent theory that φ is also deﬁned over k. This settles the Isomorphism Theorem for split reductive pairs over any ﬁeld. 2 As an application of the uniqueness aspect of the Isomorphism Theorem, we get the following corollary concerning automorphisms of connected reductive groups. For the convenience of the reader we give direct proofs over a ﬁeld, but a much more general version can be deduced from the structure of the automorphism group scheme of a connected reductive group [SGA3, XXIV, 1.7]. Corollary A.4.7 Let G be a connected reductive group over a ﬁeld k, and consider the natural left action of G/ZG on G induced by conjugation. (1) Assume G is absolutely simple and semisimple, and assume that the Dynkin diagram of Gks has no nontrivial automorphisms. Every k-automorphism of G is induced by the action of a unique element of (G/ZG )(k). (2) Let H be a smooth connected k-group equipped with a left action H ×G → G on G as a k-group scheme (i.e., a homomorphism of group functors H → Aut(G) on the category of k-schemes or k-algebras). This action uniquely factors as a k-homomorphism H → G/ZG followed by the natural action of G/ZG on G. Part (2) of this corollary implies that the “natural” algebraic structure imposed on the automorphism group of a connected reductive group G over a ﬁeld (as an extension of an étale group by G/ZG ) has the expected universal property among smooth groups acting on G. Proof. First we prove (1). In view of the uniqueness claim, we may assume k = ks . The scheme-theoretic center ZG of G represents the kernel of the homomorphism of group functors G → Aut(G), so G/ZG is a subgroup functor of Aut(G). In particular, any g ∈ (G/ZG )(k) is uniquely determined by its

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associated automorphism of G. We have to prove that all k-automorphisms of G are obtained from such g. Fix a maximal k-torus T in G. By G(k)-conjugacy of maximal k-tori (Proposition A.2.10), we can restrict attention to automorphisms f of G that satisfy f (T ) = T . Hence, f induces an automorphism of the root system of (G, T ). Fix a positive system of roots + in (G, T ). Since the group W (G, T )(k) = NG (T )(k)/T (k) acts transitively on the set of positive systems of roots in (G, T ), by composing f with conjugation by a suitable element of NG (T )(k) we can assume that f (+ ) = + . The hypothesis concerning automorphisms of the Dynkin diagram implies that the f -action on + must ﬁx the simple positive roots pointwise, and so f acts as the identity on (G, T ). Since (G, T ) generates a ﬁnite-index subgroup of X(T ), f acts as the identity on X(T ). Hence, f acts as the identity on the root datum, so by the uniqueness aspect of the Isomorphism Theorem (applied in the case of automorphisms) we see that f is the action by a point in (T /ZG )(k). This proves (1). To prove (2), let Z be the maximal central torus in G, so the H -action on G = Z · D (G) restricts to actions on Z and D (G). The automorphism scheme of a torus is étale, so the H -action on Z is trivial due to the connectedness of H . Since G and D (G) have the same maximal adjoint quotient, we can replace G with D (G) to reduce to the case when G is semisimple. Also, since G has no nontrivial automorphisms that induce the identity on its maximal adjoint quotient, it sufﬁces to treat the case when G is adjoint. We now reduce to the case when k is algebraically closed (so that we can use general theorems about Borel subgroups in smooth connected afﬁne groups). Suppose we can construct a k-homomorphism f : Hk → (G/ZG )k such that when the given natural transformation of group functors H → Aut(G) on the category of k-schemes (or k-algebras) is viewed on the category of k-schemes (or k-algebras), it factors as f

Hk → (G/ZG )k → Aut(Gk ) = Aut(G)k .

(A.4.6)

Since the second map in (A.4.6) is a subfunctor inclusion, such a map f is unique and remains unique after extension of the base ring from k to any k-algebra. In particular, over k ⊗k k we get the equality of pullback maps p1∗ ( f ) = p2∗ ( f ) along the two projections Spec(k ⊗k k) ⇒ Spec k since the composite natural transformation (A.4.6) is deﬁned over k. Thus, by descent theory, such an f must descend to a k-homomorphism and such a descent does the job. For the remainder of the argument, we now may and do assume k is algebraically closed. Let us next reduce the problem to a group-theoretic one on geometric points. Assume we can prove, over any algebraically closed ﬁeld F/k, that the map of groups H (F) → Aut(GF ) factors through the subgroup (G/ZG )(F) of automorphisms arising from the faithful action of (G/ZG )(F). Taking F = k yields

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a group homomorphism H (k) → (G/ZG )(k), and the problem is to prove that it is algebraic (i.e., arises from a k-homomorphism H → G/ZG ). Consider the generic point η ∈ H (k(H )). Applying our group-theoretic hypothesis to a geometric point over η valued in an algebraically closed extension K of k(H ), we get a point g ∈ (G/ZG )(K) whose image in Aut(GK ) = Aut(G)(K) coincides with the image of η. Since η lands in Aut(G)(k(H )), so does g. But G/ZG is a subfunctor of Aut(G), so g ∈ (G/ZG )(k(H )). (More speciﬁcally, the two pullbacks of g into (G/ZG )(K ⊗k(H ) K) coincide, so g ∈ (G/ZG )(k(H )) by descent theory.) In other words, g corresponds to a rational map of varieties from H to G/ZG over k. Let U ⊆ H be a dense open subscheme on which this rational map is deﬁned, so the composite map g×1

U × G −→ (G/ZG ) × G −→ G (the second step being the action map) coincides with the restriction of the given action map H × G → G since such equality holds after pullback along η : Spec k(H ) → U (due to how g was chosen). Hence, the group-theoretic map H (k) → (G/ZG )(k) is “algebraic” on the dense open U ⊆ H , so by translations it is everywhere algebraic as required. It remains to prove, over any algebraically closed ﬁeld k, that for each h ∈ H (k) the h-action on G arises from the action of some element of (G/ZG )(k). Since H (k) is generated by the subgroups B(k) for Borel subgroups B ⊆ H (as H is a smooth connected afﬁne k-group), it sufﬁces to treat such B in place of H . That is, we can assume H is solvable. Consider the semi-direct product G = G H using the action of H on G. Since H is solvable, we can choose a Borel subgroup B of G containing H . Hence, B = B H for a smooth connected solvable k-subgroup B of G which must be a Borel subgroup. Explicitly, B = B ∩ G inside of G , with H normalizing B. Since H is smooth, the H -action on B induces an H -action on the maximal reductive quotient B/Ru (B) that is a torus. By connectedness of H , its action on this torus quotient must be trivial. Fix a maximal torus T in B (so it is also maximal in G). For any h ∈ H (k), the automorphism of B induced by h carries T to another maximal torus h.T of B, so there exists b(h) ∈ B(k) such that b(h)Tb(h)−1 = h.T . The composition φ of the h-action on G with conjugation by b(h)−1 is an automorphism of G that preserves the pair (B, T ). The induced action by φ on T is trivial, as we can check by using equivariance with respect to the isomorphism T B/Ru (B). Thus, φ induces the identity on the root datum of (G, T ), so by the uniqueness aspect of the Isomorphism Theorem it must arise from the action of a point in (T /ZG )(k). 2 Before we turn to the Isogeny Theorem over a general ﬁeld, we give an application of Corollary A.4.7 to the behavior of maximal reductive quotients

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under passage to smooth connected normal subgroups or torus centralizers. Consider a smooth connected afﬁne group H over a ﬁeld k such that the subgroup Ru (Hk ) ⊆ Hk is deﬁned over k, or equivalently Ru,k (H )k = Ru (Hk ) (as happens when k = k). Let H be a smooth connected k-subgroup of H that is either normal or is the centralizer of a k-torus in H . (By Proposition A.2.5, every torus centralizer subgroup in H is smooth and connected.) The image of H in H red := H /Ru,k (H ) is normal when H is normal in H , and likewise the image of H in H red is a torus centralizer when H is a torus centralizer in H (Proposition A.2.8). But a smooth connected subgroup of a connected reductive group is reductive when it is normal or a torus centralizer [Bo2, 8.18, 13.19], so in both cases Ru,k (H ) has vanishing image in H red ; i.e., Ru,k (H ) ⊆ H ∩ Ru,k (H ). A ﬁner relation holds: Proposition A.4.8 Let H be a smooth connected afﬁne group over a ﬁeld k such that Ru,k (H )k = Ru (Hk ). Let H ⊆ H be a smooth connected k-subgroup. Assume either that H is normal in H or that it is the centralizer of a k-torus in H . The inclusion Ru,k (H ) ⊆ H ∩ Ru,k (H ) is an equality inside of H and is a k-descent of the subgroup Ru (H ) ⊆ H . In particular, the natural homok

k

morphism φ : H red → H red between maximal reductive quotients over k has trivial kernel. Moreover, the induced map H ss → H ss has ﬁnite central kernel. In particular, if k is perfect then the formation of the maximal adjoint semisimple quotient is an exact functor on the category of smooth connected afﬁne k-groups. Proof. Since the ﬁrst claim says exactly that H ∩ Ru,k (H ) is smooth and connected, we may and do assume that k is algebraically closed. First we treat the case when H is normal in H , so Ru (H ) is normal in H . Clearly ker φ = (H ∩ Ru (H ))/Ru (H ), so passing to H /Ru (H ) and H /Ru (H ) lets us assume that H is reductive. Consider the action of H on H by conjugation. Since H is connected, its action on H must factor as a k-homomorphism f : H → H /ZH followed by the natural action of the adjoint quotient H /ZH on H (Corollary A.4.7(2)). But f |H is the natural projection (since the H -action is through conjugation on H within H ), so f is surjective. Hence, f carries smooth normal k-subgroups to smooth normal k-subgroups, so the smooth connected unipotent normal k-subgroup Ru (H ) in H has smooth connected unipotent normal image in H /ZH under f . This image must be trivial since H /ZH is reductive, so Ru (H ) centralizes H . Therefore, the unipotent group scheme ker φ = H ∩ Ru (H ) is central in the connected reductive group H . But ZH is a group scheme of multiplicative type, so its only unipotent subgroup scheme is {1}. Thus, ker φ = 1.

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Now suppose H = ZH (S) for a torus S in H . We have to prove that the containment Ru (H ) ⊆ H ∩ Ru (H ) of unipotent normal subgroup schemes of H is an equality. It sufﬁces to show that the unipotent group scheme H ∩Ru (H ) is smooth and connected. This intersection is the centralizer in the smooth connected unipotent group Ru (H ) under the natural conjugation action by the torus S. For any torus action on a smooth connected afﬁne group, the centralizer of the action is smooth and connected (by applying Proposition A.2.5 to a suitable semidirect product). Finally, we need to show that in both cases H ss → H ss has ﬁnite central kernel. By the parts already proved, we can replace H → H with H red → H red , so H and H are reductive. The maps D (H ) → H ss and D (H ) → H ss are therefore central isogenies, so the kernel of H ss → H ss is the image of the part of D (H ) that lies in the ﬁnite central subgroup ker(D (H ) → H ss ) of D (H ). Hence, we get the required ﬁniteness and centrality. 2 Example A.4.9 To illustrate Proposition A.4.8, let H be a smooth connected afﬁne group over a ﬁeld k and let H = D (H ). Since H = D (Hk ), applying k

the proposition to Hk and H gives that D (H )ss = D (H red ) inside of H red . k

k

k

k

Here is the Isogeny Theorem over a general base ﬁeld; it is a “converse” to Example A.4.4. Theorem A.4.10 (Isogeny Theorem) Let k be a ﬁeld with characteristic exponent p 1, and let (G, T ) and (G , T ) be two split reductive pairs over k. Every p-morphism R(G , T ) → R(G, T ) between the root data has the form f (φ) for a k-isogeny φ : G → G carrying T onto T , with φ unique up to the (T /ZG )(k)-action on G . Moreover, φ is central if and only if f (φ) is central, φ is an isomorphism if and only if f (φ) is an isomorphism, and φ is a Frobenius isogeny (i.e., ker φ = ker FG/k ) if and only if f (φ) is a Frobenius isogeny. Proof. The existence and uniqueness aspects for the isogeny over an algebraically closed base ﬁeld are proved in [Spr, 9.6.5]. Also, the characterization of isomorphisms is an immediate consequence of the characterization of central isogenies. To prove the characterization of central and Frobenius isogenies over a general base ﬁeld it sufﬁces to consider the case of an algebraically closed ﬁeld. If p > 1 then it is clear that the Frobenius isogeny of G induces the Frobenius p-morphism. Thus, since an isogeny of groups over an algebraically closed ﬁeld k is determined by the isogeny of root data up to an automorphism of the target, the case of Frobenius isogenies is settled over such k. Continuing to assume k is algebraically closed, f (φ) is central if and only if φ restricts to an isomorphism between root groups; i.e., Lie(kerφ) has trivial

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T -action (equivalently, no nontrivial T -weights). Thus, if φ is a central isogeny then f (φ) is central. Conversely, assume f (φ) is central. The inﬁnitesimal (ker φ)0 has the form Spec R for a ﬁnite local k-algebra (R, m) such that the T -action on the k-algebra R is trivial on m/m2 . The T -action is therefore trivial on mi /mi+1 for all i 0, so the T -action on R is unipotent. But T is a torus, so the T -action on R is trivial. All tori of G are conjugate, so they all centralize the normal (ker φ)0 , and hence so does G (as G is generated by its maximal tori). Since G is a central quotient of Z × D (G) for a central k-torus Z of G and the connected semisimple D (G) that is its own derived group, ker φ is central in G (cf. Lemma 5.3.2). That is, φ is central when f (φ) is central. It remains to address the existence and uniqueness (up to suitable conjugation) for φ in the case of a general base ﬁeld. That is, given a p-morphism ∨

∨

f : (X , R , X , R ) → (X , R, X ∨ , R∨ ) we seek to construct a k-isogeny φ : (G, T ) → (G , T ), unique up to the (T /ZG )(k)-action, such that f (φ) = f . By the same cohomological vanishing argument with split tori as in the proof of Theorem A.4.6, it sufﬁces to treat the case of separably closed k. Let K = k, so the settled case over K provides an isogeny φ : GK → GK carrying TK onto TK and satisfying f (φ) = f . We seek t ∈ (T /ZG )(K) such that the composition of φ with the action of t is deﬁned over k. Since T and T are k-split tori, the induced K-isogeny φ : TK → TK descends to a k-isogeny T → T . By deﬁnition of f being a p-morphism, we have f (a ) = q(a)a and f ∨ (a∨ ) = q(a)a ∨ for all a ∈ R, and by Remark A.4.3 we have q(a) = q(−a) and (−a) = −a . Thus, the calculations in Example A.4.4 (especially the proof there that ca c−a = 1 for all a) allow us to use the method of proof of the Isomorphism Theorem over a general base ﬁeld (Theorem A.4.6) to prove the Isogeny Theorem over a general base ﬁeld; essentially no changes are required in the argument, apart from some intervention of q(a)-powers in some formulas. 2 Recall that a connected semisimple group G over a ﬁeld k is called simply connected if it admits no nontrivial central cover (i.e., any central k-isogeny G → G from a connected semisimple k-group is an isomorphism). Corollary A.4.11 Let k be a ﬁeld. (1) Let G be a connected semisimple k-group. There is a central isogeny π : → G from a connected semisimple k-group G that is simply connected. G The pair (G, π ) is unique up to unique isomorphism, and its formation commutes with any extension of the base ﬁeld.

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(2) If k /k is an extension of ﬁelds then any split reductive pair (G , T ) over k descends to a split reductive pair (G, T ) over k. π) is called the simply connected central cover of G. The pair (G, Proof. Part (2) is immediate from the Existence and Isomorphism Theorems, since a split reductive pair is classiﬁed up to isomorphism by its root datum. Also, in view of the claimed uniqueness up to unique isomorphism for the simply connected central cover, it sufﬁces (by Galois descent) to prove part (1) over separably closed ﬁelds. By Theorem A.4.10, a connected semisimple group is simply connected if and only if the coroots generate X ∨ . By taking the character lattice to be the Z-dual of the coroot lattice, we deduce the existence of the simply connected central cover up to an isomorphism. The isomorphism between any two simply connected central covers is unique because for any isogeny G → G between 2 smooth connected groups, there is no nontrivial G-automorphism of G . We also need a reﬁned version of the Isomorphism Theorem, resting on the notion of a pinning. Let (G, T ) be a split reductive pair over a ﬁeld k, and for each a ∈ (G, T ) let Ga denote the type-A1 subgroup of G generated by the root groups U±a . Let + be a positive system of roots in (G, T ), and let be the set of simple positive roots. Deﬁnition A.4.12 A pinning of (G, T , + ) consists of a collection of central k-isogenies {ϕa : SL2 → Ga }a∈ such that ϕa satisﬁes two properties: ϕa (diag(y, 1/y)) = a∨ (y) for all y ∈ GL1 , and ϕa carries the upper triangular unipotent subgroup U + onto the root group Ua ⊆ Ga (so ϕa : U + Ua ). Observe that if {ϕa }a∈ is a pinning then, for each a ∈ , the kernel of ϕa is either trivial or μ2 . Our deﬁnition of a pinning is not literally the same as the deﬁnition in [SGA3, XXIII, 1.1], but they encode the same information. (In one given a pinning in the above sense, the elements direction, Xa = Lie(ϕa ) 00 10 ∈ Lie(Ua ) for a ∈ are nonzero. The deﬁnition in [SGA3, XXIII, 1.1] uniquely recovers maps ϕa as above because Isom(Ga , Ua ) Isom(k, Lie(Ua )) and the action on SL2 by the diagonal torus D of PGL2 identiﬁes D(k) with Aut k (U + ). For our purposes, Deﬁnition A.4.12 is sufﬁcient.) We are interested in pinnings due to the following result, which is a reﬁnement of the Isomorphism Theorem.

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Theorem A.4.13 Let (G, T ) be a split reductive pair over a ﬁeld k. Let + ⊆ (G, T ) be a positive system of roots, and the set of simple positive roots. Let {ϕa }a∈ be a pinning. For any subﬁeld k0 ⊆ k there is a descent (G0 , T0 ) of (G, T ) to a split reductive pair over k0 such that the pinning maps ϕa : SL2 → G = (G0 )k are deﬁned over k0 for all a ∈ . That is, (T0 )k = T inside of (G0 )k = G and for each a ∈ the k0 -structure imposed on Ga from SL2 or PGL2 = SL2 /μ2 via ϕa coincides with the k0 -structure arising from the k0 -subgroup (G0 )a ⊆ G0 . Such a reductive pair (G0 , T0 ) is unique up to unique isomorphism. If G is semisimple and the maps ϕa are deﬁned over k0 relative to a k0 structure G0 on G then T automatically descends to a k0 -split maximal k0 -torus T0 in G0 : it is generated in G0 by the images of the restrictions of the ϕa to the diagonal torus in SL2 (as this same procedure in G over k recovers T , since G is semisimple). The same holds without semisimplicity if we assume the maps ϕa are deﬁned over k0 with respect to G0 and that the maximal central k0 -torus in G0 is k0 -split (as happens when k0 is separably closed). Proof. By the ﬁnal part of Corollary A.4.11, there exists a split reductive pair (G0 , T0 ) over k0 equipped with an isomorphism φ : (G, T ) ((G0 )k , (T0 )k ). Note that we are free to precompose φ with the action on G by any point in (T /ZG )(k). The problem is to ﬁnd such a φ so that each φ ◦ ϕa : SL2 → (G0 )k is deﬁned over k0 . Via our initial choice of φ we may identify the character and cocharacter groups of T with those of T0 . For each a ∈ , (G0 )a is k0 -isomorphic to SL2 or PGL2 . Fix such a k0 isomorphism ha so that it carries the k0 -split torus a∨ (GL1 ) ⊆ T0 over to the diagonal torus in SL2 or PGL2 . (We can do this since all rank-1 split tori in SL2 or PGL2 over a ﬁeld are conjugate via a rational point.) Consider the restriction of ha ◦φ ◦ϕa between the rank-1 split diagonal tori. Identify these tori with GL1 via y → diag(y, 1/y) in SL2 and y → diag(y, 1) in PGL2 , so the map ha ◦φ ◦ϕa between diagonal tori is either the standard map (i.e., the identity map of GL1 in the SL2 -case or the squaring map in the PGL2 -case) or its composite with inversion. Replacing ha by precomposition with a representative of the nontrivial element of the Weyl group of (G0 , a∨ (GL1 )) if necessary brings us to the case when ha ◦ φ ◦ ϕa is the standard map between diagonal tori for all a ∈ . The map ha ◦ φ ◦ ϕa restricts to a k-isomorphism between upper triangular unipotent subgroups. Using the standard coordinate on these copies of Ga , the k-isomorphism between them must be given by t → ca t for some ca ∈ k × . The torus T /ZG in the k-split adjoint quotient G/ZG has character group for which is a basis. Hence, there is a unique t ∈ (T /ZG )(k) such that a(t) = ca for all

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a ∈ . Composing φ with the action of t −1 leaves ha |a∨ (GL1 ) unaffected and brings us to the case ca = 1 for all a ∈ . We are now in the case that each ha ◦ φ ◦ ϕa is a central k-isogeny from SL2 to either SL2 or PGL2 such that it is the standard map between diagonal tori and the identity map on upper triangular unipotent subgroups (with their standard coordinate). It sufﬁces to check that the only such map is the standard map (which is deﬁned over the prime ﬁeld, let alone over k0 ). Passing to the quotient by the center in the PGL2 -case if necessary, this comes down to the well-known fact that a k-automorphism of SL2 or PGL2 that is the identity on the upper triangular Borel subgroup is the identity map. It remains to prove the uniqueness of (G0 , T0 ) up to unique isomorphism. Let (G0 , T0 ) be another such k0 -descent of (G, T ). We have to prove that the composite k-isomorphism φ : (G0 )k G (G0 )k is deﬁned over k0 . Since φ carries (T0 )k over to (T0 )k , and T0 and T0 are k0 -split, this isomorphism between maximal k-tori descends to a k0 -isomorphism T0 T0 . This identiﬁes X(T0 ) with X(T0 ), carrying (G0 , T0 ) to (G0 , T0 ). Since the pinning of (G, T ) is assumed to be deﬁned over k0 relative to both k0 -structures on (G, T ), for each root a ∈ (G0 , T0 ) = (G0 , T0 ) the restriction of φ between the a-root groups viewed over k is also deﬁned over k0 . Thus, the restriction of φ to the open Bruhat cells C and C associated to a common choice of positive system of roots is deﬁned over k0 . We can conclude by applying the general fact that if Y and Y are separated and geometrically integral schemes of ﬁnite type over k0 (such as G0 and G0 ) and U ⊆ Y and U ⊆ Y are dense open subschemes (such as C and C) then a k-morphism h : Yk → Yk is deﬁned over k0 if it restricts to a k-morphism hη : Uk → Uk that is deﬁned over k0 . (Indeed, the graph of h in Y × Y is the closure of the graph of hη in Uk × Uk . Thus, the graph in U × U of the k0 -descent of hη has closure in Y × Y that projects isomorphically onto Y since this can be checked over k. This closure in Y × Y is therefore the graph of a k0 -descent of h.) 2 Remark A.4.14 The method of proof of Theorem A.4.13 shows that assigning to each pinned k-split triple (G, T , {ϕa }a∈ ) the based root datum (R(G, T ), + , ∨+ ) is an equivalence of categories when we use isomorphisms on both sides. For a generalization allowing isogenies and more general base rings, see [SGA3, XXIII, 4.1]. An analogous theory over Z allows one to deﬁne ϕc for each root c by using the Weyl group action to move c into , but this involves some ambiguities in scaling by Z× = {±1} on root groups.

A.5 Weil restriction generalities Let B → B be a ﬁnite ﬂat map of noetherian rings, and X a quasi-projective B -scheme. The Weil restriction R B /B (X ) is a ﬁnite type B-scheme satisfying

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the universal property R B /B (X )(A) = X (B ⊗B A) for B-algebras A. For example, if X is the ﬁnal object Spec B then R B /B (X ) is the ﬁnal object Spec B (even if B = 0 and B = 0!). The existence and basic properties of Weil restriction are developed in [BLR, 7.6], to which we refer the reader for a proof of the existence of Weil restriction. One should think of the functor R B /B as analogous to the pushforward operation on sheaves. We will develop many properties of this functor in general, and then focus on special properties for Weil restriction relative to a ﬁnite extension of ﬁelds and for smooth afﬁne groups over a ﬁeld. (Other references for these special cases are [Pink] and [Oes, A.3] respectively.) The role of the quasi-projective (and not merely ﬁnite type) hypothesis on X is to ensure that any ﬁnite set of points in the underlying topological space of X is contained in an afﬁne open subscheme of X . Under this hypothesis, R B /B (X ) is constructed as a gluing of B-schemes R B /B (Ui ) for a suitable ﬁnite afﬁne open covering {Ui } of X . In particular, it is of ﬁnite type over B by construction. Beware that R B /B generally fails to commute with the formation of disjoint unions and Zariski-open covers; this is illustrated in Example A.5.3. Also, although R B /B does preserve quasi-projectivity, this is not seen from the construction; we address it in Proposition A.5.8. Weil restriction is right adjoint to base change along Spec(B ) → Spec(B) in the sense that there is a bijection HomB (Y , R B /B (X )) HomB (YB , X )

(A.5.1)

natural in X and any B-scheme Y (as we see by working Zariski-locally on open afﬁnes of Y ). In Proposition A.5.7 below we give an explicit description of the adjunction morphisms Y → R B /B (YB ) for quasi-projective B-schemes Y and R B /B (X )B → X for quasi-projective B -schemes X . The case of afﬁne X and Y is the one which is most essential for our purposes. However, nonafﬁne cases do naturally arise, at least in examples (such as Weil restriction of a quasi-projective coset space G/H ). If f : X → Y is a map between quasi-projective B -schemes then there is a naturally induced map R B /B ( f ) between their Weil restrictions. If f is a monomorphism them so is R B /B ( f ) (as for pushforwards of sheaves), but there are many interesting examples of surjective f for which R B /B ( f ) is not surjective. Such failure of surjectivity arises often in the study of pseudoreductive groups. The usefulness of Weil restriction in the study of groups over a ﬁeld rests on: Proposition Let k be a ﬁeld and k a nonzero ﬁnite reduced k-algebra A.5.1 (i.e., k = ki for a non-empty ﬁnite collection of ﬁnite extension ﬁelds ki /k).

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The Weil restriction G = R k /k (G ) exists as a k-scheme of ﬁnite type for any k -group scheme G of ﬁnite type, and it has a natural k-group structure. Proof. It sufﬁces to prove that G is quasi-projective over Spec(k ). Since Spec(k ) =

Spec(ki )

with {ki } the set of factor ﬁelds of k , it is equivalent to prove that G ⊗k ki is 2 quasi-projective over ki for all i. Apply Proposition A.3.5 to conclude. Returning to the case of a general noetherian base ring, some of the useful general properties of Weil restriction are recorded in the following result. Proposition A.5.2 Let B → B be a ﬁnite ﬂat map of noetherian rings, and X a quasi-projective B -scheme. (1) The formation of R B /B (X ) is naturally compatible with extension of scalars on B. (2) The B-scheme R B /B (X ) is afﬁne when X is afﬁne. (3) The formation of R B /B (X ) is compatible with ﬁber products in X ; i.e., for any pair of B -maps X1 , X2 ⇒ X3 between quasi-projective B -schemes, the natural map R B /B (X1 ×X3 X2 ) → R B /B (X1 ) ×RB /B (X3 ) R B /B (X2 ) is an isomorphism. In particular, if X is a B -group then R B /B (X ) is naturally a B-group. (4) If f : X → Y is a smooth map between quasi-projective B -schemes then the induced map R B /B ( f ) is smooth, and similarly for the property of being étale or an open immersion. In particular, if X is B -smooth then R B /B (X ) is B-smooth. Proof. The assertion in (1) follows from the functorial characterization (A.5.1), since for any B-algebra A and B-scheme Y the base change YA represents the restriction to the category of A-schemes of the functor represented by Y on the category of B-schemes (and similarly over B using B -algebras A ). Part (2) is a consequence of the construction of Weil restriction. The compatibility with ﬁber products in (3) is proved by the functorial characterization (A.5.1). The assertion in (3) concerning B -groups and B-groups follows from the categorical characterization of group scheme structures in terms of diagrams with ﬁber products and the fact that Weil restriction carries ﬁnal objects to ﬁnal objects. The veriﬁcation of smoothness and étaleness is done via the inﬁnitesimal criterion: if A is a B-algebra and J is an ideal in A with J 2 = 0 then

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in the B -algebra A := B ⊗A A the ideal J = JA satisﬁes J 2 = 0, so the map R B /B (X )(A) → R B /B (X )(A/J ) is identiﬁed with the map X (A ) → X (A /J ). This is surjective (resp. bijective) when X is smooth (resp. étale). Since an open immersion is the same thing as an étale monomorphism [EGA, IV4 , 17.9.1], the property of being an open immersion is also preserved. 2 One checks via the functorial characterization that R B /B is a left-exact functor from quasi-projective B -groups to ﬁnite type B-groups. That is, if f : G → H is a homomorphism between quasi-projective B -groups then ker(R B /B ( f )) is naturally identiﬁed with R B /B (ker f ). The behavior under quotient maps is more delicate, due to subtleties in the behavior of Weil restriction with respect to surjectivity. Before we address criteria for R B /B to interact well with certain kinds of coverings, we give an instructive example. Example A.5.3 Let k be a ﬁeld and k /k a ﬁnite separable extension of degree d > 1. Let X = Ak1 , and consider the open covering {U0 , U1 } where U0 = {t = 0} and U1 = {t = 1}. Then X = R k /k (Ak1 ) contains the Weil restrictions Ui = R k /k (Ui ) as open subschemes, but these do not cover X . Indeed, the decomposition ks ⊗k k ksd canonically identiﬁes Xks with Akds , in which (U0 )ks is the locus of points (t1 , . . . , td ) such that tj = 0 for all j and (U1 )ks is the locus deﬁned by tj = 1 for all j. Thus, points such as (1, 0, . . . , 0) in X (ks ) are not in U0 (ks ) ∪ U1 (ks ). In the opposite extreme, consider k = k d with d > 1. For any quasi projective k -scheme X with ﬁbers X1 , . . . , Xd over the points of Spec k , we have R k /k (X ) = Xi , so this is non-empty if and only if all Xi are non-empty. In particular, if Xi denotes the open locus in X with ﬁber Xi over the ith point of Spec k and empty ﬁber over the other points then {Xi } is an open cover of X but each R k /k (Xi ) is empty and so these do not cover R k /k (X ) if every Xi is non-empty. The bad behavior of Weil restriction with respect to surjectivity and covering properties are avoided in some cases: Corollary A.5.4 Let B → B be a ﬁnite ﬂat map of noetherian rings, and f : X → Y a smooth surjective map between quasi-projective B -schemes. Let X = R B /B (X ) and Y = R B /B (Y ). (1) The induced smooth map f = R B /B ( f ) : X → Y is surjective.

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(2) Assume that Spec B → Spec B is surjective and radiciel. Then R B /B commutes with the formation of ﬁnite disjoint unions, and if {Ui } is an open (resp. étale) cover of Y then {R B /B (Ui )} is such a cover of Y . (3) If X is a G -torsor over Y for a smooth quasi-projective B -group G then X equipped with its natural action by R B /B (G ) is a torsor over Y . In particular, if f : G → H is a smooth surjection between quasi-projective B -groups and K = ker f then the induced map R B /B (G )/R B /B (K ) → R B /B (H )

(A.5.2)

is an isomorphism (i.e., R B /B (H ) is a quotient of R B /B (G ) modulo the smooth normal B-subgroup R B /B (K )). Proof. To check surjectivity of f in (1) we can assume B = k for an algebraically closed ﬁeld k. Then B is a ﬁnite product of ﬁnite local k-algebras, so by passing to factor rings we can assume that B is local. In particular, its residue ﬁeld is k. The map X (k) → Y (k) is the map X (B ) → Y (B ), and to prove surjectivity of this latter map amounts to showing that for all y ∈ Y (B ) the pullback B scheme Xy admits a B -point. But Xy is smooth and surjective onto Spec B since f is a smooth surjection, so by the functorial criterion for smoothness we just need to ﬁnd a rational point over the residue ﬁeld of B . This residue ﬁeld is the algebraically closed ﬁeld k, so the required rational point exists. For (2), ﬁrst assume Y = Yi , so each Yi is open in Y . Thus, each Yi := R B /B (Yi ) is open in Y := R B /B (Y ) by Proposition A.5.2(4), and we want to prove that they are disjoint and cover Y . We may again reduce to the case when B = k is an algebraically closed ﬁeld, so the radiciel hypothesis implies that B is a ﬁnite local k-algebra (with residue ﬁeld k). The overlap Yi ∩Yj is Yi ×Y Yj , but this is R B /B (Yi ×Y Yj ) = R B /k (∅), which is empty because the only k-scheme Y for which YB is empty is the empty scheme (as B = 0). To see the covering property we compute with k-points: Y (k) = Y (B ) andYi (k) = Yi (B ), so we just need to observe that every map Spec B → Y = Yi factors through some Yi because B is local. Now consider an open cover {Ui } of Y , so {R B /B (U i )} is a collection of open subschemes of Y . Passing to a ﬁnite subcover, U := Ui is quasi-projective and R B /B (U ) is the disjoint union of the R B /B (Ui ) by what we just showed. Since U → Y is a smooth surjective map, so is its Weil restriction. Hence, the R B /B (Ui ) are an open cover of Y . The same argument works for étale covers. To prove (3), let G = R B /B (G ). Assuming that X is a G -torsor over Y , we have to show that the G-action on X makes it a torsor over Y . Since G is smooth, the map X → Y is a smooth surjection. Hence, X → Y is a smooth surjection by (1), so the desired torsor property holds provided that the natural map G × X → X ×Y X is an isomorphism. Since both sides are smooth, it sufﬁces to check this isomorphism property on ﬁbers over Spec B, so we can

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assume that B = k is an algebraically closed ﬁeld. The decomposition B = Bi into a ﬁnite product of ﬁnite local k-algebras then yields a decomposition G = Gi with Gi := R Bi /k (Gi ), where Gi denotes the restriction of G over the connected component Spec Bi of Spec B . We likewise have compatible product expressions X = Xi and Y = Yi , so we can reduce to checking on ith factors for each i separately. That is, we can assume that B is also local (with residue ﬁeld k). Hence, (2) now applies, so R B /B carries étale covers to étale covers. Since X as a G -torsor becomes trivialized étale-locally on Y , applying Weil restriction to such a trivializing cover of Y thereby reduces us to the case X = G × Y , in which case X = G × Y is a trivial G-torsor. 2

Proposition A.5.5 Let B → B be a ﬁnite ﬂat map of noetherian rings, and f : X → Y a closed immersion between quasi-projective B -schemes. The map f = R B /B ( f ) is a closed immersion. In particular, R B /B (X ) is separated. Proof. The separatedness assertion is reduced to the closed immersion claim applied to X /Y (due to the compatibility of R B /B with respect to ﬁber products). To verify the closed immersion assertion, we ﬁrst reduce to the afﬁne case, as follows. Let U → Y be an afﬁne étale cover (such as a ﬁnite disjoint union of the constituents of an afﬁne open covering of Y ). Thus, V = X ×Y U is a closed subscheme of U and is an afﬁne étale cover of X . The resulting commutative diagram R B /B (V )

/ R B /B (X )

R B /B (U )

f

/ R B /B (Y )

is cartesian with étale surjections in the horizontal direction (by Corollary A.5.4(1)), so to prove that the right side is a closed immersion it sufﬁces to do so for the left side. This reduces us to the case when Y and X are afﬁne. By choosing a closed immersion of Y into an afﬁne space over B , we can even reduce to the special case when Y is an afﬁne space over B . In this special case, the closed immersion property for f is the key ingredient in the construction of Weil restrictions in the afﬁne case. 2 In contrast with the special case of closed immersions, Weil restriction through a ﬁnite ﬂat map generally does not preserve properness unless the ﬁnite ﬂat map is étale. Conceptually, this can be predicted via the valuative criterion (non-étale ﬁnite ﬂat extensions of a discrete valuation ring are generally not Dedekind). More concretely, we can build counterexamples as follows. Let

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k /k be a nontrivial purely inseparable extension of ﬁelds and G a connected reductive k -group with B a Borel k -subgroup. For G = R k /k (G ), B = R k /k (B ), it follows from Corollary A.5.4 that G/B = R k /k (G /B ). This always fails to be proper when G is not a torus (i.e., dim G /B > 0), due to: Example A.5.6 Let k → k be a ring map such that k is locally free of ﬁnite constant rank r 1 as a k-module, and let X be a non-empty smooth quasi-projective k -scheme with pure relative dimension d . Thus, X admits an étale afﬁne cover U that is étale over Akd (e.g., a disjoint union of a suitable ﬁnite open afﬁne cover). It follows from Corollary A.5.4 that the Weil restric tion X = R k /k (X ) admits an étale cover that is étale over R k /k (Akd ). But Zariski-locally over Spec k this latter Weil restriction is isomorphic to Akrd , so X is k-smooth with pure relative dimension d = rd . We claim that if d > 0 and k is not étale over k (e.g., a nontrivial purely inseparable ﬁnite extension of ﬁelds, or k = k[t]/(t 2 )) then X is never k-proper, even if X is k -proper. A concrete example with imperfect k of characteristic p > 0 is to take X = Yk for a k-smooth Y of pure dimension d > 0 and k = k(a1/p ) for some a in k that is not a p-power in k. In this case k ⊗k k = k[t]/(t p ), so Xk = R (k[t]/(t p ))/k (Yk ⊗k (k[t]/(t p ))), which is the bundle of p-jets to Yk . For example, when p = 2 this is the tangent bundle to Yk . This is a vector bundle over Yk with nonzero rank, so it cannot be proper when Yk is proper. To establish failure of properness in all cases with d > 0 and k not k-étale, we may pass to geometric ﬁbers over Spec k so that k is an algebraically closed ﬁeld. Passing to factor rings of k allows us to assume that k is a ﬁnite local kalgebra distinct from k. Thus, we can express the dual numbers k[] as a quotient of k over k. Letting X1 denote the pullback of X along the closed immersion Spec k[] → Spec k , we get R k[]/k (X1 ) R k /k (X1 ) as closed subschemes of R k /k (X ). Hence, it sufﬁces to prove non-properness when k = k[] and d > 0. By the valuative criterion for properness, using the valuation ring k[[t]], it sufﬁces to show that the natural map of sets X (k[][[t]]) → X (k[]((t))) is not surjective, where A((t)) denotes A[[t]][1/t]. Note that this map is injective, since X is k[]-separated (as it is quasi-projective).

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Let X0 = X mod , and choose x0 ∈ X0 (k). Since X0 and X have the same underlying topological space, and X is k[]-smooth, we can lift x0 to with the x ∈ X (k[]) and then identify the complete local ring R = O X ,x0 (coordinate ring of the) completion of X along x0 . Thus, non-canonically R k[][[u1 , . . . , ud ]] as local k[]-algebras, with d > 0 and the conditions u1 = · · · = ud = 0 recovering x . Consider the k[]-algebra map R → k[]((t)) deﬁned by u1 → /t and ui → 0 for all i > 1. The associated k[]-map Spec(k[]((t))) → Spec R → X is a point η ∈ X (k []((t))), and we claim it does not extend to X (k[][[t]]). If η does extend to a point ξ ∈ X (k[][[t]]) then upon specializing along = 0, the specialization η0 ∈ X0 (k((t))) of η extends to the specialization ξ0 ∈ X0 (k[[t]]) of ξ . But by construction η0 is the composite map x0

Spec k((t)) → Spec k → X0 , and this extends to x0

Spec k[[t]] → Spec k → X0 . Hence, by separatedness of X0 over k, this latter map must be ξ0 . This says that the abstract k[]-map ξ : Spec k[][[t]] → X specializes to x0 ∈ X0 (k) at = 0, so ξ carries the closed point to x0 . Hence, ξ corresponds to a local k[]-algebra map → k[][[t]] ξ " : k[][[u1 , . . . , ud ]] = O X ,x0 carrying all ui into () and recovering the map f (u1 , . . . , ud ) → f (/t, 0, . . . , 0) after inverting t on the target ring. But k[][[t]] → k[]((t)) is injective, so ξ " must carry u1 to /t. This is a contradiction, since ξ " is valued in the subring k[][[t]] of k[]((t)). To do calculations with Weil restriction, it is extremely useful to make explicit the pair of adjunction morphisms relating Weil restriction and base change: Proposition A.5.7 Let B → B be a ﬁnite ﬂat map between noetherian rings, X a quasi-projective B -scheme, and Y a quasi-projective B-scheme. Using the bifunctorial identiﬁcation (A.5.1), let qX = qX ,B /B : R B /B (X )B → X , jY = jY ,B /B : Y → R B /B (YB ) (A.5.3)

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respectively be the canonical B -morphism corresponding to the identity map on R B /B (X ) and the canonical B-morphism corresponding to the identity map on YB . For any B -algebra A , the map qX on A -points is the map X (B ⊗B A ) → X (A ) induced by the B -algebra homomorphism B ⊗B A → A deﬁned by b ⊗a → b a . For any B-algebra A, the map jY on A-points is given by the map Y (A) → YB (B ⊗B A) = Y (B ⊗B A) induced by the B-algebra homomorphism A → B ⊗B A deﬁned by a → 1 ⊗ a. Moreover, jY is a closed immersion if Spec B → Spec B is surjective and jY is an isomorphism if Y = Spec B. Proof. It will be convenient to work with contravariant functors rather than with schemes so that we can bypass representability issues. (In the afﬁne case the following abstract proof using functors can be expressed in simpler terms.) Let C be the category of B-schemes and C be the category of B -schemes. For any contravariant set-valued functor F on C , we write FB to denote the restriction of F to C , by which we mean the composition of F with the forgetful map C → C . We call FB the base change of F. (For example, if F is represented by a B-scheme Y then FB is represented by the base change YB .) Likewise, for any contravariant set-valued functor F on C we write R B /B (F ) to denote the functor on C deﬁned by Z F (ZB ). If F is represented by a B-scheme X then representability of R B /B (F ) is equivalent to the functor A X (B ⊗B A) on B-algebras being represented by a B -scheme (which is known to occur when X is quasi-projective over B ). The bijection (A.5.1) expresses the fact that if F is the functor represented by a quasi-projective B -scheme X then its Weil restriction R B /B (F ) is the functor represented by the B-scheme R B /B (X ). This will allow us to reduce our problem to an assertion about functors, for which it is easier to see what is going on. There is a natural transformation jF : F → R B /B (FB ) deﬁned by the map F(Z) → FB (ZB ) = F(ZB ) associated to the canonical B-morphism ZB → Z for any B-scheme Z, and there is a natural transformation qF : R B /B (F )B → F deﬁned by the map F (ZB ) → F (Z ) associated to the canonical B -morphism Z → ZB = Z ×Spec(B) Spec(B ) for any B -scheme Z .

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By considering afﬁne Z and Z , when F is represented by a quasi-projective B-scheme Y (resp. when F is represented by a quasi-projective B -scheme X ) then jF corresponds to jY (resp. qF corresponds to qX ). Thus, apart from the closed immersion and isomorphism claims for jY at the end, it sufﬁces to show that in general jF and qF deﬁne inverse correspondences HomC (F, R B /B (F )) → HomC (FB , F ), HomC (FB , F ) → HomC (F, R B /B (F ))

(A.5.4)

via T → qF ◦ TB and T → R B /B (T ) ◦ jF respectively, and that if F is represented by a B-scheme Y then the resulting composite bijection R B /B (F )(Y ) = HomC (HomB (·, Y ), R B /B (F )) HomC (Hom(·, YB ), F ) = F (YB )

(A.5.5)

is the canonical identiﬁcation βF ,Y that occurs in the deﬁnition of the functor R B /B (F ). The fact that jF and qF deﬁne inverse correspondences goes exactly like the proof of adjointness of the pullback and pushforward functors between categories of sheaves on a topological space. The problem is therefore to compute (A.5.5): if F = HomB (·, Y ) for a B-scheme Y then we claim that the bijective correspondence (A.5.4) is βF ,Y . In other words, if f ∈ F (YB ) and [ f ] : FB = HomB (·, Y )|C = HomB (·, YB ) → F denotes the associated natural transformation then we claim that the composite natural transformation [ f ]

jF

F → R B /B (FB ) → R B /B (F )

(A.5.6)

is the point in R B /B (F )(Y ) corresponding to f via βF ,Y . For F = HomB (·, Y ), it follows from the deﬁnition of jF that the diagram F(Z)

jF

/ R B /B (FB )(Z) LLL LLL LLL βFB ,Z L& FB (ZB )

commutes for any B-scheme Z. In particular, for any a ∈ F(Z) the point βFB ,Z ( jF (a)) ∈ FB (ZB ) is equal to aB . (Note also that if π : ZB → Z is the natural map then the identiﬁcation FB (ZB ) F(ZB ) carries aB to

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a ◦ π.) Hence, if f ∈ F (YB ) and [ f ] : FB → F is the associated natural transformation then in the commutative diagram of sets F(Z)

jF

/ R B /B (FB )(Z)

LL LL LL LL βFB ,Z LL % FB (ZB )

[ f ]

/ R B /B (F )(Z) βF ,Z

[ f ]

/ F (ZB )

an element a ∈ F(Z) is carried to aB in FB (ZB ), and hence to [ f ](aB ) = F (aB )( f ) in F (ZB ). Setting Z = Y and a = id Y , it follows that any f = F (id YB )( f ) ∈ F (YB ) is carried by βF ,Y to the element of R B /B (F )(Y ) associated to the composite transformation (A.5.6), as desired. Finally, we check that jY is a closed immersion when Spec B → Spec B is surjective, and that it is an isomorphism when Y = Spec B. First consider the surjectivity claim. From the above functorial description of jY and consideration of points valued in any B-algebra A, if Z is a locally closed subscheme of Y then jY−1 (R B /B (Z)) = Z (using that Spec(B ⊗B A) → Spec A is surjective). Hence, we can work Zariski-locally on Y to reduce to the case when Y is afﬁne and then even the case when Y = ABn . Zariski-localization on B lets us arrange that B admits a B-basis {e1 = 1, . . . , ed }. Thus, via the description of jY it follows that R B /B (YB ) is identiﬁed with ABnd such that jY is the inclusion along the ﬁrst n coordinates. This is a closed immersion by inspection. The same calculation using n = 0 and any d 0 settles the isomorphism claim when Y = Spec B. 2 Proposition A.5.8 Let B → B be a ﬁnite ﬂat map of noetherian rings, and X a quasi-projective B -scheme. The B-scheme R B /B (X ) is quasi-projective. Proof. For a line bundle L on a scheme Z, and any s ∈ L (Z), deﬁne Zs to be the open locus where s generates L . Following [EGA, II, 4.5.3], if Z is quasi-compact and separated then L is ample if there are sections si of tensor powers L ⊗ni (ni 1) such that the Zsi are afﬁne and cover Z. Assuming Z is separated of ﬁnite type over a ring A, it is quasi-projective over A (in the sense of admitting a locally closed immersion into a projective space over A) if and only if it admits an ample line bundle [EGA, II, 5.3.2], in which case every ﬁnite subset of Z is contained in some afﬁne Zs [EGA, II, 4.5.4]. It therefore sufﬁces to construct an ample line bundle on X = R B /B (X ). Choose an ample line bundle N on X , and consider the natural map q : XB → X and the ﬁnite ﬂat map π : XB → X . The pullback L = q∗ (N ) is a line bundle on XB , and we claim that its norm L = NB /B (L ) is ample on X . Observe that L ⊗n is similarly constructed from N ⊗n for any n ∈ Z. For

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any s ∈ N (X ), deﬁne N(s ) ∈ L (X ) to be the norm of q∗ (s ) ∈ L (XB ). Since any line bundle on a semi-local ring is trivial, the locus XN(s ) is X − π(q−1 (X − Xs )). But it is immediate from functorial considerations that for any open subscheme U in X , U := X − π(q−1 (X − U )) is R B /B (U ), so XN(s ) = R B /B (Xs ). The same works for tensor powers. Also, if {Ui } is an open cover of X then the corresponding opens Ui = X − π(q−1 (X − Ui )) constitute an open cover of X provided that any ﬁnite subset of XB (e.g., the ﬁbers of π) is disjoint from some q−1 (X − Ui ). For example, if every ﬁnite subset of X is contained in some Ui then the Ui cover X . By [EGA, II, 4.5.4], there is a collection {si } of global sections of powers N ⊗ni with ni 1 such that the Xs are an open afﬁne cover of X and every i

ﬁnite subset of X is contained in some Xs . Thus, the corresponding XN(si ) are i an open afﬁne cover of X , so L is ample on X . 2

Now we specialize to the case of Weil restriction through a ﬁnite extension of ﬁelds and for group schemes. Proposition A.5.9 Let k be a ﬁeld, k a nonzero ﬁnite k-algebra, and X → Spec k a smooth surjective quasi-projective map with geometrically connected ﬁbers. The smooth k-scheme X := R k /k (X ) is non-empty and geometrically connected. The preservation of connectedness without further restrictions on the kalgebra k (e.g., it may not be étale) depends on smoothness. More speciﬁcally, for any imperfect ﬁeld k of characteristic p > 0 and purely inseparable extension k /k of degree p there exists a geometrically integral plane curve X over k smooth away from a single k -rational point such that R k /k (X ) is disconnected (and X can be chosen to be regular if [k : k p ] p2 ). We omit discussion of such examples. Proof. By extension of scalars we can assume k = k. Passing to local factor ; rings of k reduces the problem to the case when k is local. Let X0 := Xred this is the ﬁber of the k -smooth X over the residue ﬁeld k0 = k of k . We have X (k) = X (k ), and X (k ) → X0 (k0 ) = X0 (k) is surjective since X is k -smooth. Since X0 (k) is non-empty, we conclude that X is non-empty. The functor R k /k carries Zariski-open covers to Zariski-open covers, by Corollary A.5.4(2). The smooth and geometrically connected X0 must be geometrically integral, so every non-empty open in X is dense. In particular, any two non-empty open afﬁnes have a non-empty separated overlap. Since R k /k is compatible with open immersions and open covers, to prove that X is connected we can reduce to the case when X is afﬁne. By further Zariski-localization we can arrange that the k -smooth X is étale over an afﬁne space over k . That is, X is an étale cover of a dense open U ⊆ Akn for some n 1. Via the natural homeomorphism Akn Akn we see that U = Uk for a dense open U ⊆ Akn . By

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the topological invariance of the étale site [SGA1, IX, Thm. 4.10], the functor E Ek is an equivalence between the category of étale U -schemes and the category of étale U -schemes, so X = Yk for an étale U -scheme Y . Thus, we have reduced to the case when X = Yk for a smooth connected afﬁne k-scheme Y , say Y = Spec B. Now consider the natural map j : Y → R k /k (Yk ) = X that is a closed immersion (Proposition A.5.7). Since Y is connected, it sufﬁces to retract X onto Y by a homotopy. To be precise, for each y ∈ X (k) = Y (k ) there is an associated specialization y ∈ Y (k0 ) = Y (k), and we will deform y to y. The point y corresponds to a map y ∗ : B → k carrying my into the nilpotent maximal ideal of k . This latter ideal has vanishing d th power for some d 1, so y ∗ corresponds to a k-algebra map k[u1 , . . . , un ]/(u1 , . . . , un )d = B/mdy → k where {u1 , . . . , un } is a regular sequence of parameters in the k-smooth B at y. Precomposing with the scaling action t.(u1 , . . . , un ) → (tu1 , . . . , tun ) on the parameters deﬁnes a k-map Ak1 → X carrying 1 to y ∈ X (k) and 0 to y ∈ Y (k). 2 The map qX in Proposition A.5.7 has an interesting and useful surjectivity property in the smooth case: Lemma A.5.10 Let k be a ﬁeld, k a ﬁnite local k-algebra with residue ﬁeld purely inseparable over k, and X a smooth quasi-projective k -scheme. The k -map qX : R k /k (X )k → X is surjective on B -points for any k -algebra B . Proof. The residue ﬁeld hypothesis implies that A = k ⊗k k is a local kalgebra with residue ﬁeld k via the multiplication map m : A → k deﬁned by c ⊗ c → cc . The natural multiplication map π : k ⊗k B → B is induced by m via scalar extension. Since ker m has vanishing nth power for some n > 0, so does ker π. Hence, X (π ) : X (k ⊗k B ) → X (B ) is surjective due to the k -smoothness of X . But X (π ) is qX on B -points, by Proposition A.5.7, so we are done. 2 Without smoothness conditions on X , the map qX in Lemma A.5.10 can fail to be surjective. An example is X = Spec(K ) for a nontrivial ﬁnite extension K of k in characteristic p > 0 with k ⊆ k 1/p but (K ) p ∩ k not contained in k, in which case R k /k (X ) is empty. Proposition A.5.11 Let B → B be a ﬁnite ﬂat map of noetherian rings and X a smooth quasi-projective B -scheme such that X → Spec B is surjective. (1) The B-scheme X = R B /B (X ) is smooth, and the canonical map q : XB = R B /B (X )B → X

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(from (A.5.3)) is smooth and surjective. In particular, q is faithfully ﬂat. Assume X → Spec B has non-empty equidimensional ﬁbers, and that for some y ∈ Spec B the local ﬁber degrees of Spec B → Spec B at all y over y such that dim Xy > 0 have sum > 1. Then the ﬁbers of qy have dimension > 0 for all y over y. (2) Assume that X is a B -group, and that B and B are artin local rings for which the induced map on residue ﬁelds is purely inseparable. Then ker q has connected unipotent ﬁbers over Spec B . (3) If X → Spec B has geometrically connected ﬁbers then so does X → Spec B. Proof. By Proposition A.5.2(4), the smoothness of X is inherited from that of X ; likewise, the formation of X is compatible with any scalar extension on B. Thus, for the rest we may and do assume that B = k is an algebraically closed ﬁeld. Passing to factor rings of B is harmless for (1) and (3) apart from the analysis of ﬁber dimensions for q in (1), so now assume B is local. Hence, B is a ﬁnite local k-algebra with residue ﬁeld k. Now the surjectivity in (1) follows from Lemma A.5.10, and (3) follows from Proposition A.5.9. To prove the smoothness in (1), since the source and target are smooth it sufﬁces to check smoothness for the induced map q0 : X → X0 := X ⊗B k between special ﬁbers over the residue ﬁeld k of B . For smoothness of q0 it sufﬁces to check surjectivity of the induced map between tangent spaces at kpoints. Choose x ∈ X (k) and let x ∈ X (B ) be the corresponding section, with specialization x0 ∈ X0 (k) (so x0 = q0 (x)). Calculating with dual numbers shows that dq0 (x) is the natural k-linear reduction map Tanx (X ) = Tanx (X ) → Tanx0 (X0 ). This is surjective, so q is smooth. The assertion on ﬁber dimensions for q in (1) is proved in two steps, as follows. If B is local then B = k by the ﬁber-degree hypothesis, so (by consideration with dual numbers) the tangent space to R B /k (X ) at any x ∈ R B /k (X )(k) = X (B ) is the underlying k-vector space of the ﬁnite free B -module Tanx (X ). Its k-dimension is [B : k] dim(X ) > dim(X ), so the smooth ﬁber of q through x has nonzero tangent space and hence dimension > 0. For general B , we can reduce the study of q−1 (x ) to the local case when x ∈ Xy with By = k and dim Xy > 0. If By = k then qy is the projection Xy × R B /k (X ) → Xy using the product decomposition B = By × B , with X the B -ﬁber of X . Since X → Spec B must have a positive-dimensional ﬁber (due to the ﬁberdegree hypothesis, since By = k), so does qy at all points of its target. It remains to show that if X is a B -group then ker q has connected unipotent ﬁbers over Spec B . By Proposition A.5.7, the canonical map q0 : X =

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R B /k (X ) → X0 is deﬁned functorially on A-points (for a k-algebra A) by R B /k (X )(A) = X (B ⊗k A) → X ((B /m) ⊗k A) = X0 (A ) where m is the maximal ideal of B . The map q0 has now been expressed entirely in terms of the k-algebra B with residue ﬁeld k and the smooth quasi-projective B -group X . Letting Bn = B /mn+1 (n 0) and Xn = X ⊗B Bn , we see that q0 is the composition of the natural maps R Bn+1 /k (Xn+1 ) → R Bn /k (Xn ),

and it is precisely these latter maps that are studied in [Oes, A.3.5] (assuming X is afﬁne, which is never used in the proofs), and we record that result in Proposition A.5.12 below. From this it follows that ker q0 admits a composition series whose successive quotients are the vector groups Lie(X0 ) ⊗k (mi /mi+1 ). In particular, ker q0 is smooth, connected, and unipotent. 2 Here is a key fact that was mentioned in the preceding proof. Proposition A.5.12 Let k be a ﬁeld and (B, m) a ﬁnite local k-algebra with B/m = k. Let Bi = B/mi+1 for i 0. Let H be a smooth quasi-projective B-group, Hi = H ⊗B Bi , and Gi = R Bi /k (Hi ). The natural map Gi+1 → Gi is a smooth surjection and its kernel is the k-vector group corresponding to the ﬁnite-dimensional k-vector space Lie(H0 ) ⊗k mi+1 /mi+2 . Proof. This is [Oes, Prop. A.3.5]. It is assumed there that H is afﬁne, but this hypothesis is never needed in the argument. (In fact, since A ⊗k B is local for any local k-algebra A, if S is any k-scheme then every B-map SB → H lands in open afﬁnes of H Zariski-locally over S. Thus, the construction of R B/k (Y ) as a locally ﬁnite type k-scheme works for any locally ﬁnite type Bscheme Y , so even the quasi-projective hypothesis on H can be dropped from the proposition.) 2 When working with ﬁnite étale schemes over an imperfect ﬁeld, the operation of purely inseparable Weil restriction is invisible. This is made precise by: Proposition A.5.13 Let k /k be a purely inseparable ﬁnite extension of ﬁelds, X a ﬁnite étale k -scheme, and Y a ﬁnite étale k-scheme. Then X := R k /k (X ) is a ﬁnite étale k-scheme and the natural map qX : Xk → X is an isomorphism, and likewise the natural map jY : Y → R k /k (Yk ) is an isomorphism. In particular, the adjoint functors R k /k and (·) ⊗k k are naturally inverse equivalences between the categories of ﬁnite étale schemes over k and over k, and if f : G → H is an étale isogeny between smooth connected k -groups then R k /k ( f ) is an étale isogeny with the same degree as f .

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Proof. The assertion about étale isogenies is immediate from the rest via (A.5.2) with ﬁnite étale K there. The ﬁrst part of the proposition is a special case of a general fact from étale sheaf theory, but we can easily give a direct proof as follows. We can assume that k = ks , so any X is a ﬁnite disjoint union of copies of Spec k . But qYk ◦ ( jY )k is always the identity map of Yk , so it sufﬁces to prove that jY is an isomorphism for all Y . Since k /k is purely inseparable, R k /k commutes with the formation of disjoint unions (Corollary A.5.4(2)). Thus, we may assume Y = Spec k, and then the isomorphism property is a special case of the ﬁnal claim in Proposition A.5.7. 2 We conclude this section with several results concerning how Weil restriction interacts with interesting structures in the case of smooth groups of ﬁnite type over a ﬁeld. Proposition A.5.14 Let G be a nontrivial connected semisimple group over a ﬁeld k, and assume that G is simply connected. There exists a nonzero ﬁnite étale k-algebra k and a semisimple k -group G with absolutely simple ﬁbers over the factor ﬁelds of k such that G R k /k (G ) as k-groups. If (G , k /k) is another such pair then every k-group isomorphism R k /k (G ) R k /k (G ) arises via Weil restriction of a unique pair ( f , α) where α : k k is a kalgebra isomorphism and f : G G is a group isomorphism over Spec(α), so the pair (G , k /k) is unique up to unique isomorphism. Moreover, G is k-simple if and only if k is a ﬁeld. This is a reﬁnement of [BoTi1, 6.21(ii)]. Proof. We will use the concept of “simply connected datum” (G , k /k, fG ) attached to G as in Deﬁnition 4.2.3. This is a construction that makes sense for all smooth connected afﬁne k-groups, and is developed in an entirely selfcontained manner in Proposition 4.2.1 and Proposition 4.2.2. By construction, G → Spec k is semisimple and its ﬁbers are connected, absolutely simple, and simply connected. We claim that the canonical map fG : G → R k /k (G /ZG ) uniquely factors through R k /k (G ) → R k /k (G /ZG ) via a k-homomorphism f : G → R k /k (G ) that is moreover an isomorphism. Since the formation of (G , k /k, fG ) commutes with separable extension on k, to verify the isomorphism property we may assume k = ks . Due to the simply connectedness of G, it is uniquely (up to rearrangement of factors) a direct product Gi of connected semisimple groups Gi that are each absolutely simple and simply connected. (This rests on the canonical decomposition of a connected semisimple group over a ﬁeld into an almost direct product of simple factors; see [Bo2, 22.9, 22.10].) By construction of the simply connected datum, we have G = Gi over k = i k and fG corresponds (under the

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universal property of Weil restriction) to the natural map π : Gk → G /ZG over Spec k = i Spec k whose restriction over the ith factor ﬁeld k is the composition of the ith projection pi : G → Gi with the quotient map Gi → Gi /ZGi . Thus, the collection of maps pi deﬁnes a map Gk → G through which π uniquely factors (as there are no nontrivial k -homomorphisms from Gk into ZG ), and this in turn corresponds to a unique f : G → R k /k (G ) through which f uniquely factors. In other words, f is a direct product of the f is an natural maps jGi : Gi → R k/k (Gi ) which are visibly isomorphisms, so isomorphism. It remains to show that R k /k (G ) is k-simple when k is a ﬁeld, and that in general k-isomorphisms between such k-groups R k /k (G ) arise uniquely from isomorphisms between such pairs (G , k /k). First we verify the k-simplicity when k is a ﬁeld. Consider the product decomposition

R k /k (G )ks

G ⊗k ,σ ks

(A.5.7)

σ

in which the factors are indexed by the k-embeddings σ of k into ks . These factors are all ks -simple, since G is absolutely simple by hypothesis, so they are the simple factors of R k /k (G )ks . Every smooth connected normal subgroup of a connected semisimple group over a ﬁeld is a product among the simple factors (this is an immediate consequence of [Bo2, 22.10(i)]), so by transitivity of the Gal(ks /k)-action on the factors on the right side of (A.5.7) we see that there are no nontrivial proper Gal(ks /k)-stable smooth connected normal ks -subgroups of R k /k (G )ks . Hence, R k /k (G ) is k-simple. To prove the desired description of all k-group isomorphisms R k /k (G ) R k /k (G ) it sufﬁces (by Galois descent) to consider the case when k is separably closed.Then k = k n and k = k m as k-algebras, and G = ni=1 Gi and G = m k-groups Gi and Gj that are simply j=1 Gj for connected semisimple connected. In such cases R k /k (G ) = i Gi , so the ordered n-tuple {Gi } is determined as the set of k-simple factors (up to rearrangement) of R k /k (G ). In view of the uniqueness of the set of k-simple factors, we must have n = m, so the problem is to show that all k-group isomorphisms Gi Gi arise for a unique permutation t of from unique k-group isomorphisms Gi Gt(i) {1, . . . , n}. Since the k-simple factors are characterized as the minimal nontrivial smooth connected normal k-subgroups, we are done. 2 Proposition A.5.15 Let k be a ﬁeld, k a nonzero ﬁnite reduced k-algebra, G a smooth afﬁne k -group, and G = R k /k (G ).

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(1) For any smooth closed subscheme V of G and associated smooth closed subscheme V = R k /k (V ) in G we have ZG (V ) = R k /k (ZG (V )) as subschemes of G. In particular, ZG = R k /k (ZG ). (2) There is a natural bijection between the set of maximal k-tori T in G and the set of maximal k -tori T in G by the requirement that T ⊆ R k /k (T ).The maximal k-split torus T0 in T is maximal as a k-split torus in R k /k (G ) if and only if the maximal k -split torus T0 in T is maximal as a k -split torus in G . A bijection between the sets of maximal k-split tori and maximal k -split tori is likewise deﬁned by the requirement T0 ⊆ R k /k (T0 ). (3) If T is a maximal k -torus in G and T is the maximal k-torus in R k /k (T ) then ZG (R k /k (T )) = ZG (T ). In particular, if C is a Cartan k -subgroup of G then R k /k (C ) is a Cartan k-subgroup of G, and every Cartan k-subgroup of G arises in this way for a unique C . This proposition is true more generally for smooth groups of ﬁnite type (not necessarily afﬁne). All that is missing to make the argument below work in this greater generality is enough foundational material concerning tori in such groups (e.g., preservation of maximality under extension of the ground ﬁeld, behavior under smooth surjections, and rational conjugacy results in the maximal split case). Lemma C.4.4 and Proposition C.4.5 below provide the required ingredients beyond the afﬁne case. Proof. We postpone the assertions concerning split tori in (2) to the end of the proof because for the proof of the other assertions we may assume that k is separably closed. (This rests on using Galois descent for the proof of the bijectivity among maximal tori in (2).) Passing to factor ﬁelds of k lets us reduce to the case when k is a ﬁeld (purely inseparable of ﬁnite degree over k). We will assume that we are in this setting until we indicate otherwise. First consider (1). To prove the equality ZG (V ) = R k /k (ZG (V )) as closed subschemes of G = R k /k (G ) we compare A-valued points for any k-algebra A. Choose g ∈ G(A), and let g denote the corresponding point in G (A ) with A = k ⊗k A. We need to prove that g centralizes VA if and only if g centralizes VA . The identiﬁcation GA = R A /A (GA ) carries the conjugation automorphism cg : x → gxg −1 of GA over to R A /A (cg ). The natural map qG : Gk → G corresponding to G(B ) = G (k ⊗k B ) → G (B ) for k -algebras B is surjective by Proposition A.5.11(1).

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Extending scalars to A gives a quotient map (GA ) ⊗A A = (Gk ) ⊗k A GA carrying (VA ) ⊗A A onto VA (since V is k -smooth) and carrying (cg )A over to cg . The identiﬁcation of cg with R A /A (cg ) and of cg with an automorphism induced by (cg )A imply that cg is trivial on VA if and only if cg is trivial on VA . This proves that R k /k (ZG (V )) = ZG (V ) inside of G. Taking V = G then yields the relation ZG = R k /k (ZG ) between scheme-theoretic centers. Now consider the claim in (2) concerning maximal tori. For any maximal k-torus T in G, the natural surjective map Gk = R k /k (G )k G (which is even smooth, by Proposition A.5.11(1)) carries Tk onto a maximal k -torus T of G , so by the universal property of Weil restriction we have T ⊆ R k /k (T ). Since k and k are separably closed, we can apply Proposition A.2.10 to obtain the desired bijection among maximal tori. To prove (3), let V = R k /k (T ). Since T is a torus, the centralizer ZG (T ) is k -smooth (Proposition A.2.5). Thus, the centralizer ZG (V ) = R k /k (ZG (T )) is k-smooth. Letting T be the maximal k-torus in V = R k /k (T ), we wish to prove that ZG (V ) = ZG (T ). The containment ZG (V ) ⊆ ZG (T ) is clear, and to prove the reverse containment it sufﬁces to check equality on k-points since k = ks and both ZG (V ) and ZG (T ) are smooth. That is, we want any g ∈ ZG(k) (T ) to lie in ZG (V )(k) = ZG (k ) (T ). Since Gk → G carries Tk onto T , this is clear. Finally, we return to the situation over a general ﬁeld k and address the claims in (2) concerning maximal split tori. We may again assume that k is a ﬁeld. Observe that if S is a k -anisotropic torus then the k-group R k /k (S ) (which is not a k-torus if S = 1 and k /k is not separable) is k-anisotropic in the sense that there is no nontrivial k-homomorphism GL1 → R k /k (S ). Consider a matching pair (T , T ) of maximal tori, and the maximal k-split torus T0 ⊆ T and maximal k -split torus T0 ⊆ T . Since the quotient T /T0 is a k -anisotropic torus, so the k-group R k /k (T )/R k /k (T0 ) R k /k (T /T0 ) is k-anisotropic, necessarily T0 ⊆ R k /k (T0 ). If T0 is a maximal k-split torus in R k /k (G ) then since T0 ⊆ R k /k (T0 ) it follows that T0 is a maximal k -split torus in G . Indeed, assume T0 is contained in a k -split torus S ⊆ G , and let S be the maximal k-split torus in R k /k (S ). Then T0 ⊆ S, so T0 = S, but the maps Sk → S and (T0 )k → T0 are surjective, so T0 = S . Conversely, if T0 is a maximal k -split torus in G then we wish to prove that T0 is a maximal k-split torus in G. Obviously (T0 )k maps onto T0 in G since T0 is the maximal k-split torus in R k /k (T0 ), so if S0 is a k-split torus of G containing T0 then the image of (S0 )k under the canonical surjection R k /k (G )k G is a k -split torus containing T0 . Maximality of T0 then forces (S0 )k to map into

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T0 , so S0 ⊆ R k /k (T0 ). But T0 is the maximal k-split torus in R k /k (T0 ), so the containment T0 ⊆ S0 is an equality. That is, T0 is maximal as a k-split torus in R k /k (G ). Since every maximal split torus in a group over a ﬁeld is also the maximal split torus in some maximal torus over the ground ﬁeld, the preceding arguments show that for every maximal k -split torus T0 in G the maximal k-split torus T0 in R k /k (T0 ) is maximal as a k-split torus in R k /k (G ), and moreover that every maximal k-split torus T0 in R k /k (G ) arises in this way for some T0 . Observe that T0 is uniquely determined by T0 , since we have already seen that it is the image of (T0 )k under the canonical quotient map R k /k (G )k G . 2 Recall that if G is a smooth connected afﬁne group over a ﬁeld k, a Levi k-subgroup of G is a smooth connected k-subgroup L ⊆ G such that Lk Ru (Gk ) Gk via multiplication. Corollary A.5.16 Let k /k be a purely inseparable ﬁnite extension of ﬁelds, and G a connected reductive k-group. The smooth connected afﬁne k-group R k /k (Gk ) has its geometric unipotent radical deﬁned over k , and the closed k-subgroup ι : G → R k /k (Gk ) is a Levi subgroup. Proof. For any k -algebra B there is a natural quotient map k ⊗k B B as k -algebras (using the left k -structure on the source) and hence we naturally get a map of groups R k /k (Gk )k (B) = Gk (k ⊗k B) → Gk (B). By Proposition A.5.7, this is the smooth surjective map of k -groups q(Gk ) : R k /k (Gk )k → Gk whose composition with ιk is the identity. By Proposition A.5.11(2) the kernel of q(Gk ) is smooth, connected, and unipotent over k . Hence, ker q(Gk ) is a k -descent of the geometric unipotent radical of R k /k (G ), and G is a Levi k-subgroup via ι. 2

A.6 Groups without Levi subgroups In this section we construct many smooth connected afﬁne groups in nonzero characteristic not containing a Levi subgroup, even if the ground ﬁeld is algebraically closed. We never need such a construction, but we record it here for three reasons: (i) no example of this sort seems to be in the literature, (ii) it is an interesting contrast with our result (Theorem 3.4.6) that pseudoreductive groups over a separably closed ﬁeld k always admit a Levi k-subgroup, (iii) the construction involves the Greenberg functor, which is (roughly) a mixed characteristic analogue of the Weil restriction functor that we discussed in §A.5. Let k be an algebraically closed ﬁeld with char(k) = p > 0. For any noncommutative connected reductive group scheme H over the artin local ring W2 (k) of length-2 Witt vectors for k, we claim that the group H (W2 (k)) viewed

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as a k-group (in a manner that is made rigorous below) has no Levi k-subgroup. For example, SL2 (W2 (k)) viewed as a k-group has no Levi k-subgroup. The idea is to identify H (k) as the maximal reductive quotient of H (W2 (k)) and to show that if this quotient map has a k-group section (i.e., if there exists a Levi k-subgroup in H (W2 (k))) then by inspecting suitable “root groups” such a section induces a k-group section to the canonical quotient map W2 Ga . No such section exists, since Ga is p-torsion whereas W2 [p] is the preimage of αp ⊂ Ga . Hence, H (W2 (k)) viewed as a k-group has no Levi subgroup. In the instructive and interesting special case H = SL2 , most of the functorial considerations below are not necessary and so the argument simpliﬁes a lot. (The general case will essentially be reduced to this special case.) To analyze the general case rigorously without matrix calculations, we ﬁrst brieﬂy review the Greenberg functor that is analogous to Weil restriction. For n 1, let Wn denote the functor of length-n Witt vectors (w0 , . . . , wn−1 ) on k-algebras. For every k-algebra B, the ring Wn (B) is naturally a Wn (k)algebra. In view of the universal formulas for the ring scheme structure on Wn , for any afﬁne Wn (k)-scheme X of ﬁnite type, the functor B X (Wn (B)) is represented by an afﬁne k-scheme Xn of ﬁnite type. (The correspondence X Xn is the Greenberg functor on afﬁne Wn (k)-schemes of ﬁnite type. Loosely speaking, the Greenberg functor is a mixed characteristic analogue of Weil restriction through the ﬁnite ﬂat covering Spec(k[t]/(t n )) → Spec(k).) The formation of Xn is compatible with products in X over Wn (k), and Xn = Spec k when X = Spec Wn (k), so Xn is naturally a k-group scheme when X is a Wn (k)-group scheme. Example A.6.1 Let X = Ga over Wn (k). Then Xn = Wn as a k-group. In general, the k-scheme Xn is smooth when X is Wn (k)-smooth, due to the functorial criterion for smoothness and the fact that the map Wn (B ) → Wn (B) is surjective with square-zero kernel when B → B is surjective with squarezero kernel. The relationship between connectedness properties of X and Xn requires the following ﬁbration construction. Deﬁne X = X ⊗Wn (k) k, so there is a natural k-scheme map πn : Xn → X deﬁned functorially by X (Wn (B)) → X (B) = X (B)

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for k-algebras B. The map πn is a smooth surjection when X is smooth, and it is a k-homomorphism when X is a Wn (k)-group scheme. We now apply this with n = 2. Lemma A.6.2 Let X be an afﬁne W2 (k)-scheme of ﬁnite type, and choose x ∈ X (W2 (k)). Let x ∈ X (k) be its reduction. There is a natural isomorphism of k-schemes θx : π2−1 (x) Tx (X )(p) onto the Frobenius twist of the tangent space to X at x, with θx (x) = 0. In particular, if X is connected then Xn is connected. Proof. The ﬁber scheme π2−1 (x) over k represents the functor of points in X (W2 (B)) reducing to the image of x ∈ X (k) in X (B) for varying k-algebras B. We need to identify this with the functor B B ⊗k Tx (X )(p) . Let A be the coordinate ring of X , so B-points of π2−1 (x) are W2 (k)-algebra maps A → W2 (B) having the form a → a(x) + (0, h(a)) for a map of sets h : A → B (where a(x) ∈ W2 (k) is viewed in W2 (B) in the evident manner). Using the ring scheme structure on W2 , the condition that a → a(x) + (0, h(a)) is a W2 (k)-algebra map is that h(a) = h(a mod p) for a (unique) additive map h : A = A/pA → B such that h(1) = 0, h(ca) = c p h(a) for all c ∈ W2 (k)/(p) = k and a ∈ A, and h(a · a ) = a(x) p h(a ) + a (x) p h(a) for all a, a ∈ A. Such maps factor through A/m2x , vanish on k, and are klinearized after applying a Frobenius scalar extension on the source. This is precisely the set Homk (Cot x (X )(p) , B) Tx (X )(p) ⊗k B. 2 Remark A.6.3 By functoriality of the formation of πn and of tangent spaces with respect to products, θx is compatible with products in (X , x). Hence, if X is a W2 (k)-group and x = 1 then θx is a k-group isomorphism between π2−1 (1) = ker π2 and Lie(X )(p) . The functoriality of θx with respect to automorphisms of (X , x) can be made explicit: if σ ∈ Aut W2 (k) (X , x) then the induced automorphism on Tx (X )(p) via θx is dσ (x)(p) , with σ ∈ Aut k (X , x) denoting σ mod p.

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Now let H be a smooth afﬁne W2 (k)-group (such as SL2 ) and let G = H2 be the associated smooth afﬁne k-group. By Lemma A.6.2, G is connected when H is connected. By Remark A.6.3, there is an exact sequence of smooth k-groups π

1 → Lie(H )(p) → G → H → 1

(A.6.1)

where π = π2 . Since the group scheme ker π = Lie(H )(p) is commutative, the conjugation action by G on ker π factors through an action by H on Lie(H )(p) . The description of functoriality of θx with respect to automorphisms in Remark A.6.3 implies that this action by H is the Frobenius scalar extension of the adjoint action of H on Lie(H ). Assume that H is a non-commutative connected reductive W2 (k)-group. Since W2 (k) is an artin local ring with algebraically closed residue ﬁeld, H is a split reductive group scheme over W2 (k). (To avoid the theory of split reductive groups over artinian rings, the reader may prefer to work with a ﬁxed nontrivial Chevalley group, such as SL2 .) The smooth surjective map π : G H is the maximal reductive quotient of G since ker π is a vector group. Proposition A.6.4 Let k be an algebraically closed ﬁeld with char(k) > 0, and let H be a non-commutative connected reductive W2 (k)-group. Let G = H2 be the associated smooth connected afﬁne k-group. This k-group has no Levi k-subgroup. That is, the maximal reductive quotient map π : G → H has no k-group section. Proof. We assume that there is a k-group section s to π , and we seek a contradiction. The idea is to modify s so it is compatible with W2 (k)-points of a root group in G and k-points of a root group in H (relative to suitable maximal tori that are compatible via π). The root groups are Ga (over W2 (k) and k respectively), so by Example A.6.1 with n = 2, the map s would restrict to a section of the quotient map W2 → Ga . But no such section of the latter sort exists. Step 1. To get started, we will arrange compatible choices of maximal tori in G and H . Let T be a maximal W2 (k)-torus in H . Then T2 is a smooth commutative subgroup of G = H2 . It is an extension of the maximal k-torus T = T mod p of H by the vector group corresponding to the k-vector space Lie(T )(p) . Hence, the maximal k-torus T in T2 is a maximal k-torus of G. All maximal k-tori in G are G(k)-conjugate [Bo2, Cor. 11.3], so for a suitable g ∈ G(k) we may replace the section s with the section cg −1 ◦s◦cπ(g) (where cγ denotes conjugation by γ ) to arrange that the Levi k-subgroup s(H ) contains T . Under the reduction map π , the k-torus T is carried isomorphically onto the maximal torus T in H , so s(T ) = T . For any cocharacter λ : GL1 → T over W2 (k), [λ] : GL1 → T is the map between maximal tori deﬁned by the k-group map λ2 : (GL1 )2 → T2 arising from λ by functoriality. Concretely,

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[λ](t) = λ([t]) in T2 (W2 (k)) where [t] ∈ W2 (k)× is the Teichmüller lift of t ∈ k × . In particular, for each coroot c : GL1 → T we get a 1-parameter subgroup [c] : GL1 → T in G deﬁned by [c] = s ◦ c. For the associated cocharacter λ : GL1 → T over k, the restriction of s to T satisﬁes s(λ(t)) = [λ](t) for t ∈ GL1 . Step 2. Now we use root groups. The set of roots (H , T ) is nonempty since H is non-commutative. Choose a ∈ (H , T ), so the root group Ua in H is isomorphic to Ga over W2 (k). Let a ∈ (H , T ) denote the reduction of a over k, and let Ua := Ua mod p ⊆ H be the a-root group (k-isomorphic to Ga ). Consider s(Ua ) ⊆ G. We claim that this subgroup is contained in the unipotent subgroup (Ua )2 ⊆ H2 = G over k. Letting T a denote the unique codimension-1 torus in T killed by a, s(Ua ) is centralized by s(T a ), and the subgroup s(T a )(k) ⊆ T (k) ⊆ T2 (k) = T (W2 (k)) is schematically-dense in the unique codimension-1 W2 (k)-torus Ta in T that is killed by a. Thus, s(Ua ) ⊆ ZG ((Ta )2 ) inside of G by comparison of k-points (since k = k). To get ﬁner control on s(Ua ), we need to use the 1-parameter subgroup [a∨ ] : GL1 → ZG ((Ta )2 ). The maximal k-torus T in ZG ((Ta )2 ) is the almost direct product of the central torus s(T a ) and the 1-dimensional torus [a∨ ](GL1 ), so we will study the conjugation action of [a∨ ](GL1 ) on ZG ((Ta )2 ). First, we need to give an alternative description of ZG ((Ta )2 ). By the selfcontained Proposition A.8.10(1),(2) below, applied over W2 (k), the centralizer functor ZH (Ta ) inside of H is represented by a smooth closed W2 (k)-subgroup. By functorial considerations, we get ZG ((Ta )2 ) = ZH (Ta )2 . Thus, the afﬁne ﬁnite type k-group ZG ((Ta )2 ) is k-smooth since ZH (Ta ) is W2 (k)-smooth, and it is connected as well because the W2 (k)-smooth ZH (Ta ) is connected (as ZH (Ta ) mod p is the torus centralizer ZH (T a ) over a ﬁeld).

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Applying Lemma 2.1.5 to the 1-parameter subgroup [a∨ ] : GL1 → ZG ((Ta )2 ) over k provides a smooth connected unipotent k-subgroup U + := UZG ((Ta )2 ) ([a∨ ]) ⊆ ZG ((Ta )2 ) that represents the functor of points g ∈ ZG ((Ta )2 )(B) = ZH (Ta )(W2 (B)) such that [a∨ ](t) · g · [a∨ ](t)−1 → 1 in ZG ((Ta )2 )B as t → 0, with B any k-algebra. Also, Lie(U + ) is the span of the [a∨ ](GL1 )-weight spaces on Lie(ZG ((Ta )2 )) for the weights whose pairing against [a∨ ] is > 0. Step 3. Since T (k) ⊆ T (W2 (k)), and the Teichmüller lift GL1 → W2× extends to a k-scheme map A1 → W2 carrying 0 to 0, the subgroup U + (k)(⊆ G(k) = H (W2 (k))) contains (Ua )2 (k) = Ua (W2 (k)). Likewise, s(Ua ) ⊆ U + since [a∨ ] = s ◦ a∨ and a∨ (t) · u · a∨ (t)−1 → 1 as t → 0 for u ∈ Ua (k). Hence, s(Ua ) and (Ua )2 are contained in U + . We claim that s(Ua ) ⊆ (Ua )2 . It sufﬁces to show that the inclusion (Ua )2 ⊆ U + is an equality. By smoothness and connectedness, such an equality amounts to a dimension calculation: proving dim U + = 2. We shall describe Lie(U + ) inside of Lie(G) to see that it is 2-dimensional. Passing to Lie algebras on (A.6.1), Lie(G) is an extension of Lie(H ) by Lie(H )(p) . This is compatible with an action of T T that induces the adjoint action of T on the quotient Lie(H ) and the natural scalar extension of this action on the subspace Lie(H )(p) . The root system associated to (H , T ) is reduced, and the roots occurring on Lie(H )(p) are the p-powers of the ones on Lie(H ), so each root line in Lie(H ) admits a unique T -equivariant lift into Lie(G). Hence, Lie(U + ) is spanned by a 1-dimensional weight space for a p on Lie(H )(p) and a 1-dimensional weight space for a. It follows that dim U + = 2, so indeed s(Ua ) ⊆ (Ua )2 . We conclude that s restricts to a section of the reduction map (Ua )2 → Ua . But this reduction map is the natural map W2 Ga for which there is no k-group section. 2

A.7 Lie algebras and Weil restriction In the development of the structure theory of pseudo-reductive groups, Weil restriction and inﬁnitesimal group schemes are useful notions. For this reason, it is sometimes necessary to do computations with Lie algebras of Weil restrictions of group schemes of ﬁnite type over a ﬁeld, so we need to describe such Lie algebras in concrete terms. Also, in our study of exotic pseudo-reductive groups

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it will be useful to work with Lie algebras of inﬁnitesimal group schemes. Both of these topics are taken up in this section. As a preliminary step, we ﬁrst review how a Lie algebra is associated to any group scheme G over any ring k. An elegant treatment of the formalism of Lie algebras of group schemes (and even group functors) is given in [DG, II, §4, §7]. For the convenience of the reader who is more familiar with algebraic groups than algebraic geometry, we begin with a self-contained development of that formalism. (Restricting to the afﬁne ﬁnite type case below will be sufﬁcient for our needs, and it makes some of our arguments more concrete.) A natural Lie algebra structure on g = Tane (G) can be deﬁned in a couple of different ways. The most classical approach is via left-invariant global vector ﬁelds (or rather, left-invariant derivations of the structure sheaf, to allow nonsmoothness), but for some functorial arguments it will be convenient to have a more direct group-theoretic formulation near the identity. To avoid confusion, we now review both viewpoints and prove that they agree. Let us begin with the global approach through vector ﬁelds. A k-linear derivation D of the structure sheaf OG is left-invariant if, for all k-algebras B and g ∈ G(B), the B-linear derivation DB of OGB commutes with pullback along the left multiplication map g : h → gh. This condition can also be expressed in terms of the k[]-scheme automorphism D of Gk[] that is the identity on topological spaces and the k[]-algebra automorphism 1 + D on the sheaf of k[]-algebras OGk[] : Lemma A.7.1 The global derivation D is left-invariant if and only if the diagram Gk[] ×Spec k[] Gk[] 1× D

m

Gk[] ×Spec k[] Gk[]

/ Gk[]

m

D

(A.7.1)

/ Gk[]

commutes. In such cases, D is uniquely determined by its specialization D(e) ∈ g at e. Proof. The key point is the observation that to check such commutativity it sufﬁces to do so against all points g ∈ G(B) ⊆ Gk[] (B[]) in the left factor Gk[] along the left side of (A.7.1) where Spec B varies through an open afﬁne cover of G and g is the corresponding inclusion map (as then the corresponding points of Gk[] constitute an open afﬁne covering). Working more generally with any k-algebra B and any g ∈ G(B), the B[]scheme automorphisms DB[] and gB[] of GB[] trivially commute modulo ,

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and at the level of structure sheaves over B[] they commute precisely when DB and g commute on OGB . If an automorphism φ of a group functor commutes with all left translations then it must be right multiplication by φ(e) (so φ is determined by φ(e)). Hence, if D is left-invariant then it is right multiplication by D(e) = 1 + D(e) ∈ ker(G(k[]) → G(k)) = 1 + g. (Conversely, right multiplication on Gk[] by 1 + v for v ∈ g is a k[]-scheme automorphism of Gk[] that reduces to the identity modulo and so has the form D for a unique D that is necessarily left-invariant.) 2 In concrete terms, D(e) corresponds to the augmented k-algebra map k ⊕ IG /IG2 = ØG /IG2 → k[] given by a + b → a + (D( b))(e) for a local section b of IG lifting b. In other words, D(e) is the “tangent vector” along e induced by the “vector ﬁeld” D. (Keep in mind that we do not assume G is k-smooth, so the classical language of vector ﬁelds is not entirely appropriate.) This proves: Lemma A.7.2 Every element of g uniquely extends to a k-linear left-invariant derivation of the structure sheaf OG . Explicitly, if v ∈ g corresponds to Dv then right multiplication by 1 + v on Gk[] is the identity on the topological space and 1 + Dv on the structure sheaf. In view of this lemma, we can deﬁne a Lie algebra structure [·, ·]G on g over k by forming the commutator of left-invariant derivations, exactly as in the classical smooth case. By construction, this Lie algebra structure is compatible with change of the base ring in the sense that the base change map (2.1.3) is a map of Lie algebras over k . To analyze further properties of the Lie bracket, such as its functoriality in G, we now express the bracket in another way via inﬁnitesimal methods, following [DG, II, §4, 4.1–4.2]. (A related technique is used in [Ser2] to deﬁne a Lie algebra structure on smooth formal groups.) For any k-algebra B and g ∈ G(B), consider the conjugation automorphism cg of GB deﬁned by h → ghg −1 . This induces a B-linear automorphism Ad G (g) of TaneB (GB ) by functoriality. Explicitly, for X ∈ TaneB (GB ), 1 + Ad G (g)(X ) = g(1 + X )g −1

(A.7.2)

in GB (B[ ]) = G(B[ ]). Using B[] in place of B, if g ∈ G(B[]) then Ad G (g) is a B[]-linear automorphism of TaneB[] (GB[] ) = B[] ⊗B TaneB (GB )

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(equality since B[] is ﬁnite free over B) whose reduction modulo is the adjoint action of g mod . Thus, if g mod = 1 then Ad G (g) = id + · ad G (g) for a unique B-linear endomorphism ad G (g) of TaneB (GB ). Writing g = 1 + v for v ∈ TaneB (GB ), we have made a construction ad G : TaneB (GB ) → End B (TaneB (GB )) via Ad G (1 + v) = id + · ad G (v), so (v, w) → (ad G (v))(w) is B-bilinear and natural in B and G. Example A.7.3 The construction of ad G can be described in more classical terms when G is locally of ﬁnite presentation and Cot e (G) is k-ﬂat, such as when k is a ﬁeld (with G not necessarily smooth). Assume we are in such a case, so (2.1.3) is always an isomorphism and hence we may use Yonedastyle arguments with g (i.e., TaneB (GB ) = gB for any k-algebra B). Thus, g → Ad G (g) = d(cg )(e) deﬁnes a k-group map G → GL(g), and d(Ad G )(e) : g → gl(g) is ad G . Before we relate ad G and the Lie bracket, we relate ad G more directly to the inﬁnitesimal group structure. This requires introducing an additional inﬁnitesimal parameter. Lemma A.7.4 For the k-algebra k[, ] = k[] ⊗k k[] and any v, w ∈ g, the commutator (1 + v)(1 + w)(1 + v)−1 (1 + w)−1 ∈ G(k[, ]) is equal to 1 + (ad G (v))(w). Specializing this lemma along = gives nothing interesting, so it is essential that we have introduced two independent inﬁnitesimal parameters. Proof. Identifying k[, ] with k[ ][] and using scalar extension from k to k[], the point 1 + v ∈ Gk[] (k[][ ]) comes from Gk[] (k[]). Thus, the ﬁrst three terms of the commutator have product 1 + Ad G (1 + v)(w) by (A.7.2) (with B = k[], g = 1 + v, and X = w). But AdG (1 + v)(w) as a k[, ]point of G may be computed as a k[]-point of G. As such, since 1 +v is trivial modulo we have Ad G (1 + v) = id + · ad G (v). Applying this to w, we get (1 + v)(1 + w)(1 + v)−1 = 1 + w + (ad G (v))(w) = 1 + (w + (ad G (v))(w)). Since 2 = 0, this ﬁnal term is (1 + (ad G (v))(w))(1 + w).

2

Now we can prove that ad G computes the Lie algebra structure deﬁned via global derivations:

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Proposition A.7.5 The map ad G : g → gl(g) is the adjoint representation v → [v, ·]G for the Lie algebra structure on g. We only need this result for G afﬁne over a ﬁeld, but we must avoid smoothness hypotheses. Proof. Choose v, w ∈ g and consider the k[]-scheme automorphism Dv of Gk[] and the k[ ]-scheme automorphism Dw of Gk[ ] . Extending scalars from k[] and k[ ] to k[, ] via the natural maps gives the respective k[, ]-algebra automorphisms 1 + Dv and 1 + Dw of OGk[, ] (lifting the identity modulo (, )). The commutator of these latter automorphisms is (1 + Dv ) ◦ (1 + Dw ) ◦ (1 + Dv )−1 ◦ (1 + Dw )−1 = 1 + [Dv , Dw ]. Hence, it arises from right translation by 1 + [Dv , Dw ](e) (Lemma A.7.2). Passage from scheme morphisms to pullback on structure sheaves is contravariant, so this right translation computes the commutator automorphism −1 , ( Dw Dv−1 ) of Gk[, ] . For any k-algebra B and g ∈ G(B), let ρg denote right translation by g on GB . Thus, Dv = ρ1+ v on Gk[] , and Dw = ρ1+ w on Gk[ ] . It follows that on Gk[, ] we have −1 −1 −1 −1 ( Dw , Dv ) = ρ1+ w ◦ ρ1+ v ◦ ρ1+ w ◦ ρ1+ v .

But ρg ◦ ρh = ρhg , so the 4-fold composite is right translation by (1 + v)(1 + w)(1 + v)−1 (1 + w)−1 = 1 + (ad G (v))(w) ∈ G(k[, ]) due to Lemma A.7.4. We conclude that for x = (ad G (v))(w) and y = [Dv , Dw ](e) in g, right translations on Gk[ ] by 1 + x and 1 + y coincide after scalar extension to k[, ]. But G(k[ ]) → G(k[, ]) is injective since k[ ] → k[, ] is an injective map between local rings, so x = y. 2 As a consequence of Proposition A.7.5 and the obvious functoriality of ad G in G, it follows that the Lie bracket is functorial in G. Moreover, we immediately obtain the following general compatibility of Lie algebras and Weil restriction: Corollary A.7.6 Let k → k be a ﬁnite ﬂat map of noetherian rings, G a quasi-projective k -group scheme of ﬁnite type, and G = R k /k (G ). The natural isomorphism of groups G(k[]) G (k []) induces an isomorphism of subgroups Lie(G) Lie(G ) that is compatible with the structure of Lie algebra over k on both sides and is functorial in G .

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Note that the quasi-projectivity hypothesis in this corollary always holds when k is a ﬁeld, k is a ﬁnite product of ﬁelds, and G is a k -group of ﬁnite type. We will only need the corollary for such k and k . (If we work more generally with group functors then the hypotheses on the groups and rings can be weakened considerably.) A useful application of Corollary A.7.6 is in the determination of ﬁelds of deﬁnition of geometric unipotent radicals. The idea is to reduce nontrivial problems concerning ﬁelds of deﬁnition for subgroups to easier problems in linear algebra concerning ﬁelds of deﬁnition for tangent spaces. For example, consider a purely inseparable ﬁnite extension of ﬁelds k /k, a smooth connected k -group G , and the smooth connected k-group G = R k /k (G ). By Proposition A.5.11, the natural k -group map q : R k /k (G )k → G is a smooth surjection with connected unipotent kernel. Thus, ker q is a k -descent of Ru (R k /k (G )k ) when G is reductive. In [SGA3, XVII, App. III, 5.1] it is shown that if k = k and G is not unipotent then the ﬁeld of deﬁnition over k for ker q is not k. We will prove a more precise statement in such cases: the ﬁeld of deﬁnition is equal to k . In fact, even the ﬁeld of deﬁnition over k for the k -subspace ker(Lie(q)) = Lie(ker q) ⊆ Lie(R k /k (G )k ) = k ⊗k Lie(R k /k (G )) = k ⊗k Lie(G ) turns out to be k . (It is rather amazing that in these cases the ﬁeld of deﬁnition of the Lie algebra cannot be strictly smaller than the ﬁeld of deﬁnition for the group in nonzero characteristic.) The determination of this minimal ﬁeld of deﬁnition over k is of vital importance in our work, due to its role in the proof of Theorem 1.6.2, so here is a self-contained development. We begin with a lemma concerning ﬁelds of deﬁnition in linear algebra (in the sense of Remark 1.1.7). Lemma A.7.7 Let K /k /k be a tower of ﬁeld extensions, and W a k -vector space. Consider an intermediate K -subspace V as in: ker(K ⊗k W → K ⊗k W ) ⊆ V ⊆ K ⊗k W . If the ﬁeld of deﬁnition F/k for V inside of K does not contain k then V = K ⊗k W . Proof. Proceeding by contradiction, suppose V is a proper K -subspace (so W = 0). Choose a collection of k -lines {Li } that directly span W , and let Vi = V ∩ (K ⊗k Li ). For every i, clearly ker(K ⊗k Li → K ⊗k Li ) ⊆ Vi ⊆ K ⊗k Li and the ﬁeld of deﬁnition over k for Vi inside of K ⊗k Li is contained in F (and so does not contain k ). There must be some i such that Vi is a proper

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K -subspace of K ⊗k Li , so to get a contradiction we may rename such an Li as W to reduce to the case dimk W = 1. But then ker(K ⊗k W K ⊗k W ) is a K -hyperplane, so its containment in the proper K -subspace V forces it to equal V . Now we can replace the k -line W with k to reduce to a problem in ring theory: if μ : K ⊗k k → K is the natural map deﬁned by multiplication and the ideal ker μ viewed as a K -subspace is deﬁned over a subﬁeld F ⊆ K over k then we claim that F contains k . The map μ is the scalar extension by K ⊗k (·) applied to the natural map m : k ⊗k k → k . Hence, ker μ is deﬁned over the subﬁeld k ⊆ K over k, so by minimality F ⊆ k . We can therefore replace K with k to reduce to showing that if F ⊆ k is a subﬁeld over k such that ker(m) = k ⊗F I for an F-subspace I ⊆ F ⊗k k then necessarily F = k . If such an I exists then k = (k ⊗k k )/ ker(m) = k ⊗F ((F ⊗k k )/I ), so (F ⊗k k )/I is 1-dimensional over F yet its (F-isomorphic) image in k is k . Thus, [k : F] = 1, so F = k . 2 Proposition A.7.8 Let k /k be a ﬁnite extension of ﬁelds, K /k any extension ﬁeld, and G a smooth connected k -group. Let G = R k /k (G ), and let q : Gk → G be the natural surjective k -homomorphism. Consider a smooth K -subgroup scheme H ⊆ GK that contains (ker q) ⊗k K . (1) Let F ⊆ K be the ﬁeld of deﬁnition over k for H . If H = GK then F contains k . (2) If G is afﬁne and not unipotent and k /k is purely inseparable then the ﬁeld of deﬁnition over k for Ru (Gk ) coincides with the ﬁeld of deﬁnition over k for Ru (G ) inside of k. The same holds for geometric radicals if k G is afﬁne and not solvable. Proof. We ﬁrst deduce (2) from (1), so assume k /k is purely inseparable. In particular, k has a unique k-embedding into k. By Proposition A.5.11(2), ker q is a smooth connected unipotent k -group. Hence, (ker q) ⊗k k is contained inside of Ru (Gk ). The exact sequence 1 → ker q → Gk → G → 1

(A.7.3)

implies that if G is not unipotent then the smooth connected afﬁne k-group G cannot be unipotent. Thus, when G is afﬁne and non-unipotent we can apply (1) to K = k and H = Ru (Gk ). It follows that in such cases the ﬁeld of deﬁnition F/k for H contains k . In other words, F is also the ﬁeld of deﬁnition over k for H . But Gk / ker q = G and over k this carries H /(ker q)k over to Ru (G ), so for any subﬁeld L of k over k we see that H is deﬁned over L (as k

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a subgroup of GL ) if and only if Ru (G ) is deﬁned over L (as a subgroup of k GL ). This settles (2) for unipotent radicals, and the assertion for radicals in the non-solvable case goes the same way. To prove (1) we will show that if F does not contain k then H = GK . Since G is k-smooth and connected and H is K -smooth, it sufﬁces to show that Tane (H ) = Tane (GK ). Observe that V := Tane (H ) is a K -subspace of Tane (G)K , and its ﬁeld of deﬁnition over k is certainly contained in F. Since V contains Lie(ker q)K = ker(Lie(q)K ), and (by Corollary A.7.6) Lie(q) is identiﬁed with the kernel of the natural multiplication map k ⊗k Lie(G ) → Lie(G ), we are reduced to a problem in linear algebra that is settled by LemmaA.7.7.

2

The next issue to address is a criterion for R k /k (G ) to be its own derived group, with G a connected semisimple k -group and k /k a ﬁnite extension of ﬁelds that may not be separable. It is instructive to ﬁrst give an example in which R k /k (G ) admits a nontrivial commutative quotient. Example A.7.9 Let k /k be a purely inseparable ﬁnite extension of ﬁelds and G1 → G2 a central isogeny between connected semisimple k -groups with kernel μ . The exact sequence of fppf group sheaves (1.3.3) shows that the k-group R k /k (G1 )/R k /k (μ ) is a normal subgroup of R k /k (G2 ), and that the cokernel Q by this subgroup is commutative. Thus, R k /k (G2 ) is not its own derived group if dim Q > 0, which is to say that dim R k /k (μ ) > 0. This occurs when k = k and μ is not k -étale, by Example 1.3.2. For example, R k /k (PGLp ) is not its own derived group when char(k) = p > 0 and k /k is a nontrivial purely inseparable ﬁnite extension. We now establish a general criterion for a Weil restriction to be its own derived group, and then apply it in the simply connected semisimple case. Proposition A.7.10 Let k be a ﬁeld and A a nonzero ﬁnite k-algebra. Let G be a smooth afﬁne A-group such that each geometric ﬁber of G over Spec A is its own derived group and acts on its Lie algebra with vanishing space of coinvariants. The smooth afﬁne k-group R A/k (G) is its own derived group, or equivalently it has no nontrivial commutative quotient over k. In our study of exotic pseudo-reductive groups, this result is used in the generality in which it is stated (without reductivity hypotheses on G and with A possibly non-reduced). Before we prove Proposition A.7.10, we make some remarks on its hypotheses. First, the vanishing hypothesis on ﬁbral coinvariants implies the ﬁbral perfectness hypothesis when the ﬁbers are connected. Second, the afﬁne hypothesis on G can be relaxed to ﬁnite type because the only role of

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afﬁneness is to ensure that R A/k (G) exists as a scheme. Existence only requires that any ﬁnite set of points of G lies in an afﬁne open subscheme, and this latter property holds in the ﬁnite type case since Gred is quasi-projective over the ﬁnite product of ﬁelds Ared (due to Proposition A.3.5). Proof. By extending scalars from k to k and from A to A ⊗k k, we may assume that k is algebraically closed. In this case the problem is to show that the abstract group G(A) = R A/k (G)(k) is its own derived group. Passing to the local factor rings of A allows us to assume that A is local. Let G0 be the special ﬁber of G. We may induct on the length of A, the case of length 1 being obvious. In the general case with A of length larger than 1, there is an ideal I of A that is killed by the maximal ideal of A and is of dimension 1 over k. Thus, the result holds for R (A/I )/k (G mod I ) by induction, so G(A/I ) has no nontrivial commutative quotient. Observe that the reduction map π : G(A) → G(A/I ) is surjective by Asmoothness of G. We claim that the image of ker π in any commutative quotient of G(A) is trivial, so any commutative quotient of G(A) is also a commutative quotient of G(A/I ) and thus is trivial. To prove the triviality of the image of ker π in any commutative quotient of G(A), ﬁrst observe that ker π is naturally identiﬁed with the underlying additive group of I ⊗k Lie(G0 ) in such a way that the natural conjugation action by G(A/I ) on ker π factors through the natural adjoint action by G0 (k) on I ⊗k Lie(G0 ). Recall also that dimk I = 1. Hence, in any commutative quotient of G(A) the image of ker π is a quotient of the additive group Lie(G0 ) in which the image of every element of the form g0 .v−v vanishes (for g0 ∈ G0 (k) acting on v ∈ Lie(G0 ) by the adjoint action). The differences g0 .v − v depend k-linearly on v and so generate Lie(G0 ) as an additive group since we assume Lie(G0 ) has vanishing space of coinvariants for the adjoint action. Thus, ker π has vanishing image in every commutative quotient of G(A), as desired. 2 Corollary A.7.11 Let k be a ﬁeld and k a nonzero ﬁnite reduced k-algebra. Let G be a semisimple k -group whose ﬁbers over the factor ﬁelds of k are connected and simply connected. Then R k /k (G ) is its own derived group. Proof. We may and do assume that k is a ﬁeld. By PropositionA.7.10, it sufﬁces is a connected and simply connected semisimple group over to prove that if G on Lie(G) has vanishing space of a ﬁeld K then the adjoint representation of G coinvariants. contains a maximal K-torus We can assume that K is separably closed, so G is simply connected, any pair of opposite root groups T that is K-split. Since G relative to T generates an SL2 (rather than PGL2 ). These copies of SL2 generate and their Lie algebras span Lie(G) since G is simply connected, so it sufﬁces G, to check the vanishing of the coinvariants for SL2 acting on sl2 by the adjoint

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action. This is a straightforward calculation (even in characteristic 2, where the analogue for PGL2 acting on pgl2 = gl2 /z is false). 2 Remark A.7.12 In view of the conclusion of Corollary A.7.11, the k -group G2 in Example A.7.9 must violate the vanishing hypothesis on coinvariants in Proposition A.7.10 when k = k and μ is not k -étale. To verify the nonvanishing directly, consider the G2 -equivariant quotient g = coker(Lie(G1 ) → Lie(G2 )) of Lie(G2 ). This is nonzero for dimension reasons, since Lie(μ ) = 0, To check that g has trivial G2 -action, we may assume that k is separably closed. Hence, G1 (k ) has Zariski-dense image in G2 , so it is enough to show that the G1 (k )-action on g is trivial. But there is a G1 (k )-equivariant identiﬁcation of g with a subgroup of the degree-1 fppf cohomology H1 (Spec(k []), μk [] ) over the dual numbers k [], and the G1 (k )-action on this cohomology is trivial due to the centrality of μ in G1 . Now we shift our attention from the Lie bracket to a Lie algebra structure that is speciﬁc to prime characteristic. Let p be a prime. For any Lie algebra g over an Fp -algebra k, a p-Lie algebra structure on g is a set-theoretic map X → X [p] from g to itself such that ad(X [p] ) = ad(X ) p , (cX )[p] = c p X [p] for all c ∈ k, and (X + Y )[p] satisﬁes an identity whose precise formulation we omit here; see [Bo2, 3.1] or [DG, II, §7, 3.1–3.3] for the complete deﬁnition. One example of such a structure is the p-power operation in any associative k-algebra (equipped with its standard Lie algebra structure via commutators). Thus, for any group scheme G over k, Lie(G) admits a p-Lie algebra structure via the p-power operation on left-invariant global k-linear derivations of OG (within the associative algebra of all k-linear endomorphisms of OG ). We will write Liep (G) when we wish to think of Lie(G) equipped with its p-Lie algebra structure arising from G. Lemma A.7.13 The p-Lie algebra structure is functorial in G. Moreover, if G = R k /k (G ) for a ﬁnite ﬂat map k → k of noetherian rings and a quasiprojective k -group G then the natural identiﬁcation Lie(G) Lie(G ) of Lie algebras over k respects the p-Lie algebra structures. Proof. In the spirit of the proof of Proposition A.7.5, we give an inﬁnitesimal description of the p-operation, from which functoriality and the compatibility with Weil restriction will be immediate. Consider the p-fold tensor power k[1 , . . . , p ] = k[]⊗p . i it is clear that 1, s, s2 , . . . , s p−1 and Since each i2 vanishes, for s = e := i linearly span the k-subalgebra of Sp -invariant elements. As a kalgebra, this is k[s, e] with relations s p = se = e2 = 0. In particular, there is

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a unique k-algebra map k[s, e] k[] satisfying s → 0 and e → . For any v ∈ g, the elements 1 + i v ∈ G(k[1 , . . . , p ]) pairwise commute (set w = v in Lemma A.7.4 and use Proposition A.7.5). Thus,

(1 + i v) ∈ G(k[1 , . . . , p ])Sp = G(k[s, e]),

so specialization along s → 0 and e → yields an element 1 + [v] for some [v] ∈ g. It sufﬁces to prove that v → [v] is the p-operation. Right translation by 1+[v] on Gk[] is the specialization along k[s, e] k[] of right translation by (1 + i v) ∈ Gk[s,e] (k[s,e]). Thus, on the level of structure sheaves, 1 + D[v] is the specialization of (1 + i Dv ). But this latter specialization is clearly p p p 1 + Dv , so D[v] = Dv . By deﬁnition, Dv[p] = Dv , so we are done. 2 When G is afﬁne, Lemma A.7.13 allows us to compute the p-operation on Liep (G) using a k-subgroup inclusion of G into GLn (since the p-operation is the p-power map L → L p on Mat n (k) = gln , as follows from applying the proof of Lemma A.7.13 to G = GLn ). The usefulness of the p-operation for our purposes is due to the following result. Proposition A.7.14 Let k be a ﬁeld with char(k) = p > 0 and consider the functor G Liep (G) from locally ﬁnite type k-groups to p-Lie algebras over k. This restricts to an equivalence on the full subcategory of ﬁnite k-group schemes G for which the relative Frobenius morphism FG /k : G → G (p) vanishes. Moreover, if B is any k-algebra and G is a ﬁnite k-group with vanishing relative Frobenius morphism then the natural map HomB (GB , GB ) → Hom(Liep (G )B , Liep (GB ))

(A.7.4)

is bijective for any locally ﬁnite type k-group G. Proof. This is a special case of [SGA3, VIIA , 7.2, 7.4] (which works more generally with group schemes that may not arise via base change from a ﬁeld). 2 Deﬁnition A.7.15 Let k be a ﬁeld with char(k) = p > 0. A k-group scheme G has height 1 if G is k-ﬁnite and the relative Frobenius morphism FG /k vanishes. Example A.7.16 Let G be a k-group locally of ﬁnite type and n a p-Lie subalgebra of g = Lie(G). Let N be the unique k-group of height 1 such that Lie(N ) = n as p-Lie algebras. By (A.7.4), the inclusion n → g of p-Lie algebras arises from a unique k-homomorphism f : N → G. The k-group ker f has height 1 (as this is inherited from N ), and Lie(ker f ) = ker(n → g) = 0.

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Hence, ker f = 1, so f is a closed immersion (PropositionA.2.1). In other words, n uniquely “exponentiates” to a closed k-subgroup N ⊆ G of height 1. We claim that N is (scheme-theoretically) normal in G if and only if n is stable under the adjoint action of G on g. Indeed, normality says that NB is carried back into itself under G(B)-conjugation for all k-algebras B, whereas stability under the adjoint action says that nB is stable under the adjoint action of G(B) on gB for all k-algebras B. By the bijectivity of (A.7.4), these two conditions are equivalent. In case n is G-stable and G is a smooth afﬁne k-group, there are two quotients of G naturally associated to n: the quotient G/N in the sense of group schemes, and the quotient G/n in the sense of Borel [Bo2, §17]. We never use this second point of view, but since it has the advantage of enabling one to study quotients by normal inﬁnitesimal subgroup schemes entirely in the language of smooth groups and Lie algebras, we now prove that these two quotients coincide. To prove the equality of G/N and G/n as quotients of G, we ﬁrst note that (by construction) the quotient map G → G/n is a purely inseparable isogeny whose Lie algebra kernel is n and it is universal among k-homomorphisms G → H to smooth afﬁne k-groups such that the tangent map at the identity kills n. In particular, G/n uniquely dominates G/N as quotients of G. By Proposition A.7.14, the quotient map G → G/n kills N since it kills n = Lie(N ) on Lie algebras, so G/N uniquely dominates G/n as quotients of G. This proves that G/N = G/n as quotients of G.

A.8 Lie algebras and groups of multiplicative type We now turn to the properties of centralizers under the action of a torus, and more generally under the action of a group scheme of multiplicative type. For example, the study of exotic pseudo-reductive groups involves working with centralizers of a μp -action in characteristic p ∈ {2, 3}. Also, centralizers under a torus action on a possibly non-smooth group are useful in our work with pseudo-reductive groups in all nonzero characteristics. Before we prove results concerning centralizers for the action of a group scheme of multiplicative type over a ﬁeld, it will be convenient to review some basic results concerning the classiﬁcation of such group schemes and their ﬁnite-dimensional linear representations. The ﬁrst key fact gives a useful interpretation of character groups: Proposition A.8.1 Let k be a ﬁeld. The functors Hom(, GL1 ) and M X(M ) := Hom(M , GL1 ) are quasi-inverse anti-equivalences between the category of k-groups M of multiplicative type and the category of étale k-groups whose group of ks -points is a ﬁnitely generated Z-module. Proof. By Galois equivariance, it sufﬁces to treat the case when k is separably closed. It is straightforward to check (by induction on dimension, along with

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Cartier duality for ﬁnite group schemes in the 0-dimensional case) that the ksubgroup schemes of GLn1 are in natural bijection with quotients of X(GLn1 ) = Zn , and that the functors in question restrict to an anti-equivalence between the category of M that are a k-subgroup of some GLn1 and the category of ﬁnitely generated Z-modules. The problem is therefore to prove that every k-subgroup scheme of GLn1 is deﬁned over k. (The argument will work over any ﬁeld, not just a separably closed ﬁeld.) Let M be such a subgroup scheme. We claim that the action of some element of GLn (Z) carries M over to GLr1 × μ for some r 0 and a ﬁnite k-subgroup μ ⊆ GLn−r 1 . This would reduce our problem to the case of k-ﬁnite M , and to verify it we only need to work over k. 0 is a nontrivNow taking k to be algebraically closed, if dim M > 0 then Mred n ial torus, so it contains a GL1 . Any GL1 inside of GL1 over k can be converted into the ﬁrst factor by applying a suitable element of GLn (Z) to GLn1 . Passing to the quotient by this and proceeding by induction on dim M then yields the claim. Now returning to a general k, we may assume M is k-ﬁnite, so M is m-torsion for some m 1. Hence, M is a k-subgroup scheme of μnm . We therefore just have to show that such subgroup schemes are deﬁned over k. Cartier duality converts this into the analogous assertion for quotients of the ﬁnite constant k-group (Z/mZ)n , which is clear. 2 Corollary A.8.2 Let M be a group scheme of multiplicative type over a ﬁeld. Every k-subgroup scheme of M is of multiplicative type, and the closed subscheme Mred ⊆ M is a smooth k-subgroup. 0 is a k-torus. In particular, Mred Proof. Since Mk is a subgroup scheme of a torus, every k-subgroup scheme of M is certainly of multiplicative type (by deﬁnition). To analyze Mred , we may assume that k is separably closed. By applying the structure theorem for ﬁnitely generated Z-modules to X(M ) and dualizing back, we conclude that M GLr1 × μ for a ﬁnite multiplicative group μ (dual to an ordinary ﬁnite commutative group). Hence, Mred = GLr1 × μred . This reduces the problem to the case M = μ, so by passing to primary parts we can separately treat the cases when M is étale or inﬁnitesimal, each of which is clear. 2 Example A.8.3 The ﬁnal claim in Corollary A.8.2 is speciﬁc to the case of groups of multiplicative type because over any imperfect ﬁeld k, (i) the underlying reduced scheme of an afﬁne k-group scheme G of ﬁnite type need not be a k-subgroup scheme, (ii) even when Gred is a k-subgroup, it may not be k-smooth.

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The main issue is that for schemes X and Y of ﬁnite type over an imperfect ﬁeld, the inclusion (X × Y )red ⊆ Xred × Yred may not be an equality. For an example of (i), assume k is imperfect of characteristic p > 0 and consider the isomorphism H1 (k, μp ) k × /(k × ) p provided by fppf Kummer theory. This cohomology group classiﬁes isomorphism classes of commutative p-torsion k-group extensions of Z/pZ by μp . (For the convenience of the interested reader, here is a proof. The δ-functor H• (k, ·) on p-torsion fppf abelian sheaves over Spec k is the derived functor of (k, ·) = Homk (Z/pZ, ·). Thus, for p-torsion abelian sheaves F , the group H1 (k, F ) is naturally isomorphic to the group Ext 1k (Z/pZ, F ) classifying isomorphism classes of extensions E of F by Z/pZ in the category of p-torsion abelian sheaves over Spec k. Since we can view E as a Z/pZ-torsor over F , by descent theory for afﬁne maps it follows that E is represented by a ﬁnite k-group scheme when F is, such as for F = μp . The notion of short-exactness for ﬁnite commutative k-groups matches that for fppf abelian sheaves over Spec k.) Consider such an extension G of Z/pZ by μp that is is classiﬁed by a mod (k × ) p for some a ∈ k × that is not a pth power in k. The ﬁber of G Z/pZ over i ∈ Z/pZ is Spec k[x]/(x p − a[i] ) for any representative [i] ∈ Z of i. Hence, G 0 = μp and G − G 0 is reduced, so Gred is the disjoint union of G − G 0 and the identity point. This is étale at the origin but is not étale over k, so Gred admits no k-group scheme structure. To give an example of (ii), let k /k be a purely inseparable extension of degree p = char(k) > 0 and let H = R k /k (GL1 ). Consider G = H [p], so Gk = μp × Ru (Hk ). Thus, G is not smooth. Writing k = k(α) with α p = a ∈ k − k p , we have G(B) = {x0 + α ⊗ x1 + · · · + α p−1 ⊗ xp−1 ∈ k ⊗k B | p

p

p

xi ∈ B, x0 + ax1 + · · · + a p−1 xp−1 = 1} functorially in k-algebras B. In other words, G is the hypersurface in afﬁne p-space over k deﬁned by the vanishing of the polynomial p

p

p

x0 + ax1 + · · · + a p−1 xp−1 − 1. This polynomial is irreducible over k because a ∈ k − k p . We conclude that Gred = G. Remark A.8.4 Instances of (i) (i.e., examples of afﬁne k-group schemes of ﬁnite type whose underlying reduced scheme is not a subgroup scheme) naturally arise as the group scheme of p-torsion in a Tate elliptic curve when k is a Laurent series ﬁeld of characteristic p with parameter a.

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Example A.8.5 Let k be a nonzero commutative ring with connected spectrum. For any m 1 we have Homk (μm , GL1 ) = End k (μm ) = Z/mZ by Cartier duality. Likewise, we claim that End k (GL1 ) = Z. Connectedness reduces this to a problem over the local rings of k, for which we can reduce to the noetherian case. Scalar extension to the completion and comparison against the special ﬁber reduces the problem to the case of artin local k, in which case an elementary induction on the length of k does the job. We conclude that it is reasonable to deﬁne a k-split group of multiplicative type to be a k-group that is isomorphic to a ﬁnite product of copies of GL1 and k-groups μm for various m 1. The interest in this notion is that it is “dual” to the category of ﬁnitely generated Z-modules via the functors M X(M ) := Homk (M , GL1 ), Dk () := ⊗ GL1 (where ⊗ GL1 denotes the afﬁne k-group representing the sheaﬁﬁcation of B ⊗Z B× for the étale topology). This duality is veriﬁed by using product decompositions to reduce to the special cases M = GL1 and M = μm , as well as = Z and = Z/mZ, with m 1. If we drop the connectedness hypothesis on Spec k in Example A.8.5 then we can go in the reverse direction. That is, given a ﬁnitely generated Z-module we can construct the afﬁne k-group Dk () := ⊗ GL1 . However, X(Dk ()) is the global sections of the constant sheaf for the Zariski topology on Spec k, so this is not if Spec k is not (non-empty and) connected. Nonetheless, for our purposes the following deﬁnition is sufﬁcient. Deﬁnition A.8.6 Let k be a ring. A k-split group of multiplicative type is a kgroup M that is k-isomorphic to Dk () for some ﬁnitely generated Z-module . (Equivalently, M is a ﬁnite product of copies of GL1 and k-groups μm for various m 1.) Example A.8.7 Let k be a commutative ring with connected (non-empty) spectrum, M a split k-group of multiplicative type, and X = X(M ). For every χ ∈ X , the k-homomorphism χ : M → GL1 is identiﬁed with an element eχ in the coordinate ring k[M ] of M . Since (eχ eχ )(m) = χ (m)χ (m) = (χ χ )(m) for all points m of M , we have eχ eχ = eχ χ . Likewise, e1 is the constant function 1, and the comultiplication is eχ → eχ ⊗ eχ since χ (mm ) = χ (m)χ (m ).

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The group algebra k[X ] over k has a natural structure of commutative and co-commutative Hopf algebra, and we have just constructed a natural map k[X ] → k[M ]. Since X(M × M ) = X(M ) ⊕ X(M ) and M is a product of copies of GL1 and k-groups μm , we verify that k[X ] → k[M ] is an isomorphism by reducing to a direct calculation when M = GL1 and M = μm (m 1). This procedure works without connectedness hypotheses on Spec k provided that we begin with the Z-module rather than the k-group. That is, if is a ﬁnitely generated Z-module and M = Dk () is the associated k-group of multiplicative type then the natural map from to X(M ) deﬁnes a map of Hopf algebras k[] → k[M ] over k. This is seen to be an isomorphism by reducing to the special cases = Z and = Z/mZ. It is an important fact in the theory of split tori over a ﬁeld that their ﬁnitedimensional linear representations are direct sums of characters (and so are completely reducible). The following lemma records the generalization for split groups M of multiplicative type over any ring k, using functorial linear representations of M on a k-module V ; this means a B-linear representation of M (B) on B ⊗k V functorially in k-algebras B. Lemma A.8.8 Let k be a commutative ring, and a ﬁnitely generated Zmodule. Let M = Dk () be the corresponding k-split k-group of multiplicative type. There is a natural bijection between the set of functorial linear representations ρ of M on a k-module V and the set of k-linear -gradings on V . The correspondence is that if V = λ∈ Vλ then ρ(t)

vλ = λ(t)vλ .

In particular, all such ρ are semisimple when k is a ﬁeld. Proof. Let End(V ) denote the linear endomorphism functor of V ; this associates to any k-algebra B the B-algebra End B (B ⊗k V ) of B-linear endomorphisms of B ⊗k V . (This is generally not representable, unless V is locally free of ﬁnite rank over k.) For any afﬁne k-group scheme G, to give a functorial linear representation ρ of G on V is to give a k[G]-valued point of End(V ) that specializes to the identity automorphism of V at the identity of G and in general is multiplicative. That is, we get a k[G]-linear endomorphism of the k[G]-module V ⊗k k[G], or equivalently a k-linear map αρ : V → V ⊗k k[G], satisfying a multiplicative action condition and an identity element condition. Concretely, if αρ (v) = vi ⊗ fi then for any k-algebra B and g ∈ G(B) we have ρ(g)(v) =

fi (g)vi

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in VB := V ⊗k B. Likewise, if αρ (vi ) = g, g ∈ G(B) we compute ρ(gg )(v) =

j∈Ji

vij ⊗ hij then for any B and

fi (gg )vi , ρ(g)(ρ(g )(v)) = ρ(g) fi (g )vi = hij (g)fi (g )vij . i

j∈Ji

Thus, if m∗G : k[G] → k[G] ⊗k k[G] denotes the comultiplication then the multiplicative action condition ρ(gg ) = ρ(g) ◦ ρ(g ) says exactly that the elements vi ⊗ fi = vi ⊗ m∗G ( fi ), (id V ⊗ m∗G ) (αρ ⊗ id k[G] ) vi ⊗ fi = ( vij ⊗ hij ) ⊗ fi i

j∈Ji

in V ⊗k k[G] ⊗k k[G] have the same pullback to VB under all pairs (g, g ) ∈ G(B) × G(B) for all B, which is to say that they coincide in V ⊗k k[G] ⊗k k[G]. Since vi ⊗ fi = αρ (v) with v ∈ V arbitrary (and vi and fi depending on v), the action condition says exactly that the diagram of k-linear maps αρ

/ V ⊗k k[G]

V αρ

id V ⊗m∗G

V ⊗k k[G]

αρ ⊗id k[G]

/ V ⊗k k[G] ⊗k k[G]

(A.8.1)

commutes. Now consider the special case when G = M = Dk () is k-split of multiplicative type, so the k-algebra k[M ] is naturally identiﬁed with ⊕λ∈ keλ where eλ (t) = λ(t) for all points t of M , so eλ eλ = eλλ . The comulti∗ plication is mM : eλ → eλ ⊗ eλ . The map αρ can be uniquely written as v → fλ (v) ⊗ eλ for k-linear maps fλ : V → V (a ﬁnite sum for each v ∈ V ), so ρ(g)(v) = λ λ(g)fλ (v) for points g of G valued in any k-algebra. The diagram (A.8.1) encoding the action condition t.(t .v) = (tt ).v says that λ

eλ ⊗ eλ ⊗ fλ (v) =

eλ ⊗ eλ ⊗ fλ ( fλ (v)),

λ,λ

or in other words that fλ ◦ fλ vanishes if λ = λ and is equal to fλ if λ = λ. That is, the linear maps fλ are projectors onto mutually independent submodules Vλ ⊆ V .

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Since eλ (1) = 1 for all λ, the condition 1.v = v says that v = λ fλ (v) for all v (a ﬁnite sum), so V = ⊕Vλ . Since ρ(g)(v) = λ λ(g)fλ (v) (a ﬁnite sum for any speciﬁc v ∈ V ), the M -action on V restricts to the action on each Vλ through λ. 2 Remark A.8.9 If V is a ﬁnite free k-module of rank n 1 then we can interpret the map αρ in the preceding proof very concretely in terms of a kbasis {v1 , . . . , vn } of V by taking Ji = {1, . . . , n} and vij = vj in the calculations (as we may). The action condition gives the identity fi (g g) = j hji (g )fj (g) for all g , g ∈ G(B) with a k-algebra B. This says that αρ (gv) = vi ⊗ fi (xg) for all v ∈ VB , or in other words that the map αρ : V → V ⊗ k[G] is G-equivariant if we use ρ on the source and id V ⊗rG on the target, with rG the right regular representation of G on k[G]. That is, upon ﬁxing an ordered k-basis of V , αρ is a G-equivariant embedding of V into k[G]n with each factor k[G] endowed with the right regular G-action. This embedding encodes the representation space V in terms of its “matrix coefﬁcients”. For any μp -action on a group scheme G locally of ﬁnite type over a ﬁeld of characteristic p > 0, the centralizer of μp on G can be studied via Lie algebra centralizers. This viewpoint is used by Borel in [Bo2, 9.1–9.2], where inﬁnitesimal group schemes are avoided, but for our purposes it is useful to have both viewpoints available. The third part of Proposition A.8.10 below uniﬁes them. Before stating the result, we recall that an afﬁne group scheme H of ﬁnite type over an algebraically closed ﬁeld F is linearly reductive if its ﬁnite-dimensional algebraic representations are completely reducible. For example, by Lemma A.8.8 linear reductivity holds if the identity component H 0 is of multiplicative type and the component group H /H 0 has order not divisible by char(F). We also observe that if H is a ﬂat afﬁne group scheme over a ring k and M is a ﬁnitely presented k-module equipped with a functorial linear left H action (i.e., an R-linear left action of H (R) on MR naturally in k-algebras R) then the linear dual M ∨ = Hom(M , k) inherits a natural linear left H -action. Indeed, to naturally deﬁne an H (R)-action on (M ∨ )R naturally in all k-algebras R it sufﬁces (by the proof of Yoneda’s Lemma) to treat ﬂat R since k[H ] is k-ﬂat, and for such R we have (M ∨ )R = HomR (MR , R) on which there is an evident R-linear left H (R)-action. The submodule (M ∨ )H of elements ∈ M ∨ such that the element R ∈ (M ∨ )R is H (R)-invariant for all R consists of those linear forms : M → k that are H -equivariant (in the functorial sense), again since it sufﬁces to consider ﬂat R (such as R = k[H ]). A case of much interest is M = Coty (Y ) for a k-scheme Y locally of ﬁnite presentation that is equipped with a left H -action, with y ∈ Y (k) ﬁxed by the H -action; in such cases M ∨ = Tany (Y ).

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Proposition A.8.10 Let Y be a separated scheme over a ring k, and let H be an afﬁne ﬂat k-group scheme equipped with a left action on Y . Assume that for some faithfully ﬂat extension k → k the coordinate ring k [Hk ] = k ⊗k k[H ] is a projective k -module. (1) The invariant locus of the H -action on Y (i.e., the functor assigning to any k-algebra R the subset of points y ∈ Y (R) such that HR acts trivially on y) is represented by a closed subscheme Y H in Y , with Y H locally of ﬁnite type over k if Y is. If Y is locally of ﬁnite presentation over k then Tany (Y H ) = Tany (Y )H for any y ∈ Y H (k). (2) Assume k is noetherian, Y and H are locally of ﬁnite type, and the geometric ﬁbers of H → Spec k are linearly reductive. Let Y be another separated k-scheme locally of ﬁnite type equipped with a left H -action. If f : Y → Y is a smooth H -equivariant k-map then the induced map Y H → Y H is smooth. In particular, if Y is smooth then Y H is smooth. (3) Assume k is a ﬁeld with char(k) = p > 0, M := H is of multiplicative type and p-torsion, and G := Y is equipped with a k-group structure respected by the M -action. Consider the induced action of M on g = Lie(G) and the adjoint representation ρ : m = Lie(M ) → gl(g). The representation ρ determines G M and Lie(G M ) via the formulas G M = {g ∈ G | (Ad(g))(ρ(X )) = ρ(X ) for all X ∈ m}, Lie(G M ) = {v ∈ g | ρ(X )(v) = 0 for all X ∈ m}. Before we prove this result, we make some remarks on its hypotheses. The separatedness hypothesis on Y is satisﬁed when k is a ﬁeld and Y is a k-group scheme. Two interesting cases for which the projectivity hypothesis on k[H ] is satisﬁed after a faithfully ﬂat scalar extension are when k is a ﬁeld and when H is a group of multiplicative type in the sense of Deﬁnition A.8.6, or more generally a k-group of multiplicative type in the sense of [SGA3, IX, 1.1]. (By the deeper [RG, I, 3.3.3–3.3.5], the homogeneity arising from the k-group structure on H implies that k[H ] is projective over k whenever H is ﬁnitely presented and afﬁne over k with connected ﬁbers. We do not use this fact, but it shows that the projectivity hypothesis on H in Proposition A.8.10 is satisﬁed in most cases of “practical” interest.) Finally, as a special case of (3), if dim m = 1 (e.g., M = μp ) and M is a subgroup scheme of G acting via conjugation, then for a nonzero X in m we get expressions for G M and its Lie algebra in terms of the effect on X of the adjoint actions of G and g on g. More speciﬁcally, if G is smooth then G M = ZG (M ) and the formulas in (3) show that in such cases ZG (M ) and Lie(ZG (M )) respectively coincide with ZG (X ) and zg (X ) in the sense of [Bo2, 9.1].

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Proof of Proposition A.8.10 To prove (1), by descent theory it is harmless to make a faithfully ﬂat scalar extension on k, so we can assume that k[H ] is a projective k-module. Consider the map α : H × Y → Y × Y deﬁned by (h, y) → (h.y, y). Since the diagonal Y /k is closed in Y ×Y (as Y is separated), the locus D := α −1 (Y /k ) of points (h, y) such that h.y = y is a closed subscheme in H × Y . If Y is locally of ﬁnite type over k then so is D. Associated to D we wish to deﬁne an “ideal of coefﬁcients” J on Y , ﬁnitely generated if Y is locally of ﬁnite type. This rests on the following general observation. If R is a commutative ring, P is a projective R-module and M is any R-module, then for any f ∈ P ⊗ M and any expression xj ⊗ mj for f we can consider the submodule of M generated by the mj . We claim that among the submodules of M obtained in this way, there is a unique one that is contained in all others; this is the module of coefﬁcients C( f ) ⊆ M . To prove this, consider a free R-module of the form F = P ⊕ Q with an R-basis {e i }. Identifying P ⊗ M as a submodule of F ⊗ M , we can uniquely write f = ei ⊗ ci with ci = 0 for all but ﬁnitely many i. Letting π : F → P be the projection, we then have f = π(ei ) ⊗ ci in P ⊗ M . But for any expression x ⊗ m for f in P ⊗ M j j as a ﬁnite sum of elementary tensors, we can write xj = rij ei with rij ∈ R, so f =

ei ⊗ ( rij mj ).

i

j

Thus, ci = j rij mj , so deﬁning C( f ) to be the R-span of the ci proves the existence of a module of coefﬁcients. For each open afﬁne U in Y , D ∩ (H × U ) corresponds to an ideal IU in k[H × U ] = k[H ] ⊗k k[U ] with k[H ] a projective k-module. Deﬁne JU be the k[U ]-linear span of the k-modules of coefﬁcients of all elements f ∈ IU . It is clear that JU is an ideal and we claim its formation is compatible with localization on U , so the ideals JU globalize to an ideal J on Y . This assertion only involves k[H ] through its underlying k-module structure, so by k-projectivity we can replace it with a free k-module. Then the claim is clear via a k-basis. For any k-algebra R and y ∈ Y (R), the condition that y is centralized by the HR -action on YR is that the section σy : HR → HR × YR deﬁned by h → (h, y) lies entirely in DR . We claim that this is the same as J being killed by pullback along y : Spec R → Y . To prove it we can work Zariski-locally on R to reduce to the case when y factors through an afﬁne open U ⊆ Y , so if k[H ] → F is a k-linear split injection into a free k-module and {ei } is a k-basis of F then for any f ∈ IU we have uniquely f = ei ⊗ ai, f with {ai, f }i, f a generating of JU using modules of coefﬁcients). Hence, set of JU (by the construction σy∗ ( f ) = ei ⊗ y∗ (ai, f ) in k[H ] ⊗k R = R[HR ], so σy factors through D precisely when all y∗ (ai, f ) vanish in R, which is to say that y pulls JU back to 0 in R. This proves that the zero scheme of J in Y represents Y H . By evaluating

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on points valued in the ﬂat algebra of dual numbers over k, for any y ∈ Y H (k) we thereby obtain the description of Tany (Y H ) as the H -invariants in Tany (Y ). To prove (2) we ﬁrst note that taking Y = Spec k yields the second part as a special case of the ﬁrst. To prove the smoothness of Y H → Y H in general, we may assume k is artin local with algebraically closed residue ﬁeld κ and it sufﬁces to check the inﬁnitesimal lifting criterion for smoothness using points valued in ﬁnite local k-algebras. That is, if R R0 is a surjection between ﬁnite local k-algebras and y0 ∈ Y H (R0 ) has image y0 = f (y0 ) ∈ Y H (R0 ) which is equipped with a lift y ∈ Y H (R) then we just have to construct a lift y ∈ Y H (R) of y0 such that f (y) = y . By induction on the length of R, we can assume that I := ker(R R0 ) is killed by mR . Let y0 = y0 mod mR ∈ Y H (κ) and y0 = f (y0 ) = y0 mod mR ∈ Y H (κ), so the ﬁber Yy0 = f −1 (y0 ) through y0 is smooth since f is smooth. Smoothness of f ensures that for any R-algebra A and the κ-algebra A0 = A ⊗R κ = A/mR A, the map (A.8.2) Y (A) → Y (A0 ) ×Y (A0 ) Y (A) is surjective. The ﬁber of (A.8.2) over ((y0 )A0 , yA ) is a torsor over the A0 -module F(A) = IA ⊗κ Tany0 (Yy0 ) = IA ⊗A0 (A0 ⊗κ Tany0 (Yy0 )) naturally in A; this torsor structure is denoted v + ξ for v ∈ F(A). For any h ∈ H (A) with reduction h0 ∈ H (A0 ) = Hκ (A0 ) and any v ∈ F(A), we have h.(v + ξ ) = h0 .v + h.ξ . Thus, for a point y ∈ Y (R) over (y0 , y ) the H (A)-action on yA for varying A satisﬁes h.yA = c(h) + yA for a map of set-valued functors c : HR → F (depending on the initial choice of y) that satisﬁes the 1-cocycle condition c(h h) = h .c(h) + c(h ). The notion of a 1-coboundary is deﬁned in the evident manner, namely maps of the form h → h.v − v for some v ∈ F(R), and c modulo 1-coboundaries is independent of the initial choice of y. The problem of ﬁnding an HR -invariant y is precisely the condition that c is a 1-coboundary. Letting V denote the vector group over κ corresponding to the ﬁnite-dimensional κ-vector space F(R) = I ⊗κ Tany0 (Yy0 ), since IR[H ] = I ⊗R R[H ] (due to the R-ﬂatness of R[H ]) we get the equality F(R[H ]) = IR[H ] ⊗κ Tany0 (Yy0 ) = I ⊗κ κ[H ] ⊗κ Tany0 (Yy0 ) = V (κ[H ]) which carries c over to a 1-cocycle c0 : Hκ → V . This respects the notion of 1coboundary on both sides, so the proof of (2) is reduced to verifying the triviality of the quotient set H1 (Hκ , V ). Such cohomology classes correspond exactly to isomorphism classes of Hκ -equivariant linear extensions of Ga by V , as is readily checked by a computation with a linear splitting. By hypothesis all linear representations of Hκ over κ are completely reducible, so any Hκ -equivariant linear extension of Ga by V is a split extension and hence (2) is proved.

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467

Finally, consider (3). We can assume k = k, so M is a power of μp . Since the formation of Lie algebras commutes with scheme-theoretic intersections and G M ×M = G M ∩ G M for any direct product decomposition M = M × M , we may reduce to the case M = μp . Thus, we assume for the rest of the argument that M = μp and k = k with characteristic p > 0. Also, if we let G = G M be the semidirect product deﬁned by the action then G M = G M M inside of G and Lie(G M ) = Lie(G M ) ⊕ m inside of Lie(G ) = g ⊕ m, so the proposed formulas for G M and its Lie algebra would follow from those for G M and its Lie algebra. Hence, we may and do rename G as G to reduce to the case when M is a k-subgroup of G acting via conjugation. For a k-algebra R and g ∈ G(R), the MR -action is trivial on g precisely when the R-group map MR → GR deﬁned by t → gtg −1 is the identity map. Since M = μp , by Proposition A.7.14 any R-group map MR → GR is uniquely determined by its associated map on Lie algebras. In particular, the R-group map MR → GR deﬁned by g-conjugation on MR is the canonical inclusion if and only if the induced map on Lie algebras over R is the canonical inclusion. But the map on Lie algebras is X → (Ad(g))(X ). Setting such a map equal to the inclusion map X → X is a Zariski-closed condition on g ∈ G, and by using a single X that spans m, we get a closed k-subgroup scheme of G that must be G M and is given by the desired recipe in (3). To compute Lie(G M ) in terms of the Lie-theoretic description of G M , consider the k-subgroup Fix(m) ⊆ GL(g) of linear automorphisms of g that restrict to the identity on the subspace m. The preceding alternative description of ZG (M ) gives a cartesian square GM

/ Fix(m)

G

/ GL(g) Ad

Since the formation of Lie algebras is compatible with ﬁber products, we therefore get a cartesian square of Lie algebras Lie(G M )

/ zg (m)

g

/ gl(g) ad

from which we see that Lie(G M ) is as desired.

2

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Corollary A.8.11 Let G be a connected group scheme of ﬁnite type over a ﬁeld k, and let H be an afﬁne k-group scheme of ﬁnite type equipped with a left action on G over k. If Hk is linearly reductive then the H -action on G is trivial if and only if the induced action of H on Lie(G) is trivial. Proof. The action is trivial if and only if G H = G. Since Lie((G H )0 ) = Lie(G H ) = Lie(G)H , it sufﬁces to prove that (G H )0 = G when their Lie algebras coincide. We may also assume k = k, so Gred is smooth and connected, hence irreducible. It sufﬁces to prove that G H and G coincide formally at the identity; i.e., their complete local rings at the identity coincide. Indeed, in such cases the closed G H would contain a (dense) open subscheme around the identity in the irreducible G, so G(k)-translations would imply G H = G. Since the H -action on G preserves the identity section, it acts on the inﬁnitesimal neighborhoods of the identity (even when H is not smooth). In particular, if R is the ﬁnite local k-algebra of such a neighborhood then H acts algebraically on R viewed as a k-algebra with ﬁnite dimension as a k-vector space and it sufﬁces to prove that this action is trivial for each such R (as then G H would contain all inﬁnitesimal neighborhoods of the identity in G, so G H and G would coincide formally at the identity). Let m be the nilpotent maximal ideal of R, so k = R/m. The Lie algebra hypothesis implies that H acts trivially on m/m2 , so it acts trivially on each successive quotient mi /mi+1 for i 1. Hence, H acts on the ﬁnite-dimensional k-vector space R through a k-group map H → GL(R) factoring through a unipotent subgroup of GL(R). But H is linearly reductive, so the k-group map H → GL(R) is trivial. This gives the triviality of the H -action on Spec(R). 2 The following result is of independent interest, and it will be used in the proof of Proposition A.8.14 below. Proposition A.8.12 Let G be a connected reductive group over a ﬁeld k, and H an afﬁne k-group scheme of ﬁnite type equipped with a left action on G. Assume that Hk is linearly reductive. Then the identity component of the schemetheoretic ﬁxed locus G H is a connected reductive k-group. This result was originally proved in [PY1, 2.4] for ﬁnite étale H with char(k) #H . Proof. We may and do assume that k = k. Proposition A.8.10(1),(2) ensures that G H is a smooth closed k-subgroup of G. In general, the identity component of a smooth closed k-subgroup G of G is reductive if G/G is afﬁne. Indeed, we can assume G is connected, and then the reductivity of G follows from [Bo1, Thm. 1.1(i)] (whose proof via étale cohomology works for connected G ) or [Ri, Thm. A] (whose proof uses Mumford’s conjecture, proved by Haboush). Thus, it sufﬁces to prove that G/ZG (H ) is afﬁne.

A.8 Lie algebras and groups of multiplicative type

469

Deﬁne G H to be the semi-direct product against the H -action on G, and let π : G H → H be the natural projection. Let R be the functor whose value on a k-algebra R is the set of R-group sections ρ to πR . There is a natural left action of G on R via conjugation, and for the canonical point ρ0 ∈ R(k) corresponding to the natural section of π the G-stabilizer of ρ0 is represented by G H . More generally, if ρ ∈ R(k) is any k-group section to π then the functorial stabilizer Gρ in G is represented by a closed subscheme (as we can use ρ to rewrite G H as a semidirect product for the action of H on G through ρ-conjugation, and Gρ is the H -centralizer for this new action), so it sufﬁces to prove that the quotients G/Gρ are afﬁne for all ρ. A mild complication is that R is generally not representable (apart from special cases, such as when H is étale), but we can approximate it by representable subfunctors as follows. such Lemma A.8.13 There is a directed system of G-stable subfunctors Ri in R that each Ri is represented by an afﬁne k-scheme of ﬁnite type and R = Ri as functors on k-algebras. Proof. In general, for any pair of afﬁne k-group schemes G and G of ﬁnite type, consider the functor H = Hom(G , G ) of group scheme homomorphisms (i.e., its value on any k-algebra R is the set of R-group maps GR → GR ). The coordinate ring A of G is the directed union of ﬁnite-dimensional k-subspaces that generate A as a k-algebra and are stable under the left and right action of G on A (by viewing A as a G × G -representation space via (g1 , g2 ).a : g → a(g1−1 gg2 )); the same holds for A and G . For any such pair of subspaces V ⊆ A and V ⊆ A we deﬁne the subfunctor HV ,V ⊆ H to have as R-points the R-homomorphisms f : GR → GR such that the pullback map f ∗ carries VR into VR . It is clear that H (R) is the directed union of its subsets HV ,V (R) for all R, and that each subfunctor HV ,V is stable under the natural G -action on H via conjugation. We claim that each HV ,V is represented by an afﬁne k-scheme of ﬁnite type. Since A is a quotient of the symmetric algebra Sym(V ) by a ﬁnitely generated ideal J = (h1 , . . . , hn ), if d is the maximal degree of the hi and we denote by V d ⊆ A the k-linear span of monomials in V of degree at most d then for any R-linear map T : VR → VR the unique R-algebra extension [T ] : ) . Thus, it is easy to check that Sym(V )R → AR carries the hi into (V d R the condition [T ](J ) = 0 is a Zariski-closed condition on the scheme of linear maps Hom(V , V ), and for such T the condition that the resulting R-algebra map AR → AR corresponds to an R-group homomorphism is an additional Zariskiclosed condition. This proves the asserted representability of each HV ,V . In the same way, the functor of group scheme sections to a ﬁxed k-homomorphism G → G is a directed union of subfunctors that are represented by afﬁne k-schemes of ﬁnite type.

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Applying this to π : G H → H , we get the desired G-stable subfunctors Ri in R. 2 Returning to the proof of Proposition A.8.12, since each ρ0 ∈ R(k) lies in some Ri0 (k) we see that the G-orbit of ρ0 lies in Ri0 . Thus, to prove that G/Gρ0 is afﬁne for each ρ0 it sufﬁces to work with each Ri separately. Fix i and consider the left action of G on Ri . We claim the action map α : G × Ri → Ri × Ri deﬁned by (g, ρ) → (g.ρ, ρ) is smooth. Consider the smooth projection map q : G × Ri × Ri → Ri × Ri and the H -action on the source by h.(g, ρ, ρ ) = (ρ(h)gρ (h)−1 , ρ, ρ ). This makes q an H -equivariant map (using trivial action on the target), so by Proposition A.8.10(2) the induced map between subschemes of H -invariants is smooth. But the H -invariants of the source consist of triples (g, ρ, ρ ) such that g.ρ = ρ, so smoothness implies that there is no obstruction to inﬁnitesimal lifting of points of G conjugating one point of Ri into another. This is precisely the functorial criterion for smoothness of α. For each ρ0 ∈ Ri (k), the pullback of α to ρ0 -ﬁbers under second projection to Ri is the orbit map G → Ri deﬁned by g → g.ρ0 , so this map is smooth since α is smooth. Thus, all G-orbits in Ri are open and smooth. But the orbits are pairwise disjoint, so it follows that the afﬁne k-scheme Ri is smooth with only ﬁnitely many orbits, and hence all orbits are afﬁne. For each ρ0 ∈ Ri (k), smoothness of the orbit map for ρ0 implies that the afﬁne orbit is identiﬁed with G/Gρ0 , so G/Gρ0 is afﬁne. 2 Proposition A.8.14 Let G be a smooth connected afﬁne group over a ﬁeld k and H an afﬁne k-group of ﬁnite type equipped with a left action on G. Assume that Hk is linearly reductive. (1) If a normal k-subgroup N in G is preserved by H then (G H )0 → ((G/N )H )0 is surjective. (2) Assume that the H -action preserves Ru,k (G). Then (Ru,k (G) ∩ (G H )0 )0 = Ru,k ((G H )0 ). In particular, the natural map (G H )0 → G/Ru,k (G) factors as (G H )0 → (G H )0 /Ru,k ((G H )0 ) ((G/Ru,k (G))H )0

(A.8.2)

with the second map an étale unipotent isogeny (which is an isomorphism when k is perfect). Also, if G is pseudo-reductive over k then (G H )0 is pseudo-reductive over k.

A.8 Lie algebras and groups of multiplicative type

471

(3) If G is pseudo-reductive over k, T is a maximal k-torus in G, and H is a k-subgroup of T acting on G via conjugation, then ((G H )0ks , Tks ) is the set of roots in (Gks , Tks ) that are trivial on Hks . Note that the hypothesis in (2) that H preserves Ru,k (G) is satisﬁed in two interesting cases: H is smooth (e.g., a torus) or H is a k-subgroup of G acting via conjugation. Proof. The H -equivariant map Lie(G) → Lie(G/N ) is surjective since N is smooth, so it induces a surjection between subspaces of H -invariants because this can be checked over k (with Hk linearly reductive). Since Lie((G H )0 ) = Lie(G H ) = Lie(G)H , and similarly with G/N in place of G, (1) is proved. In particular, the second map in (A.8.2) is surjective with kernel (Ru,k (G) ∩ (G H )0 )/Ru,k ((G H )0 ). To prove (2), ﬁrst note that since (by hypothesis) H preserves Ru,k (G), we have Ru,k (G) ∩ G H = Ru,k (G)H . In particular, using Proposition A.8.10(2) when char(k) > 0, this intersection is smooth. It is visibly unipotent and normal in G H , so (Ru,k (G) ∩ G H )0 ⊆ Ru,k ((G H )0 ). To prove the reverse inclusion (which will imply that the second map in (A.8.2) is an étale unipotent isogeny), it sufﬁces to prove that Ru,k ((G H )0 )k ⊆ Ru (Gk ) (as then the smooth connected normal k-subgroup of G generated by Ru,k ((G H )0 ) would be unipotent and hence would be contained in Ru,k (G)). Since Ru,k (G )k ⊆ Ru (G ) for any smooth connected afﬁne k-group G , k

by taking G = (G H )0 we see that it sufﬁces to consider the case when k is algebraically closed. In other words, to prove (2) apart from the isomorphism claim in (A.8.2) for perfect k, we just need to show that if k is algebraically closed then the image of (G H )0 in G red is reductive. But we can apply (1) to the quotient map G → G red to identify the image of (G H )0 in G red with ((G red )H )0 , so we can assume that G is reductive. Then the reductivity of (G H )0 is Proposition A.8.12. When k is perfect, the ﬁnal map in (A.8.2) must be an isomorphism. Indeed, in this case it is an étale unipotent isogeny between smooth connected afﬁne k-groups, and the target group is reductive, so both groups are reductive. But

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in a connected reductive group over any ﬁeld there are no nontrivial étale unipotent normal subgroups (as an étale normal subgroup must be central, due to connectedness considerations, and the center of a connected reductive group is of multiplicative type). This completes the proof of (2). By (2), the k-group (G H )0 in (3) is pseudo-reductive over k and contains T as a maximal k-torus. Thus, the identiﬁcation of Lie((G H )0 ) with Lie(G)H proves (3). 2

Appendix B Tits’ work on unipotent groups in nonzero characteristic

In our study of pseudo-reductive groups, we need to use some fundamental results of Tits concerning the structure of smooth connected unipotent groups and torus actions on such groups over an arbitrary (especially imperfect) ground ﬁeld of nonzero characteristic. These results were presented by Tits in a course at Yale University in 1967, and lecture notes [Ti1] for that course were circulated but never published. Much of the course was concerned with general results on linear algebraic groups that are available now in many standard references (such as [Bo2], [Hum2], and [Spr]). The only previously available written account (with proofs) of Tits’ structure theory of unipotent groups is his unpublished Yale lecture notes, though a summary of the statements of his results is given in [Oes, Ch. V]. In this appendix we give a self-contained development of the theory, with complete proofs, and in many parts we have simply reproduced arguments from Tits’ lecture notes. We go beyond what we need in order that this appendix may serve as a useful general reference on Tits’ work on this topic. Throughout this appendix, k is an arbitrary ﬁeld with characteristic p > 0.

B.1 Subgroups of vector groups A smooth connected solvable k-group G is k-split if it admits a composition series whose successive quotients are k-isomorphic to Ga or GL1 . In the case of tori this is a widely used notion, and it satisﬁes convenient properties, such as: (i) every subtorus or quotient torus (over k) of a k-split k-torus is k-split, (ii) every k-torus is an almost direct product of its maximal k-split subtorus and its maximal k-anisotropic subtorus. However, in contrast with the case of tori, it is not true for general smooth connected solvable G that the k-split property is inherited by smooth connected normal k-subgroups:

473

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Tits’ work on unipotent groups in nonzero characteristic

Example B.1.1 Suppose that k is imperfect and choose a ∈ k−k p . The k-group U := {y p = x − ax p } is a k-subgroup of the k-split G = Ga2 and it becomes isomorphic to Ga over k(a1/p ) but it is not k-isomorphic to Ga as a k-scheme, let alone as a k-group. Indeed, the regular compactiﬁcation of U has a unique point at inﬁnity, but that point has residue ﬁeld k(a1/p ) so it is not k-rational (in contrast with Ga ). Tits introduced an analogue for unipotent k-groups of the notion of anisotropicity for tori over a ﬁeld. This rests on a preliminary understanding of the properties of subgroups of vector groups, so we take up that study now. The main case of interest to us will be imperfect ground ﬁelds. Deﬁnition B.1.2 A polynomial f ∈ k[x1 , . . . , xn ] is a p-polynomial if every pj

monomial appearing in f has the form cij xi for some cij ∈ k; that is, f = pj ∈ k[xi ]. (In particular, fi (0) = 0 for fi (xi ) with fi (xi ) = j cij xi all i. Together with the identity f = fi (xi ), this uniquely determines each fi in terms of f . Note that f (0) = 0.) Proposition B.1.3 A polynomial f ∈ k[x1 , . . . , xn ] is a p-polynomial if and only if the associated map of k-schemes Gan → Ga is a k-homomorphism. Proof. This is elementary and is left to the reader.

2

A nonzero polynomial over k is separable if its zero scheme in afﬁne space is generically k-smooth. Proposition B.1.4 Let f ∈ k[x1 , . . . , xn ] be a nonzero polynomial such that f (0) = 0. Then the subscheme f −1 (0) ⊆ Gan is a smooth k-subgroup if and only if f is a separable p-polynomial. Proof. The “if” direction is clear. For the converse, we assume that f −1 (0) is a smooth k-subgroup and we denote it as G. The smoothness implies that f is separable. To prove that f is a p-polynomial, by Proposition B.1.3 it sufﬁces to prove that the associated map of k-schemes Gan → Ga is a k-homomorphism. Without loss of generality, we may assume that k is algebraically closed. For any α ∈ G(k), f (x + α) and f (x) have the same zero scheme (namely, G) inside of Gan . Thus, f (x + α) = c(α)f (x) for a unique c(α) ∈ k × . Consideration of a monomial term appearing in f implies that c is a polynomial function of α (over k), and clearly c(α + β)f (x) = f (x + α + β) = c(β)f (x + α) = c(β)c(α)f (x). Hence, c : G → GL1 is a k-homomorphism. But G is a unipotent k-group, so c = 1. That is, f (x + α) = f (x) for all α ∈ G(k).

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475

Pick β ∈ k n , so f (β + α) − f (β) = 0 for all α ∈ G(k). Thus f (β + x) − f (β) vanishes on G, so f (β + x) − f (β) = g(β)f (x) for a unique g(β) ∈ k. Note in particular that g(0) = 1. It follows that for any γ ∈ k n , we have f (β + γ + x) − f (β + γ ) = g(β + γ )f (x) and f (β + γ + x) = f (β) + g(β)f (γ + x) = f (β) + g(β)f (γ ) + g(β)g(γ )f (x). Comparing highest degree terms yields that g(β + γ ) = g(β)g(γ ), so g : Gan → Ga satisﬁes g(β)p = g(0) = 1 for all β ∈ k n . Therefore g = 1, so f (β) + f (x) = f (β + x) for all β ∈ k n . This says that f is additive. 2 Corollary B.1.5 Let G ⊆ Gan be a smooth k-subgroup of codimension 1. Then G is the zero scheme of a separable nonzero p-polynomial in k[x1 , . . . , xn ]. Proof. Since G is smooth of codimension 1 in Gan , it is the zero scheme of a separable nonzero polynomial f ∈ k[x1 , . . . , xn ]. By Proposition B.1.4, f is a p-polynomial. 2 n Deﬁnition B.1.6 If f = i=1 fi (xi ) is a p-polynomial over k in n variables with fi (0) = 0 for all i, then the principal part of f is the sum of the leading terms of the fi . Lemma B.1.7 Let V be a vector group of dimension n 1 over k, and let f : V → Ga be a k-homomorphism. Then the following are equivalent: (1) there exists a non-constant k-scheme morphism f : Ak1 → V such that f ◦ f = 0; (2) for every k-group isomorphism h : Gan V , the principal part of the p-polynomial f ◦ h ∈ k[x1 , . . . , xn ] has a nontrivial zero in k; (3) there exists a k-group isomorphism h : Gan V such that f ◦ h “only depends on the last n − 1 coordinates” (i.e., ker( f ◦ h) contains the ﬁrst factor of Gan ). In this lemma, it is not sufﬁcient in (2) to consider just a single choice of h. For example, if k is imperfect and a ∈ k − k p , then f = y p − (x + ax p ) has principal part y p − ax p with no zeros on k 2 − {0}. Composing f with the k2 automorphism (x, y) → (x, y + x p ) yields the polynomial y p + x p − (x + ax p ) 2 whose principal part is y p + x p , which has zeros on k 2 − {0}. Proof. We will show that (1) ⇒ (2) ⇒ (3) ⇒ (1). For (1) ⇒ (2), assume that (1) holds and let ϕ = h−1 ◦ f . Let ϕi : Gan → Ga be the ith component of ϕ, and ai t si denote the leading term of ϕi (t), with si = 0

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when ϕi = 0. For some i we have si > 0, since some ϕi is non-constant (as ϕ pmi is non-constant, because of the same for f ). Let ni=1 ci xi be the prinicipal part of f ◦ h, so 0 = f (h(ϕ(t))) =

n

pmi si pimi

ci ai

t

+ ...

i=1

since f ◦ h ◦ ϕ = f ◦ h ◦ h−1 ◦ f = f ◦ f = 0. Let N = maxi {si pmi } > 0. Deﬁne bi = ai if si pmi = N (so bi = 0), and bi = 0 if si pmi < N . Since the coefﬁcient pmi of the term of degree N in f (h(ϕ(t))) must be zero, we have ni=1 ci bi = 0 with bi ∈ k and some bi is nonzero, so (2) holds. To prove (2) ⇒ (3), assume (2) holds and let h : Gan V be any k-group isomorphism. We may assume f = 0, so the principal part of f ◦ h is nonzero. The proof will proceed by induction on the sum d of the degrees of nonzero p mi terms of the principal part ni=1 ci xi of f ◦ h. If cr = 0 for some r, we are done by interchanging xr and x1 . So we may assume that all ci are nonzero and, upon permuting the coordinates, that m1 · · · mn 0. By (2), there pmi exists (a1 , . . . , an ) ∈ k n − {0} such that ni=1 ci ai = 0. Let r 0 be minimal such that ar = 0. Deﬁne the k-group isomorphism h : Gan Gan by h (y1 , . . . , yn ) = (x1 , . . . , xn ) with x1 = y1 , . . . , xr−1 = yr−1 , xr = ar yr ,

pmr −mr+1

xr+1 = yr+1 + ar+1 yr

pmr −mn

, . . . , xn = yn + an yr

.

Thus, f ◦ h ◦ h is a p-polynomial with principal part i=r

pmi ci y i

+

n i=1

pmi

ci ai

pmr

· yr

=

pmi

ci yi

i=r

pmi since ni=1 ci ai = 0. The sum of the degrees of the nonzero terms of the principal part of f ◦ h ◦ h is strictly smaller than d since cr = 0, so the induction hypothesis applies. Finally, we assume (3) and prove (1). Let h : Gan → V be a k-isomorphism such that ker( f ◦ h) contains the ﬁrst factor of Gan . Deﬁne ϕ : Ga → Gan by ϕ(t) = (t, 0, 0, . . . , 0). Finally, let f = h ◦ ϕ. Then f ◦ f = f ◦ h ◦ ϕ = 0. 2 Lemma B.1.8 If K/k is a Galois extension and a p-polynomial of the form n pmi over k has a zero in K n − {0} then it has a zero in k n − {0}. i=1 ci xi

B.1 Subgroups of vector groups

477

Proof. The proof is by induction on n. The terms may be ordered so that m1 pm1 m2 · · · . If n = 1, then since c1 a1 = 0 with a1 ∈ K × we see that c1 = 0, pm1

so c1 x1

has a zero in k × .

pmi Now suppose n > 1 and that ni=1 ci ai = 0 with ai ∈ K not all zero. Let a = (a1 , . . . , an ). If an = 0 then the theorem is true by the induction hypothesis. pmn −mi for all i. If an = 0, we may assume an = 1 by replacing ai with ai /an pmi For all σ ∈ Gal(K/k), the point a − σ (a) is a zero of ci xi . If not all ai belong to k then a − σ (a) = 0, so since an − σ (an ) = 0 we may again apply the inductive hypothesis. 2 Lemma B.1.9 Let V be a vector group over k, K/k a Galois extension, and f : V → Ga a k-homomorphism. The equivalent conditions (1), (2), and (3) of Lemma B.1.7 hold over K if and only if they hold over k. Proof. It is clear that if (1) holds over k, it also holds over K. On the other hand, by Lemma B.1.8, (2) is true over k if it is true over K. 2 Lemma B.1.10 Every smooth p-torsion commutative afﬁne k-group G embeds as a k-subgroup of a vector group over k. Moreover, G admits an étale isogeny onto a vector group over k, and if G is connected and k = k then G is a vector group over k. Proof. We ﬁrst construct the embedding into a vector group over k, and then at the end use this to make the étale isogeny. Consider the canonical ksubgroup inclusion G → R k /k (Gk ) for any ﬁnite extension ﬁeld k /k. Since

R k /k (Ga ) Ga[k :k] , it is harmless (for the purpose of ﬁnding an embedding into a vector group over k) to replace k with a ﬁnite extension. If Gk embeds as a subgroup of GaN over k, the embedding descends to a ﬁnite extension k /k inside of k. Hence, for the construction of the embedding into a vector group we can now assume that k is algebraically closed. The component group G/G 0 is a power of Z/pZ. Thus, since G is commutative and p-torsion, the connected-étale sequence of G splits. That is, G = G 0 × (Z/pZ)n for some n 0. The ﬁnite constant k-group Z/pZ is a k-subgroup of Ga , so we can assume that G is connected. We shall prove that G is a vector group. Since k = k, G has a composition series whose successive quotients are Ga . By induction on dim G, it sufﬁces to prove that a commutative extension U of Ga by Ga over k is a split extension if p · U = 0. Let W2 be the additive k-group of Witt vectors of length 2, so there is a canonical exact sequence of k-groups 0 → Ga → W2 → Ga → 0. It is a classical fact (see [Ser1, Ch. VII, §2, Lemma 3]) that every commutative extension U of Ga by Ga over k is obtained by pullback of this Witt vector

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extension along a (unique) k-homomorphism f : Ga → Ga . The p-torsion subgroup of such a pullback is W2 [p] ×Ga , f Ga . But W2 [p] is an extension of αp by Ga , and the k-group scheme preimage f −1 (αp ) in Ga is ﬁnite unless f = 0. Hence, by dimension considerations, the only case in which the pullback extension U is p-torsion is when f = 0, in which case U is a split extension of Ga by Ga . Now return to the setting of a general ground ﬁeld k, and ﬁx a k-subgroup inclusion of G into a vector group V , say with codimension c. Choose a linear structure on V (in the sense of Deﬁnition A.1.1). Then W → Lie(W ) is a bijection between the set of linear subgroups of V and the set of linear subspaces of Lie(V ). Hence, if we choose W so that Lie(W ) is complementary to Lie(G) then the natural map G → V /W is an isomorphism on Lie algebras, so it is an étale isogeny. Since W is a linear subgroup of V , the quotient V /W is a vector group over k. 2 Proposition B.1.11 Let V1 , . . . , Vn be k-groups isomorphic to Ga , and let V = n V . Let U be a smooth k-subgroup of V such that Uks is the ks -subgroup i i=1 of Vks generated by images of ks -scheme morphisms Ak1s → Vks that pass through 0. There exists a k-group automorphism h : V V such that h(U ) is the direct product of some of the Vi inside of V . In particular, U is a vector group over k and is a k-group direct factor of V . Proof. The proof is by induction on n and is trivial for n = 1. Now consider n > 1. The case U = V is trivial, so we can assume dim U n − 1. First assume that dim U = n − 1 > 0. By Corollary B.1.5, U is the kernel of a k-homomorphism f : V → Ga . By hypothesis, there exists a non-constant ks scheme morphism Ak1s → Uks , so by Lemma B.1.7 (applied over ks ) and Lemma B.1.9 there exists a k-group automorphism h of V such that h (U ) ⊇ V1 . But then of h (U ) into n h (U ) = V1 × U , where U denotes the projection V = i=2 Vi . Applying the induction hypothesis to V and U , we are done. Suppose now that dim U < n − 1, and let U denote the projection of U into the product V as deﬁned above. By the inductive hypothesis, after relabeling V2 , . . . , Vn there exists a k-group automorphism h1 : V → V such that h1 (U ) = ri=2 Vi for some r < n. Setting h = id V1 × h1 : V V , we then have h (U ) ⊆ is now complete.

r

i=1 Vi ,

and we can again apply induction. The proof 2

Corollary B.1.12 In a smooth p-torsion commutative afﬁne k-group G, every smooth k-subgroup that is a vector group is a k-group direct factor.

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Proof. This is a consequence of Proposition B.1.11, provided that G is a ksubgroup of a vector group. Such an embedding is provided by Lemma B.1.10. 2 The following proposition is a useful reﬁnement of Lemma B.1.10. Proposition B.1.13 Let k be an inﬁnite ﬁeld of characteristic p > 0 and let U be a smooth p-torsion commutative afﬁne k-group. Then U is k-isomorphic to a k-subgroup of codimension 1 in a k-vector group. In particular, U is isomorphic (as a k-group) to the zero scheme of a separable nonzero p-polynomial over k. This proposition is also true for ﬁnite k if U is connected since then it is a vector group; see Corollary B.2.7. Proof. By Lemma B.1.10, U can be identiﬁed with a k-subgroup of a k-vector group V . Let m = dim V −dim U . If m 1 then we are done by Corollary B.1.5, so we assume m > 1. We will show that U can be embedded in a k-vector group W with dim W = dim V − 1, which will complete the argument via induction on m. The vector group W will arise as a quotient of V . The k-linear subspace Lie(U ) in Lie(V ) has codimension m. Fix a choice of linear structure on V (in the sense of Deﬁnition A.1.1). Since m 2, the Zariski closure Ga .U (⊆ V ) of the image of the multiplication map Ga × U → V is a closed subscheme of V with nonzero codimension. By irreducibility of V , the union Lie(U ) ∪ (Ga .U ) inside of V is a proper closed subscheme of V . Since V (k) is Zariski-dense in V (as k is inﬁnite), there exists v ∈ V (k) with v∈ / Lie(U ) ∪ (Ga .U ). Let L ⊆ V be the k-subgroup corresponding to the line kv ⊆ V (k). Consider the canonical k-homomorphism φ : V → W := V /L, and let ψ = φ|U . We shall prove ker ψ = 1, from which it follows that ψ identiﬁes U with a k-subgroup of W . It sufﬁces to show that Lie(ψ) is injective (so ker ψ is étale) and that ψ|U (k) is injective. The map Lie(ψ) has kernel L ∩ Lie(U ) = {0}, so it is indeed injective. If ψ|U (k) is not injective then the line L would lie in Ga .U since Ga .U is stable under the Ga -multiplication on V . But the point v ∈ L(k) does not lie in (Ga .U )(k), due to how we chose v, so indeed ψ|U (k) is injective. 2

B.2 Wound unipotent groups It is natural to consider a smooth connected unipotent k-group U to be analogous to an anisotropic torus if U does not contain Ga as a k-subgroup. This concrete viewpoint is inconvenient for developing a general theory, but eventually we will prove that it gives the right concept. A more convenient deﬁnition to get

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the theory off the ground requires going beyond the category of k-groups, as follows. Deﬁnition B.2.1 A smooth connected unipotent k-group U is k-wound if every map of k-schemes Ak1 → U is a constant map to a point in U (k). Remark B.2.2 Note that the deﬁnition of k-wound also makes sense in characteristic 0, where it is only satisﬁed by U = 1 (since a nontrivial smooth connected unipotent group in characteristic 0 always contains Ga as a subgroup over the ground ﬁeld). Thus, although in this appendix we only work with ground ﬁelds of nonzero characteristic, it is convenient in practice (for handling some trivialities) to make the convention that “wound” means “trivial” for smooth connected unipotent groups in characteristic 0. Whereas anisotropicity for a torus over a ﬁeld is insensitive to purely inseparable extension of the ground ﬁeld but is often lost under a separable algebraic extension of the ground ﬁeld, the k-wound property will behave in the opposite manner: it is insensitive to a separable extension on k (such as scalar extension from a global ﬁeld to a completion) but is often lost under a purely inseparable extension on k. Example B.2.3 Assume k is imperfect and choose a ∈ k − k p . The k-group {y p = x − ax p } becomes isomorphic to Ga over k(a1/p ) but it is k-wound; i.e., over k it cannot be the target of a non-constant k-scheme morphism Ak1 → U . Indeed, such a map would be dominant and so would extend to a ﬁnite surjective map between the regular compactiﬁcations, contradicting that the unique point at inﬁnity for Ak1 is k-rational but the one for U is not (its residue ﬁeld is k(a1/p )). The k-group U is a k-wound subgroup of the k-split group Ga2 . In [Oes, Ch. V, 3.5] there is an example over any imperfect ﬁeld k of a 2-dimensional k-wound smooth connected p-torsion commutative afﬁne group G admitting a 1-dimensional (necessarily k-wound) smooth connected k-subgroup G such that G/G Ga as k-groups. Thus, k-wound groups can admit smooth quotient maps onto nonzero k-split groups. Example B.2.4 Assume k is inﬁnite. By Corollary B.1.5, smooth p-torsion commutative afﬁne k-groups G are precisely the zero schemes of separable nonzero p-polynomials f over k. Since G is connected if and only if it is geometrically irreducible (as for any k-group scheme of ﬁnite type), we see that G is connected if and only if f is irreducible over k, as well as if and only if f is geometrically irreducible over k. Assume G is connected. If the principal part fprin of f has no zero on k n − {0} then by Lemma B.1.7 it follows that G is k-wound. The converse is false, as we saw following the statement of Lemma B.1.7. However, if fprin has a zero on k n − {0} then the

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calculation in the proof of (2) ⇒ (3) in Lemma B.1.7 (taking h to be the identity map of Gan ) shows that we can ﬁnd a p-polynomial F ∈ k[x1 , . . . , xn ] having zero scheme k-isomorphic to G as a k-group (so F is geometrically irreducible over k) with the sum of the degrees of the monomials appearing in Fprin strictly less than the corresponding sum for fprin . Continuing in this way, we eventually arrive at a choice of f having zero scheme G (as a k-group) such that fprin has no zeros on k n − {0}. In this sense, the zero schemes of geometrically irreducible p-polynomials f over k for which fprin has no nontrivial k-rational zero are precisely the p-torsion commutative k-wound groups (up to k-isomorphism). Theorem B.2.5 Every smooth connected p-torsion commutative afﬁne k-group U is a direct product U = V × W of a vector group V and a smooth connected unipotent k-group W such that Wks is ks -wound. In this decomposition, the subgroup V is uniquely determined: Vks is generated by the images of ks -scheme morphisms ϕ : Ak1s → Uks passing through the identity. Proof. By Galois descent, there is a unique smooth connected k-subgroup V of U such that Vks is generated by the images of ks -scheme morphisms ϕ : Ak1s → Uks that pass through the identity. By Lemma B.1.10, we can identify U with a k-subgroup of a vector group over k. Thus, by Proposition B.1.11, V is a vector group over k and (by Corollary B.1.12) we have U = V × W as k-groups for some k-subgroup W of U . Since U is a smooth connected unipotent k-group, so is its direct factor W . Clearly, Wks is ks -wound (due to the deﬁnition of V ). Now we prove that V in this decomposition is unique. Consider any decomposition of k-groups U = V × W , where V is a vector group over k and W is a smooth connected unipotent k-subgroup of U such that Wks is ks -wound. The image of any ks -scheme morphism ϕ : Ak1s → Uks passing through the identity is contained in Vks because otherwise the composite of ϕ and the canonical projection Uks → Wks would be a non-constant ks -scheme morphism from Ak1s to Wks (contradicting that Wks is assumed to be ks -wound). Hence, V ⊆ V , so V = V × V1 with V1 the image of the vector group V under the projection U W . Since Wks is ks -wound and V is a vector group, V1 = 0. That is, 2 V = V. Corollary B.2.6 A smooth connected p-torsion commutative afﬁne k-group U is k-wound if and only if Uks is ks -wound, and also if and only if there are no nontrivial k-homomorphisms Ga → U . The k-group U is a vector group over k if and only if Uks is a vector group over ks . Proof. This is immediate from Theorem B.2.5.

2

Corollary B.2.7 If k is perfect then a smooth connected p-torsion commutative afﬁne k-group is a vector group.

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Proof. By Corollary B.2.6, we may assume that k is algebraically closed. This case is part of Lemma B.1.10. 2 To get results on k-wound groups beyond the commutative p-torsion case, we need to study smooth connected p-torsion central k-subgroups in a general smooth connected unipotent k-group U . This is taken up in the next section.

B.3 The cckp-kernel In a smooth connected unipotent k-group U , any two smooth connected p-torsion central k-subgroups generate a third such subgroup. Hence, the following deﬁnition makes sense. Deﬁnition B.3.1 The cckp-kernel of U is its maximal smooth connected ptorsion central k-subgroup. Note that if U = 1 then its cckp-kernel is nontrivial, since the latter contains the cckp-kernel of the last nontrivial term of the descending central series of U . By Galois descent and specialization (as in the proof of Proposition 1.1.9(1)), the formation of the cckp-kernel commutes with any separable extension on k. Proposition B.3.2 Let U be a smooth connected unipotent k-group, and let k /k be a separable extension. Let F denote the cckp-kernel of U . Then U is k-wound if and only if Uk is k -wound, and the quotient U /F is k-wound whenever U is k-wound. Also, the following conditions are equivalent: (1) U is k-wound, (2) U does not have a central k-subgroup k-isomorphic to Ga , (3) the cckp-kernel of F of U is k-wound. This proposition implies that U is k-wound if and only if U admits no nontrivial k-homomorphism from Ga . Such a characterization of the k-wound property is analogous to the characterization of anisotropic tori over a ﬁeld in terms of homomorphisms from GL1 over the ground ﬁeld. Proof. Obviously (1) ⇒ (2). By Theorem B.2.5, (2) and (3) are equivalent. Also, by specialization (as in the proof of Proposition 1.1.9(1)), if UK is not K-wound for some separable extension K/k then the same holds with K/k taken to be some ﬁnite separable extension. Thus, to prove the equivalence of (1), (2), and (3) and the fact that Uk is k -wound whenever U is k-wound, it sufﬁces to show that if Uks is not ks -wound then the cckp-kernel F of U is not k-wound. Thus let ϕ : Ak1s → Uks be a non-constant ks -scheme morphism. Composing with a U(ks )-translation if necessary, we may assume ϕ(0) = 1. We may choose

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such a ϕ so that ϕ(Ak1s ) is central. Indeed, suppose ϕ(Ak1s ) is non-central, so U is not commutative and there exists g ∈ U (ks ) not centralizing ϕ(Ak1s ). The ks -scheme morphism ϕ : Ak1s → Uks deﬁned by ϕ (x) = g −1 ϕ(x)−1 gϕ(x) carries 0 to 1, so it is then non-constant, and its image lies in derived group D (Uks ) = D (U )ks . The k-subgroup D (U ) has smaller dimension than U and is nontrivial since the smooth connected k-group U is not commutative. Hence, by iteration with the descending central series (whose ﬁrst term is D (U )), the required non-constant ϕ with ϕ(Ak1s ) central is eventually obtained. We may also assume that ϕ(Ak1s ) is p-torsion by replacing the original ϕ with pe · ϕ for some e 0. The nontrivial ks -subgroup generated by ϕ(Ak1s ) then lies in the ccks p-kernel of Uks ; i.e., it lies in Fks . Thus Fks is not ks -wound, so by Corollary B.2.6 the k-group F is not k-wound. It remains to show that if U is k-wound then U /F is k-wound. For this we may, in view of the preceding conclusions, assume that k = ks . Suppose that U is k-wound and U /F is not k-wound. Thus, there exists a central k-subgroup A of U /F that is k-isomorphic to Ga . Let π denote the canonical homomorphism U → U /F. The k-subgroup scheme π −1 (A) in U is an extension of A by F, so it is smooth, connected, and unipotent. We claim that π −1 (A) is central in U . If not, we get a non-constant k-scheme morphism ϕ : Ak1 → F (contradicting that U is k-wound) as follows. Choose g ∈ U (k) not centralizing π −1 (A) (recall k = ks ), identify Ga with A = π −1 (A)/F, and deﬁne ϕ : π −1 (A)/F → F by xF → gxg −1 x−1 . Thus, π −1 (A) is central in U . Similarly, π −1 (A) is p-torsion because otherwise we would get a non-constant k-scheme morphism ψ : Ak1 → F via ψ(xF) = x p . We have shown that π −1 (A) lies in the cckp-kernel F of U , so the given inclusion F ⊆ π −1 (A) is an equality. Hence, A = 1, which is absurd since A Ga . 2 Corollary B.3.3 Let U be a k-wound smooth connected unipotent k-group. Deﬁne the ascending chain of smooth connected normal k-subgroups {Ui }i0 as follows: U0 = 1 and Ui+1 /Ui is the cckp-kernel of the k-wound group U /Ui for all i 0. These subgroups are stable under k-group automorphisms of U , their formation commutes with any separable extension of k, and Ui = U for sufﬁciently large i. In particular, if H is a smooth k-group equipped with an action on U then H carries each Ui back into itself. Proof. The well-posedness of the deﬁnition (e.g., that U /U1 is k-wound) and the compatibility with separable extension on k follow from Proposition B.3.2. By dimension considerations, Ui = U for sufﬁciently large i since the cckpkernel of a nontrivial smooth connected unipotent k-group is nontrivial. Finally, if H is a smooth k-group acting on U then we need to prove that H carries each Ui into itself. For this we may extend scalars to ks , so k is

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separably closed. Then the H -stability of Ui is equivalent to the H (k)-stability of Ui , and this latter property is a special case of each Ui being stable under all k-automorphisms of U . 2 We will now prove the following structure theorem that is analogous to the canonical presentation of a torus over a ﬁeld as an extension of an anisotropic torus by a split torus. Theorem B.3.4 Let U be a smooth connected unipotent k-group. Then there exists a unique smooth connected normal k-split k-subgroup Usplit of U such that U /Usplit is k-wound in the sense of Deﬁnition B.2.1. The subgroup Usplit contains the image of every k-homomorphism from a k-split smooth connected unipotent k-group into U . Also, the kernel of every k-homomorphism from U into a k-wound smooth connected unipotent k-group contains Usplit , and the formation of the k-subgroup Usplit is compatible with any separable extension of k. Proof. The proof is by induction on dim U . If U is k-wound then Usplit = {1} satisﬁes the requirements and is unique as such. Assume that U is not kwound, and let A be a smooth central k-subgroup isomorphic to Ga (Proposition B.3.2). Let H = U /A. By induction, there exists a smooth connected normal k-subgroup Hsplit in H with the desired properties in relation to H (in the role of U ). Let Usplit be the corresponding subgroup of U containing A. It is k-split, and U /Usplit H /Hsplit is k-wound. Let U be a smooth connected unipotent k-group having a composition series U = U0 ⊇ U1 ⊇ · · · with successive quotients k-isomorphic to Ga , and let ϕ : U → U be a khomomorphism. There exists a minimal i such that ϕ(Ui ) ⊆ Usplit . If i > 0 /U → U /U then there is induced a k-homomorphism Ga Ui−1 split with i nontrivial image. This contradicts that U /Usplit is k-wound. Thus, i = 0; i.e., ϕ(U ) ⊆ Usplit . It follows in particular that Usplit is unique. Also, for any k-homomorphism ϕ : U → U into a k-wound smooth connected unipotent k = {1}. This says that ker ϕ contains U group U we have ϕ(Usplit ) ⊆ Usplit split . The last assertion of the theorem follows from Proposition B.3.2. Indeed, if k /k is a separable extension and U := Uk then (Usplit )k ⊆ Usplit and the k -split quotient Usplit /(Usplit )k is a k -subgroup of the k -group (U /Usplit )k that is k -wound (by Proposition B.3.2). This forces Usplit = (Usplit )k . 2 Corollary B.3.5 Let G be a smooth connected afﬁne k-group. The maximal k-split smooth connected unipotent normal k-subgroup Rus,k (G) in G is equal to Ru,k (G)split . In particular, Ru,k (G)/Rus,k (G) is k-wound and the formation of Rus,k (G) commutes with separable extension on k.

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Proof. By Galois descent, Rus,ks (Gks ) descends to a smooth connected unipotent normal k-subgroup of G. Such a descent is k-split, since the k-split property of smooth connected unipotent k-groups is insensitive to separable extension on k (due to Theorem B.3.4). Thus, the descent is contained in Rus,k (G), so the inclusion Rus,k (G)ks ⊆ Rus,ks (Gks ) is an equality. In other words, the formation of Rus,k (G) is compatible with separable algebraic extension on k. Hence, to prove the compatibility with general separable extension on k and the agreement with the maximal k-split smooth connected k-subgroup of Ru,k (G), we may assume that k = ks . But Ru,k (G)split is a characteristic k-subgroup of G, so it is normal due to the Zariski-density of G(k) in G when k = ks . This proves that Ru,k (G)split ⊆ Rus,k (G). The reverse inclusion is clear. The compatibility of the formation of Rus,k (G) now follows from such a compatibility for two constructions: the formation of Usplit in Theorem B.3.4, and the formation of Ru,k (G) (by Proposition 1.1.9(1)). 2

B.4 Torus actions on unipotent groups Consider the action of a k-torus T on a smooth connected unipotent k-group U . This induces a linear representation of T on Lie(U ), so if T is k-split then we get a weight space decomposition of Lie(U ). If U is a vector group then it is natural to wonder if this decomposition of Lie(U ) can be lifted to the group U . When dim U > 1, the T -action may not respect an initial choice of linear structure on U (in the sense of Deﬁnition A.1.1) since char(k) = p > 0, so we ﬁrst seek a T -equivariant linear structure. For example, if U = Ga2 with its usual linear structure and T = GL1 with the action t.(x, y) = (tx, t p x p + ty) then the T -action is not linear and the action on Lie(U ) = k 2 has the single weight given by the identity character of T . But note that if we compose the T -action with the additive automorphism (x, y) → (x, y − x p ) of U then the action becomes t.(x, y) = (tx, ty), which is linear. Tits proved rather generally that if a k-split T acts on U with only nontrivial weights on Lie(U ), then there are nontrivial constraints on the possibilities for U as a k-group and that (after passing to a suitable characteristic composition series for U ) the action can always be described in terms of linear representations of T . To explain his results in this direction, we begin with the following proposition that generalizes Lemma B.1.10 by incorporating a torus action. Proposition B.4.1 Let U be a smooth p-torsion commutative afﬁne k-group equipped with an action by a k-torus T . There exists a linear representation of T on a ﬁnite dimensional k-vector space V and a T -equivariant isomorphism of U onto a k-subgroup of V . Proof. Let G = T U , and choose a k-subgroup inclusion of G into GLn for some n. Inside of the k-vector group Mat n of n × n matrices, let A be the linear

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span of U − 1 = {u − 1 | u ∈ U }. (In other words, A is the smooth additive k-subgroup of Mat n generated by the image of the multiplication map of kschemes Ga × (U − 1) → Mat n , using the natural linear structure on Mat n .) For any integer i 1, let Ai be the linear span of the points a1 · · · ai , with a1 , . . . , ai ∈ A. (This is the smooth additive k-subgroup of Mat n generated by the image of the multiplication map Ga × (A × · · · × A) → Mat n using i copies of A.) Observe that each Ai is stable under T -conjugation on Mat n . Since a p = 0 for every a ∈ A, clearly A j = {0} for j > (p − 1) dim A. Thus, it makes sense to let ν be the smallest integer 1 such that Aν = {0}. From the identity (v − 1)(w − 1) = (vw − 1) − (v − 1) − (w − 1) in any associative ring with identity, we obtain that A = A1 ⊇ A2 ⊇ · · · ⊇ Aν = {0}. Since Ai ⊆ A for all i and a p = 0 for all a ∈ A, the translate 1 + A of A inside of Mat n is a smooth commutative k-subgroup of GLn (using the geometric series expansion for (1 + a)−1 truncated in degree below p). Let us interpret 1 + A in more ring-theoretic terms. The scheme product A := Ga ⊕ A is naturally a commutative ring scheme (with identity) over k that is a commutative subalgebra of Mat n and contains A as an ideal whose νth power vanishes. The kgroup 1+A is precisely the k-group of 1-units in A (i.e., A × = GL1 ×(1+A)). Concretely, A (k) is a ﬁnite local k-algebra with maximal ideal A(k) whose ith power is Ai (k). The k-group 1 + A is a commutative k-subgroup of GLn stable under the T conjugation action. By construction, U is a k-subgroup of 1 + A which is stable under conjugation by T . It sufﬁces to T -equivariantly identify the multiplicative k-group 1+A with the additive k-group A, as this will then yield a T -equivariant identiﬁcation of U with a k-subgroup of a linear representation space for T . We will construct such an identiﬁcation using truncated exponentials, but some care is needed since the maximal ideal A(k) of A (k) may not have vanishing pth power (as an ideal). Each quotient Ai /Ai+1 for i 1 is a linear representation space for T , so over ks it admits a decomposition into weight spaces according to some characters of Tks . These characters are a union of orbits of Gal(ks /k) on X(Tks ), and Gal(ks /k) compatibly permutes the corresponding weight spaces. Thus, we can choose a ks -basis of each (Ai /Ai+1 )(ks ) = Ai (ks )/Ai+1 (ks ) consisting of a union of Gal(ks /k)-orbits of Tks -eigenvectors. By complete reducibility of linearrepresentations of T over k, the T -module Ai is isomorphic to the direct sum ji A j /A j+1 for all i 1. In this way, we get a Gal(ks /k)-stable ks -basis

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{a1 , . . . , ar } of A(ks ) consisting of Tks -eigenvectors such that Ai is spanned by the last dim Ai vectors of the basis for all i 1. Consider the map of ks -schemes Aks → 1 + Aks deﬁned by i

x i ai →

exp (xi ai ),

i

p−1 j where exp (a) := j=0 a /j!. This map is clearly Tks -equivariant, and by commutativity of 1 + A and the Gal(ks /k)-stability of {ai } it is Gal(ks /k)equivariant. Thus, this map uniquely descends to a T -equivariant map of kschemes ϕ : A → 1 + A. It sufﬁces to prove that this is a k-group isomorphism, so we may now extend scalars to ks to reduce to the case k = ks . Since a p = 0 for any point a of A, for any points x and y of Ga we have exp (xa) · exp (ya) = exp ((x + y)a). In view of the commutativity of 1 + A, it follows that ϕ is a k-homomorphism. Since 1 + A is the group of 1-units of the ring scheme A = Ga ⊕ A, the natural identiﬁcation of Lie(A × ) with Lie(A ) (as for the linear algebraic unit group of any ﬁnite-dimensional associative k-algebra) restricts to an identiﬁcation of Lie(1 + A) with Lie(A). Under this identiﬁcation, Lie(ϕ) is identiﬁed with the identity map of Lie(A), so Lie(ϕ) is an isomorphism. Hence, ϕ is étale. Thus, ϕ is surjective (since 1 + A is connected) and has ﬁnite étale kernel C. It remains to prove C = 1, for which it sufﬁces to show that C(k) = 1. In view of how ϕ is deﬁned in terms of the ai , and how the ordered basis of elements ai was chosen in accordance with the ﬁltration of A by the k-subgroups A j , the vanishing of the subset C(k) − 1 ⊆ A(k) = A1 (k) is seen by studying it in A/A2 (to show it lies in A2 ), then in A2 /A3 (to show it lies in A3 ), and so on. 2 We now apply our work with wound groups to analyze the structure of smooth connected unipotent k-groups equipped with a sufﬁciently nontrivial action by a k-torus. Proposition B.4.2 Let T , U , and V be as in Proposition B.4.1, and let V = V0 ×

Vi

i∈I

be a T -equivariant decomposition with a k-vector subspace V0 on which T acts trivially and lines Vi in V on which T acts through a nontrivial k-rational character χi .

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There exists a T -equivariant k-automorphism α of the additive k-group V such that α(U ) = (α(U ) ∩ V0 ) ×

Vj

j∈J

for a suitable subset J of I . In particular, if T has only nontrivial weights on V then the k-group U is a vector group admitting a T -equivariant linear structure. Proof. Since V0 = ZV (T ), clearly U0 := U ∩ V0 is ZU (T ). This is smooth since U is smooth. Step 1. By semisimplicity of the T -action on V , we can uniquely write V = V0 × V as linear representations of T . We claim that the canonical map of k-groups ι : U0 × (U ∩ V ) → U is an isomorphism (so in particular, U ∩ V is smooth). Once this is proved, we can replace V with V and replace U with U ∩ V to reduce to the case V = V (i.e., ZV (T ) = 0). To establish that ι is an isomorphism,we may assume that the ground ﬁeld is algebraically closed. Note that V = i∈I Vi , with T acting on each line Vi through a nontrivial character χi . We can choose s ∈ T (k) such that χi (s) = 1 for every i ∈ I . Consider the linear map f : V → V deﬁned by f (v) = s · v − v. It is obvious that f maps V onto V := i∈I Vi with ker f = ZV (s) = V0 and that the restriction of f to V is a linear automorphism. The image f (U ) is a smooth k-subgroup of V , and it lies in U due to the T -stability of U inside of V . By deﬁnition, V has a T -equivariant composition series whose successive quotients are 1-dimensional vector groups with a nontrivial T -action. Hence, all T -stable k-subgroup schemes of V are connected. In particular, f (U ) is connected. Since U0 ∩ f (U ) = 0 (as V0 ∩ V = 0), under addition we have that U0 × f (U ) is a k-subgroup of U . Thus, f : U → f (U ) is a map onto a k-subgroup of U and the restriction of this map to f (U ) is therefore an endomorphism f (U ) → f (U ) with trivial kernel. But f (U ) is smooth and connected, so this endomorphism is an automorphism. In other words, f : U f (U ) is a projector up to an automorphism of f (U ). Since U ∩ ker f = U ∩ V0 = U0 , this shows that the k-subgroup inclusion U0 × f (U ) → U is an isomorphism, so f (U ) = U ∩ V . This completes the proof that ι is an isomorphism. Step 2. We now may and do assume V = V . In particular, U is connected. We may also assume V = 0 (i.e., I is non-empty). For each k-rational character χ of T , let Cχ = (ker χ )red . Note that Cχ might be disconnected, but it is smooth.

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489

We shall say that two k-rational characters χ and χ of T are equivalent if Cχ = Cχ inside of T . Since char(k) = p > 0, this equivalence condition says a

exactly that χ p = χ εp for some ε ∈ {±1} and some nonnegative integers a and a . Let m be the number of equivalence classes among the χi . Choose an enumeration and let V 1 , . . . , V m be the partial subproducts of these equivalence classes, j of Vi deﬁned by taking V to be the product of the Vi for which χi in the jth equivalence class. Let Cj denote the common Cχi for χi in the jth equivalence class. We need to choose the enumeration of the equivalence classes so that an additional property holds: Cj is not contained in Cj for j > j . To see that this can be done, observe that every Cχ is of codimension 1 in T , so a necessary condition for Cχ to contain Cχ is that Cχ0 = Cχ0 . Since T /Cχ0 is a 1-dimensional torus, its ﬁnite étale k-subgroup Cχ /Cχ0 is cyclic and is determined by its order. Hence, to get an enumeration of the desired type it sufﬁces to arrange the Cj for which Cj0 is a ﬁxed k-subtorus of T so that the order of the component group of Cj is strictly increasing in such j. A key property of such a reﬁned choice of enumeration of the equivalence classes is that no Cj is contained in the union (inside of T ) of the Cj for j < j. Indeed, by dimension considerations we only need to consider those Cj for which Cj0 = Cj0 , and then passing to component groups inside of a

(T /Cj0 )(ks ) = ks× reduces us to the obvious fact that a ﬁnite subgroup of ks× cannot be covered by a union of proper subgroups (due to the cyclicity of these ﬁnite groups).

j Step 3. We claim that the natural map of k-group schemes m j=1 (U ∩V ) → U j is an isomorphism (from which it follows that each U ∩ V is smooth, due to the smoothness of U ). To verify this isomorphism property we may assume that k is algebraically closed. We proceed by induction on m. If m = 1 then there is nothing to do, so we can assume m > 1. As we have observed above, Cm is not covered by {Cj }j<m , so we can choose t ∈ Cm (k) not contained in j<m Cj . Deﬁne the map g : V → V by g(v) = t · v − v. The way we chose t gives that ker g = V m , g(V ) = j<m V j , and the restriction of g to the subspace g(V ) is an automorphism of g(V ). Consider the restriction of g to U . Since U is a T -stable subgroup of V , g restricts to an endomorphism of U and moreover ker(g|U ) = U ∩ ker g = U ∩ V m . Thus, ker(g|U ) ∩ g(U ) ⊆ V m ∩ g(V ) = 0, so addition deﬁnes a subgroup inclusion ker(g|U ) × g(U ) → U . Exactly as in Step 1, the map g : U → g(U ) is a projector up to an automorphism of g(U ) by smoothness and connectedness ofg(U ), so it follows that U = ker(g|U ) × g(U ). This forces g(U ) = U ∩ ( j<m V j ), so in particular this intersection is smooth. Induction on m may then be applied by replacing U with g(U ) and replacing V with j<m V j .

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Step 4. Now we can assume that V = V 1 , which is to say that the χi are equivalent to each other. Fix a 1-parameter k-subgroup λ : GL1 → T such that for some weight χ of T on Lie(U ) ⊆ Lie(V ) = V we have λ, χ = 0. The same then holds for all χ since V = V 1 . Let I + = {i ∈ I | λ, χi > 0}, I − = I − I + ,

V± =

Vi .

i∈I ±

We now use the general construction UG (±λ) for the solvable smooth connected k-group G = U GL1 with GL1 acting on U through λ and the T -action on U . This provides smooth connected k-subgroups ZG (λ) ⊇ GL1 , U + := UG (λ) ⊇ U ∩ V + , U − := UG (−λ) ⊇ U ∩ V − .

(B.4.1)

The containments in (B.4.1) are equalities on Lie algebras, due to the weight space decomposition of Lie(G) with respect to T , and hence these are equalities of groups since U ± and ZG (λ) are smooth and connected. It also follows that the product scheme U − × GL1 × U + is Zariski-open in G via multiplication in G, so U − ×U + is open in U via multiplication. But U is smooth, commutative, and connected, so U = U − × U + . Our task is to prove the existence of an automorphism α of the k-group V = V + × V − such that α(U ) = j∈J Vj for a suitable subset J of I . For this, ± ± it will sufﬁce to prove that there exists an automorphism α of V and a subset ± ± ± ± J of I such that α (U ) = j∈J ± Vj . Since we can replace λ with −λ, it sufﬁces to prove the existence of α + . That is, we may replace V with V + and U with U + to arrange that I = I + ; i.e., λ, χi > 0 for all i ∈ I . We identify the e index set I with {1, . . . , n} so that χi = χ p i for a k-rational character χ of T and 0 e1 · · · en . The proof of the existence of the desired automorphism α will be by induction on n. The case n 1 is trivial, so we can assume n > 1. We can also assume that U is nontrivial. Let us ﬁrst assume that U contains a nontrivial k-rational point u. (Later we will prove that this is always the case.) Identify each Vi with Ga as a k-group, so in V = ni=1 Vi = Gan we have u = (0, . . . , 0, aj , . . . , an ) ∈ k n for some aj ∈ k × . Applying the k-linear T -equivariant automorphism of scaling by 1/aj on V brings us to the case aj = 1. The transformation α : (x1 , . . . , xn ) → (x1 , . . . , xj , yj+1 , . . . , yn )

B.4 Torus actions on unipotent groups

p

491

em −ej

with ym = xm − am xj for m > j is a T -equivariant k-automorphism of the k-group V , and it maps u to the point 1 in the factor group Ga = Vj . Hence, the k-subgroup α (U ) in V meets Vj in a nonzero T -stable subgroup scheme, and the only such subscheme is Vj (as T acts on Vj = Ga by a nontrivial character). We conclude a k-subgroup of V with U equal that α (U ) = Vj × U as to α (U ) ∩ i=j Vi . The direct factor U of α (U ) inherits smoothness from α (U ), and U is obviously T -stable. Induction on n can therefore be applied to U and i=j Vi to complete the argument, provided that U (k) = 0 whenever U = 0. It remains to show that if U is nontrivial, then U (k) = 0. Such a property is obvious when k is separably closed (as U is smooth), so the proposition is proved over ks . In particular, for any k, the ks -group Uks is a vector group over ks . Hence, by Corollary B.2.6, U is a vector group over k. Thus, U (k) = 0 when U is nontrivial. 2 Theorem B.4.3 Let T be a k-torus and U a smooth p-torsion commutative afﬁne k-group. Suppose that there is given an action of T on U over k. Then U = U0 × U with U0 = ZU (T ) and U a T -stable k-subgroup that is a vector group admitting a linear structure relative to which T acts linearly. Proof. By Proposition B.4.1, there is a T -equivariant embedding of U into a linear representation space V for T . Let V0 × V be the unique decomposition of V into a direct product of linear representations for T such that V0 has the trivial T -action and ZV (T ) = 0 (i.e., Vks has no occurrence of the trivial character of Tks in its weight space decomposition). Note that U0 = U ∩ V0 . If k = ks , or more generally T is k-split, then V is a direct product of ﬁnitely many 1-dimensional linear representations of T , so the desired result follows from Proposition B.4.2. Hence, in general the desired decomposition holds for Uks over ks . The argument in Step 1 of the proof of Proposition B.4.2 shows that U = U0 × (U ∩ V ), so we can replace (V , U ) with (V , U ∩ V ) to reduce to the case where U is connected and ZV (T ) = 1 (so ZU (T ) = 1). In this case, Vks has no occurrence of the trivial character in its weight space decomposition under Tks , so likewise for Lie(U )ks . By the settled case of a separably closed ground ﬁeld, there is a Tks equivariant isomorphism between Uks and Lie(U )ks = Lie(Uks ) (viewed as a vector group with Tks -action over ks ). We will show that there exists an isomorphism which is Gal(ks /k)-equivariant (and hence descends to a k-group isomorphism U Lie(U ) respecting the T -actions) using Galois cohomology. Let G = Aut Tks (Vks ) denote the abstract group of Tks -equivariant automorphisms of the additive ks -group Vks (ignoring the linear structure on Vks ). The group G is equipped with a natural action by Gal(ks /k), and it sufﬁces to prove that H1 (k, G) = 1.

492

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Recall that we have reduced to the case in which all Tks -weights on Vks are nontrivial. For any such weight χ, let Vχ denote the χ -weight space in Vks . Also, let S be the set of such weights that do not have the form ψ p for another Tks -weight ψ on Vks . Observe that S is Gal(ks /k)-stable inside of X(Tks ) − {0}. For each χ ∈ S, deﬁne V (χ ) =

Vχ pei

i e

e

where 0 = e1 e2 . . . and {χ p i } is the set of powers χ p with e 0 such e that χ p is a Tks -weight on Vks . Any Tks -equivariant automorphism of the additive ks -group Vks must carry V (χ ) back into itself. To see this, ﬁrst note that the restriction of such an automorphism to any line in Vχ pe is a Tks -equivariant ks -homomorphism e e Ga (χ p ) → Vks (where Ga (χ p ) denotes Ga equipped with the Tks -action e through χ p ). A ks -endomorphism of Ga is given by a p-polynomial in one variable over ks , so upon choosing a ks -basis of Vks consisting of Tks -eigenvectors, pe it follows that the homomorphism Ga (χ ) → Vks has image contained in e particular, V (χ ) is indeed stable under the action of G on ei e Vχ p i . In Vks . Thus, G = χ∈S Gχ where Gχ is the group of Tks -equivariant automorphisms of the additive ks -group V (χ ). Decomposing this product according to the Gal(ks /k)-action on S, we may assume that S consists of a single Gal(ks /k)-orbit. Fix a χ ∈ S. Let K/k be the ﬁeld of deﬁnition of χ as a character of Tks (equivalently, the ﬁnite separable extension in ks corresponding to the subgroup of Gal(ks /k) ﬁxing χ ). The group Gχ has a natural action by Gal(ks /K), and G = Ind kK (Gχ ) as groups equipped with a Gal(ks /k)-action. (Here, for a group endowed with an action by Gal(ks /K), we deﬁne Ind kK () to be the group of functions f : Gal(ks /k) → such that f (hg) = h f (g) for all g ∈ Gal(ks /k) and h ∈ Gal(ks /K). It is endowed with a Gal(ks /k)-action via (g f )(g ) = f (g g).) By a non-commutative version of Shapiro’s Lemma in degree-1 Galois cohomology, we have H1 (k, G) = H1 (K, Gχ ). (This is stated in [Ser3, Ch. I, 5.8(b)]. Also see [Oes, Ch. IV, 2.3] for a related discussion for Weil restriction of algebraic groups.) Thus, we may reduce to the case when V = V (χ ) for a k-rational character χ of T . But then the linear action of T on V is obtained through projection to the quotient χ (T ) GL1 (followed by a suitable linear action of this quotient on V ), so we may assume T = GL1 and that χ : T → GL1 is the identity map. Each linear subspace e e V pe in Vks is the ﬁxed scheme for the action of χ μpe on Vks , so it is deﬁned over k and is Tks -stable. The Gal(ks /k)-equivariant ﬁltration by such linear subspaces is preserved by G, and passing to the successive quotients by this ﬁltration deﬁnes a Gal(ks /k)-equivariant composition

B.4 Torus actions on unipotent groups

493

series for G whose successive quotients are the groups of Tks -equivariant automorphisms of the additive ks -groups Vχ pei and the group G0 of g ∈ G whose induced automorphism on every Vχ pei is trivial. Hence, it sufﬁces to prove that H1 (k, G0 ) = 1 and to treat the case when T acts on V through a single k-rational character. In this latter case the functor of T -equivariant automorphisms of the k-group V is the same as the linear automorphism functor of V , which is to say G = GL(V ). The vanishing of H1 (k, GL(V )) is well known. It remains to prove that H1 (k, G0 ) = 1. But G0 visibly has a Gal(ks /k)-stable composition series whose successive quotients are the vector spaces of Tks -equivariant homomorphisms from Vχ pe to V pe χ

for e e. Since these weight spaces are deﬁned over k, such successive quo tients are the ks -vector spaces of ks -homomorphisms f : Gan → Gan satisfying a p f (cx) = c f (x) for all c ∈ GL1 and some a 0. This is the group Homks (ks ⊗Frak

s

, ks

ksn , ksn ) = ks ⊗k Homk (k ⊗Frak ,k k n , k n )

with its natural Gal(ks /k)-action, where Fr L : L → L denotes the p-power map of a ﬁeld L of characteristic p. The additive group ksN with its standard Gal(ks /k)-action has vanishing degree-1 cohomology for Gal(ks /k) for any N 0, so we are done. 2 Corollary B.4.4 Let T be a k-torus and U a k-wound smooth connected unipotent k-group. The only T -action on U is the trivial one. Proof. Our aim is to prove that the k-subgroup scheme ZU (T ) is equal to U . For the k-group G = U T , we have that the torus centralizer ZG (T ) is equal to ZU (T ) T . But ZG (T ) is smooth and connected, so the same holds for ZU (T ). Since ZU (T ) is a scheme-theoretic centralizer, Lie(ZU (T )) is the T -centralizer in Lie(U ). Hence, to prove that ZU (T ) = U it sufﬁces (by smoothness and connectedness of U ) to prove that T acts trivially on Lie(U ). By Corollary B.3.3 there is a T -equivariant composition series of U consisting of smooth connected normal k-subgroups such that the successive quotients of the ﬁltration are k-wound, commutative, and p-torsion. It sufﬁces to prove the corollary for the successive quotients in the role of U , so we may thereby reduce to the case when U is p-torsion and commutative. An application of Theorem B.4.3 then concludes the argument. 2

Appendix C Rational conjugacy in connected groups

Let G be a smooth connected afﬁne group over a ﬁeld k. In [BoTi2], Borel and Tits announced (without proof) some remarkable results generalizing important theorems when G is reductive. Among these are the G(k)-conjugacy of maximal k-split k-tori, maximal k-split smooth connected unipotent k-subgroups, and minimal pseudo-parabolic k-subgroups, as well as the Bruhat decomposition for G(k) (relative to a choice of minimal pseudo-parabolic k-subgroup). In this appendix we use §§2.1–3.5 and Appendix B to prove these results, following the ideas outlined in [Ti3, §§2–3] (with some scheme-theoretic improvements). We also give some generalizations in §C.4 for group schemes locally of ﬁnite type over a ﬁeld.

C.1 Pseudo-completeness A key observation is that the coset space G/P modulo a pseudo-parabolic k-subgroup P satisﬁes the following variant of the valuative criterion for properness. Deﬁnition C.1.1 A scheme X over a ﬁeld k is pseudo-complete over k if it is ﬁnite type and separated and X (R) = X (K) for any discrete valuation ring R over k with fraction ﬁeld K and residue ﬁeld separable over k. For any pseudo-complete X , if C is a smooth curve over k and c ∈ C is a closed point such that k(c)/k is separable then any k-morphism C − {c} → X uniquely extends to a k-morphism C → X . Proposition C.1.2 Let X be a scheme over a ﬁeld k. (1) If X is pseudo-complete over k then so is any closed subscheme. (2) If k /k is an extension of ﬁelds then X is pseudo-complete over k if Xk is pseudo-complete over k . The converse holds when k /k is separable. 494

C.1 Pseudo-completeness

495

(3) In the deﬁnition of pseudo-completeness for X , it sufﬁces to consider complete discrete valuation rings over k with residue ﬁeld that is separably closed and separable over k. Proof. If Z is closed in X then Z(R) = Z(K) ∩ X (R) for any domain R and K = Frac(R), so (1) follows. To prove (3), we may assume that X is separated and of ﬁnite type. In particular, X (R) ⊆ X (K) for any discrete valuation ring R over k with K = Frac(R). If R → R is a local extension of discrete valuation rings over k with K → K the corresponding extension of fraction ﬁelds then X (R) = X (R ) ∩ X (K) inside of X (K ) since R = R ∩ K inside of K. Hence, X (R) = X (K) if X (R ) = X (K ). Taking R to be the completion of a strict henselization of R, (3) follows. Since the properties of being separated and of ﬁnite type descend through a ground ﬁeld extension, to prove (2) we have to show that if X is separated and of ﬁnite type then it satisﬁes the valuative criterion deﬁning pseudo-completeness over k if Xk satisﬁes the analogous criterion over k , and conversely when k /k is separable. We ﬁrst check preservation under separable extension on k, so let k /k be separable and consider a discrete valuation ring R over k with fraction ﬁeld K and residue ﬁeld separable over k . By transitivity of separability in ﬁeld extensions, this residue ﬁeld is also separable over k, so by the assumed valuative criterion for X over k we have X (R ) = X (K ). But X (A) = Xk (A) for any k -algebra A, so Xk (R ) = Xk (K ). This proves the “converse” part of (2). To prove the ﬁrst part of (2) (i.e., pseudo-completeness descends from Xk over k to X over k for any k /k), we shall adapt the argument used to prove (3). Consider a discrete valuation ring R over k whose residue ﬁeld F is separable over k. We will construct a discrete valuation ring R over k with residue ﬁeld F separable over k and a local injection R → R over k → k . Granting this, if K → K is the induced extension of fraction ﬁelds then X (R ) = Xk (R ) = Xk (K ) = X (K ). Thus, as in the proof of (3), we deduce that X (R) = X (R ) ∩ X (K) = X (K ) ∩ X (K) = X (K). To construct such an R over k , let t be a uniformizer of R. Consider the ring A = R ⊗k k . The quotient A/tA = F ⊗k k is a reduced ring since F/k is separable. Thus, for a minimal prime ideal p of F ⊗k k , the local ring F = (F ⊗k k )p is a ﬁeld. This ﬁeld is separable over k since F/k is separable. Let P be the corresponding prime ideal of A, so R := AP is a local k -algebra in which t is not a zero divisor and such that R /tR = F . Thus, t generates the maximal ideal of R , so if R is noetherian (e.g., if k /k is ﬁnitely generated) then it is a discrete valuation ring with residue ﬁeld F and uniformizer t such that R → R is a local injection over k → k . This settles the case when k /k is ﬁnitely generated. Since a direct limit of separable ﬁeld extensions is separable and a direct limit of discrete valuation rings with a common uniformizer is a discrete valuation ring with the same uniformizer, the general case follows by

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expressing k as a direct limit of subﬁelds ki ﬁnitely generated over k and using that a minimal prime of F ⊗k k contracts to a minimal prime of each F ⊗k ki (by ﬂatness). 2 It follows from Proposition C.1.2(3) that pseudo-completeness is equivalent to properness when k is perfect, since properness can be checked using the valuative criterion with complete discrete valuation rings having an algebraically closed residue ﬁeld. Hence, pseudo-completeness is of interest primarily for imperfect k. Also, the separability hypothesis on k /k in Proposition C.1.2(2) is essential, as the following example illustrates. Example C.1.3 Let k /k be any ﬁnite extension of ﬁelds, and X a projective k scheme. The Weil restriction X = R k /k (X ) exists as a k-scheme of ﬁnite type, and it is separated since the functor R k /k preserves separatedness (as follows by using consideration of diagonal maps and closed immersions). However, such an X is never proper when k /k is not separable and X is non-empty and smooth with pure dimension d > 0, as we saw in Example A.5.6. In particular, Xk is not proper over k in such cases. But X is always pseudo-complete over k. Indeed, if B is a discrete valuation ring over k with residue ﬁeld F and fraction ﬁeld K then by deﬁnition X (B) = X (B ) for the ﬁnite ﬂat B-algebra B = B ⊗k k , and likewise X (K) = X (K ) for K = K ⊗k k . The quotient of B by a uniformizer of B is the ring F ⊗k k , and if F/k is separable then this tensor product is a ﬁnite product of ﬁelds. In such cases the localization of B at each maximal ideal is therefore a discrete valuation ring. Hence, if F/k is separable then B is regular of pure dimension 1 (and so B is a ﬁnite product of Dedekind domains) with total ring of fractions K , so by the usual valuative criterion of properness for the projective X we deduce that X (B ) = X (K ). This proves that X is pseudo-complete over k. Remark C.1.4 Let X be a separated k-scheme of ﬁnite type. In [Ti3, §2.3], Tits uses only R = ks [[t]] in his deﬁnition of pseudo-completeness for X (which he calls relative completeness), so his deﬁnition may initially seem weaker than Deﬁnition C.1.1. In fact, these two deﬁnitions are equivalent, and they are also equivalent to the geometric condition that k-morphisms to X from punctured smooth curves over ks extend across ks -rational punctures. Below we prove the equivalence of these deﬁnitions, using Chow’s Lemma and approximation arguments, but we will never use the equivalences. One advantage of Deﬁnition C.1.1 is its insensitivity to arbitrary separable extension on k (which is not obvious for either of the other two deﬁnitions). But such strength of the deﬁnition is precisely because it requires the consideration of many discrete valuation rings R. In Proposition C.1.6 we will directly verify this stronger-looking deﬁnition for some natural examples. First we prove the equivalence between Tits’ deﬁnition and the deﬁnition using punctured smooth curves over ks . We can assume k = ks , so the problem

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497

is to show that if the valuative criterion using k[[t]] fails then it also fails using the local ring at a k-rational point on a smooth curve over k. The henselization B = k{t} of k[t](t) is a direct limit of such “algebraic” local rings, and B = k[[t]]. = Let K = B[1/t] denote the fraction ﬁeld of B and K B[1/t] = k((t)). It sufﬁces then X (B) = X (K). Consider any x ∈ X (K) not to show that if X ( B) = X (K) in X ( B). We seek to make an analogous point with (B, K) replacing ( B, K). Without loss of generality we can replace X with the closure of the image of x, so X is irreducible and reduced with x mapping onto its generic point. Since X is separated over k, by Chow’s Lemma there is a proper birational map π : X → X with X a quasi-projective integral k-scheme, so x uniquely and clearly x ∈ X ( B). By quasi-projectivity, X is open in lifts to x ∈ X (K) a projective k-scheme X , so by the valuative criterion for properness x does extend to a point x ∈ X ( B). The map x : Spec B → X identiﬁes the closed point with a k-rational point x0 ∈ X (k) − X (k). By Artin approximation (or an earlier version due to Greenberg [Gre]) applied to the excellent henselian discrete valuation ring B (and X B ), there is a point x ∈ X (B) lifting x0 ∈ X (k) that is as “close” as we wish to x (using the t-adic topology). Since x carries into X , the radical coherent ideal of X − X in X has x -pullback in Spec K B equal to (t r ) for some r 1. Hence, by choosing x sufﬁciently close to x we can arrange that the restriction xη : Spec K → X of x lands in X . The π composite k-map Spec K → X → X cannot extend to a B-point, for if it did then this would uniquely lift to a B-point of X extending xη (due to properness of π), contradicting that the B-point x ∈ X (B) − X (B) extending xη is unique in X (B) (due to the separatedness of X over k). In a similar manner we can show that Tits’ notion of relative completeness implies the valuative criterion for pseudo-completeness (the converse being trivial). In other words, if this valuative criterion fails for X using some discrete valuation ring B over k with residue ﬁeld F separable over k then we claim that it already fails for ks [[t]]. We can replace B with the completion of a strict henselization, so B is complete and F is separably closed. Let ks be the separable closure of k in F. By Proposition C.1.2(2), we can rename ks as k. Since F/k is a separable extension of ﬁelds, there is a k-algebra isomorphism B F[[t]]. (Indeed, by separability, F/k is formally smooth in the sense of commutative algebra [Mat, Thm. 26.9]. Hence, the quotient map B/m2 F admits a kalgebra section, and continuing inductively there is a k-algebra map F → B lifting the identity on residue ﬁelds. Any uniformizer of B may be taken as t.) Thus, we have a point x ∈ X (F((t))) not in X (F[[t]]). As above, we may replace X with a closed subscheme so that X is integral and x dominates the generic point of X , and after applying Chow’s Lemma we may also assume that X is quasi-projective over k. Let X be a projective compactiﬁcation of X , so x ∈ X (F((t))) uniquely extends to a point x ∈ X (F[[t]]). Using Artin/Greenberg approximation as

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above, we get a point y ∈ X (F{t}) whose generic point lies in X but whose special point does not. In other words, the radical ideal of X − X has y-pullback in F{t} equal to (t r ) for some r 1. By direct limit considerations, we can assume that F is ﬁnitely generated over k, so F = Frac(A) for a smooth kalgebra domain A. Since F is a direct limit of such algebras A, for some such A there is an étale map Spec R → Spec(A[t]) hitting t = 0 such that y ∈ X (R) with the radical ideal of X − X pulling back to (t r ) in R. Specialization at a k-point of A then gives a point in X (k{t}) whose generic point lies in X and special point does not. Passing to the completion, we have constructed a point in X (k((t))) not lying in X (k[[t]]). This completes the proof of the equivalence of the three deﬁnitions of pseudo-completeness over k. Pseudo-complete k-schemes satisfy an analogue of the Borel ﬁxed point theorem: Proposition C.1.5 Let k be a ﬁeld and X a pseudo-complete k-scheme such that X (k) = ∅. Assume that X is endowed with a left action by a smooth connected afﬁne k-group H that is k-split solvable. There is a point in X (k) ﬁxed by H . At the start of §B.1 we review the notion of a k-split solvable smooth connected afﬁne k-group. Proof. We may assume H = 1, so by deﬁnition of being k-split there is a k-split solvable smooth connected normal k-subgroup N of H such that H /N is k-isomorphic to Ga or GL1 . By induction on dim H , there is some x ∈ X (k) ﬁxed by N . Consider the subfunctor of X whose B-points for any k-algebra B consist of those x ∈ X (B) ﬁxed by NB . This is represented by a closed subscheme X in X . Indeed, by Galois descent we can assume k = ks , and then N (k) is Zariski-dense in N . Thus, this subfunctor of X is represented by

{x ∈ X | nx = x}.

n∈N (k)

Since X is closed in X , it inherits the pseudo-completeness property from X . Also, induction gives that X (k) is non-empty. Thus, we may replace X with X and replace H with H /N to reduce to the case where H is either Ga or GL1 . Choose x ∈ X (k) and consider the orbit map H → X deﬁned by g → gx. By pseudo-completeness, this extends to a map f : P1k → X . The natural action of H on itself extends to an action on P1k ﬁxing the point ∞ at inﬁnity, and f is H -equivariant. Hence, f (∞) ∈ X (k) is ﬁxed by H . 2 Proposition C.1.6 Let G be a smooth connected afﬁne group over a ﬁeld k, and P a pseudo-parabolic k-subgroup of G. The coset space G/P is pseudocomplete over k.

C.1 Pseudo-completeness

499

Proof. By Proposition 2.2.10, we may assume that G is pseudo-reductive over k. To prove the pseudo-completeness of G/P it is harmless to make a separable extension on k, so we can assume that k = ks . Hence, G(k) is Zariski-dense in G. Let G = G red . By Proposition 3.5.4, the image P of Pk in G is a parabolic k

subgroup, so G /P is proper over k. To check pseudo-completeness of G/P over k, choose a discrete valuation ring A over k with residue ﬁeld F that is separable over k. Let K = Frac(A) and let t ∈ A be a uniformizer. Since k = ks , F := F ⊗k k is a ﬁeld. Thus, A := A ⊗k k is a discrete valuation ring with uniformizer t, residue ﬁeld F , and fraction ﬁeld K = K ⊗k k. Pick x ∈ (G/P)(K), and consider its image x ∈ (G /P )(K ) = (G /P )(A ). This specializes to a point x0 ∈ (G /P )(F ). Since G(k) is Zariski-dense in G, it has Zariski-dense image in (G /P )F . Hence, by applying a suitable left G(k)-translation to x (which is harmless for the purpose of checking if x ∈ (G/P)(A)) we can arrange that x0 lies in U (F ) for any desired non-empty open subset U ⊆ G /P . It would then follow that x ∈ U (A ), since an open subset of Spec A containing the closed point is the entire space. We will make a useful choice of such a U shortly. Since G is pseudo-reductive, there is a 1-parameter k-subgroup λ : GL1 → G such that P = PG (λ). Let λ : GL1 → G be the induced homomorphism over k, so PG (λ ) is the image of Pk (i.e., P = PG (λ )) and UG (−λ ) is the image of UG (−λ)k (Corollary 2.1.9). Since the multiplication maps UG (−λ) × P → G and UG (−λ )×P → G are open immersions (Proposition 2.1.8(3)), it follows that UG (−λ) and UG (−λ ) are naturally identiﬁed with dense open subschemes U and U of G/P and G /P respectively. By applying a suitable left G(k)-translation to x we may assume that x ∈ U (A ). Using the Zariski topology on G(k), we claim that there is a dense open locus of g ∈ G(k) such that gx ∈ U (A ). This follows from considering the closed point of Spec A and using the general fact that if X is a k-scheme in which X (k) is Zariski-dense and if U is an open subscheme of XK for an extension ﬁeld K/k then X (k) is Zariski-dense in XK and the set X (k) ∩ U := {x ∈ X (k) | xK ∈ U (K) inside X (K)} is open in X (k). These assertions concerning X (k) are consequences of the more general assertion that XK → X is open. To prove such openness, we may work Zariski-locally on X so that it is afﬁne, and then we may assume K = Frac(R) for a domain R of ﬁnite type over k such that U = UK for an open U ⊆ XR . The map XR → X is open [EGA, IV2 , 2.4.6], and each ﬁber meets XK , so U has the same image as U . We likewise see that there is a dense open locus of g ∈ G(k) such that gx ∈ U (K). Choosing g in the overlap of these two dense open subsets, we may arrange that x ∈ U (K) and x ∈ U (A ). It now sufﬁces to prove that under the map U (K) → U (K ), the preimage of U (A ) is U (A). Pick a maximal k-torus T of G containing λ(GL1 ), so Tk

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is carried isomorphically onto a maximal torus T in G containing λ (GL1 ). Since U (A) = Uk (A ) ∩ U (K) inside of Uk (K ), we are led to focus attention on the quotient map of k-groups Uk → U . The kernel of this map has no nontrivial points of U (k). Indeed, this kernel is contained in Ru (Gk ), so if it contains a nontrivial element of U (k) then it would contain inﬁnitely many such elements due to T (k)-equivariance (since U = UG (−λ)), contradicting pseudo-reductivity of G over k. By Corollary 3.3.12 (applied with −λ in place of λ), U is directly spanned by the subgroups Ua = U(a) for the non-divisible a ∈ (G, T ) such that a, λ < 0, and the quotient map Gk G carries each (U(a) )k onto the corresponding 1 in G . If we choose a T -equivariant linear structure dimensional root group U(a) on U(a) , as we may always do, then restricting the quotient map (U(a) )k U(a) to a basis of lines arising from T -stable lines of U(a) yields endomorphisms of Ga over k given by monomials (due to the T -equivariance). Recalling that U (k) → U (k) is injective, we are reduced to the following special situation. Let f be a p-polynomial in several variables x1 , . . . , xn over k (in the sense of Deﬁnition B.1.2, so f has vanishing constant term) and assume that f has no nontrivial k-rational zero. In particular, every xi occurs in f or else we could construct a nontrivial k-rational zero of f . Assume that in fact each variable qi × occurs in a single monomial: f = ci xi for some ci ∈ k and some qi 1 that is a power of the characteristic exponent of k. Our problem is to show that if ξ1 , . . . , ξn ∈ K and f (ξ ) ∈ A then ξi ∈ A for all i. Since each variable q xi intervenes in f only in a term of the form ci xi i with qi 1, if some ξi vanishes then we can drop the corresponding xi -term from f without affecting the hypothesis of no nontrivial k-rational zeros and preserving the condition that f (ξ ) = 0. Hence, we may assume that ξi = 0 for all i. Let mi = −ord A (ξi ) ∈ Z, and assume that mi 1 for some i (i.e., ξi ∈ A for some i). Thus, N := maxi mi qi 1. Let ai = t mi ξi mod t ∈ A/tA = F, so ai = 0 for all i and the element t N f (ξ ) = 0 in A/tA = F is equal to q ci ai i t N −mi qi mod t. Hence, a nontrivial F-rational zero of f is given by the point (a1 , . . . , an ) ∈ F n with ai = ai when mi qi = N and ai = 0 otherwise. qi We conclude that the p-polynomial f = ci xi over k acquires a nontrivial zero in the separable extension F/k when f is viewed as a polynomial over the ﬁeld F ⊗k k. Such a nontrivial zero occurs in a subﬁeld L of F that is ﬁnitely generated over k, and hence it occurs in Rk for a smooth ﬁnitely generated k-subalgebra R of F. Since the k-rational points of Spec R are Zariski-dense (by smoothness, as k = ks ), by viewing f as a polynomial over Rk we can specialize at some k-rational point of Spec R to get a nontrivial zero of f over k. This is a contradiction. 2 We warn the reader that the converse to Proposition C.1.6 is false. To explain this, it is convenient to use the following terminology:

C.1 Pseudo-completeness

501

Deﬁnition C.1.7 For a smooth connected afﬁne group G over a ﬁeld k, a smooth k-subgroup Q is quasi-parabolic if the coset space G/Q is pseudo-complete over k. By Proposition C.1.6, pseudo-parabolic k-subgroups Q in G are quasiparabolic. Since pseudo-completeness is insensitive to separable extension on k (Proposition C.1.2(2)), the same holds for quasi-parabolicity. Also, any quasiparabolic k-subgroup Q in G contains every k-split solvable smooth connected normal k-subgroup H of G. (Proof: We can assume k = ks , so then by Proposition C.1.5, some G(k)-conjugate of H is contained in Q. By normality, H ⊆ Q, as desired.) It follows that a k-split smooth connected solvable k-group has no proper quasi-parabolic k-subgroup, and that any quasi-parabolic k-subgroup of a smooth connected unipotent k-group U contains the (normal) maximal k-split smooth connected k-subgroup Usplit . In contrast with pseudo-parabolicity, a smooth k-subgroup of G containing a quasi-parabolic k-subgroup can fail to be quasi-parabolic. For example, in Lemma C.1.8 below we will show that if char(k) > 0 then {1} is quasi-parabolic in any k-wound smooth connected unipotent k-group U , yet over any imperfect ﬁeld k there are examples (see [Oes, Ch. V, 3.5]) of 2-dimensional k-wound U admitting Ga as a quotient by a smooth connected normal k-subgroup U , so U is not quasi-parabolic in U even though {1} is. Such examples are of limited interest, since we are most interested in pseudo-reductive G. It is natural to wonder: is every quasi-parabolic k-subgroup actually pseudoparabolic when G is pseudo-reductive? For example, in the reductive case this amounts to asking if the quasi-parabolic k-subgroups are parabolic (by Proposition 2.2.9). The answer to the question is negative in general over imperfect ﬁelds, though afﬁrmative in the reductive case (over any ground ﬁeld). First we give a result in the unipotent case. Lemma C.1.8 Let U be a smooth connected unipotent group over a ﬁeld k. There exist proper quasi-parabolic k-subgroups in U if and only if U is not k-split, and if U is k-wound then it is pseudo-complete over k. Proof. We have seen above that Usplit is contained in any quasi-parabolic k-subgroup of U , so by passing to U /Usplit it sufﬁces to treat the case p = char(k) > 0 and k-wound U . That is, we aim to prove that U is pseudo-complete when it is nontrivial and k-wound. By Proposition B.3.2 and Proposition C.1.2(2), we may assume k = ks . Applying Proposition B.3.2 again, there is a composition series {Ui } for U consisting of smooth connected normal k-subgroups such that each Ui /Ui+1 is k-wound, commutative, and p-torsion. We will induct on the length of this composition series. Granting the case of length 1 for a moment, we may assume

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that there is an exact sequence 1 → U → U → V → 1 with U pseudo-complete and V a p-torsion commutative k-wound smooth connected unipotent k-group. By the base case for the induction (that we are assuming for now), V is also pseudo-complete over k. To prove that U is pseudocomplete over k, it sufﬁces (by Proposition C.1.2(3)) to check the valuative criterion for complete discrete valuation rings A over k with residue ﬁeld that is separably closed and separable over k. Let K be the fraction ﬁeld of A and consider the commutative diagram 1

/ U (A)

/ U (A)

/ V (A)

1

/ U (K)

/ U (K)

/ V (K)

/ 1

The bottom is exact, and the top is also exact because A is complete with separably closed residue ﬁeld and U → V is a smooth surjection. The outer vertical maps are isomorphisms due to the pseudo-completeness of U and V over k, so the middle vertical map is also an isomorphism. It only remains now to treat the case where U is commutative and p-torsion. In this case it is the zero locus in Akn of a nonzero separable p-polynomial f ∈ k[x1 , . . . , xn ] (Proposition B.1.13). We need to show that if {f = 0} is k-wound then any solution to f = 0 in K n actually lies in An . By Lemma B.1.7, the k-wound property implies that after a suitable additive k-automorphism of Akn (which we may certainly apply without loss of generality) it can be arranged that the principal part fprin of f has no nontrivial zero on k n . This is the only property of fprin that we shall use (together with the fact that fprin (0) = 0). In qi particular, f involves each variable (since f (0) = 0), so fprin = ci xi for some ci ∈ k × and p-power qi 1. We now adapt the calculation at the end of the proof of Proposition C.1.6 to get a contradiction if f vanishes at a point ξ ∈ K n not in An . For any i0 , dropping the xi0 -terms from f yields a polynomial in n − 1 variables having no nontrivial zero on k n−1 . Thus, we can assume ξi = 0 for all i. Let t be a uniformizer of A and F = A/tA the residue ﬁeld. Deﬁne mi = −ord A (ξi ) ∈ Z, so mi 1 for some i. Hence, N = maxi mi qi 1. The residue class zi := t mi ξi mod t ∈ F is nonzero for all i, and the vanishing residue class t N f (ξ ) mod t ∈ A/tA = F q is also equal to ci zi i t N −mi qi mod t. This is precisely fprin (z1 , . . . , zn ) with zi = zi when mi qi = N and zi = 0 otherwise. Hence, fprin acquires a nontrivial zero in the separable extension F/k.

C.1 Pseudo-completeness

503

Since k = ks , it now sufﬁces to prove that if h is a polynomial over a separably closed ﬁeld k with h(0) = 0 and if h acquires a nontrivial zero in a separable extension F/k then h has a nontrivial zero in k. We may shrink F to be ﬁnitely generated over k, so F = Frac(A) for a k-smooth domain A. For any nonzero a ∈ A, the localization A[1/a] is k-smooth and hence admits a k-point since k = ks . Thus, we can specialize at a k-point of A to get a nontrivial zero of f over k. 2 The preceding lemma is a key ingredient in the following general result concerning quasi-parabolic subgroups of pseudo-reductive groups. Theorem C.1.9 Let G be a pseudo-reductive group over a separably closed ﬁeld k. The quasi-parabolic k-subgroups of G are precisely the pseudoparabolic k-subgroups if and only if the Cartan k-subgroups of G are tori. In particular: (1) if G is reductive then its quasi-parabolic k-subgroups are precisely its parabolic k-subgroups, (2) if G is not reductive (so k is imperfect) and char(k) = 2 then G contains a quasi-parabolic k-subgroup that is not pseudo-parabolic, (3) if k is imperfect of characteristic 2 then there exist non-reductive standard pseudo-reductive k-groups G in which all quasi-parabolic k-subgroups are pseudo-parabolic. Since quasi-parabolicity and pseudo-parabolicity are insensitive to separable extension on k, part (1) of the theorem is valid over any ﬁeld k. Proof. Let P be a minimal pseudo-parabolic k-subgroup of G, and T a maximal k-torus in P. Let U = Ru,k (P), so U is k-split because G is pseudo-reductive (Corollary 2.2.5). Since P is minimal and k = ks , it follows from Proposition 3.3.15 that P = ZG (T ) · U . In particular, H = T U is a normal k-subgroup of P. Consider the smooth ﬁbration G/H → G/P that is a torsor for the group P/H = ZG (T )/T . This quotient is unipotent, and it is commutative since ZG (T ) is a Cartan k-subgroup of the pseudo-reductive G (Proposition 1.2.4). Since ZG (T ) is pseudo-reductive and T is k-split, it follows that P/H is k-wound (as any commutative k-group extension of Ga by GL1 is a split extension). Thus, by Lemma C.1.8, P/H is pseudo-complete over k. The quotient space G/P is pseudo-complete since P is pseudo-parabolic, and the structure group P/H for the ﬁbration π : G/H → G/P is pseudo-complete. We claim that this implies that G/H is also pseudo-complete (i.e., H is quasiparabolic in G). To check this, it sufﬁces to verify the valuative criterion using a complete discrete valuation ring A over k with residue ﬁeld that is separably closed and separable over k. Let K denote the fraction ﬁeld of A, and choose any x ∈ (G/H )(K). Let y = π(x) ∈ (G/P)(K). Since (G/P)(K) = (G/P)(A), we

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view y as an A-point and pull back the ﬁbration π along y to get a P/H -torsor E over Spec A. The k-group P/H is smooth and A is complete with separably closed residue ﬁeld, so such a torsor must be trivial. That is, E (P/H )A as A-schemes. In particular, E(A) = E(K). But x ∈ E(K), so x extends to an A-point of E, which in turn deﬁnes an A-point of G/H that extends x. Thus, H is indeed quasi-parabolic in G. Now assume that the quasi-parabolic k-subgroups of G are precisely the pseudo-parabolic k-subgroups. By construction, H is contained in the minimal pseudo-parabolic k-subgroup P, so H = P. Since ZP (T ) = ZG (T ), we conclude that ZH (T ) = ZG (T ). But H = T U with T acting on U = Ru,k (P) with only nontrivial weights on Lie(U ). Thus, the inclusion T ⊆ ZH (T ) between smooth connected k-groups is an equality on Lie algebras, so it is an equality of groups. This proves that the Cartan k-subgroups of G are tori. Conversely, assume that the Cartan k-subgroups of G are tori and let H be any quasi-parabolic k-subgroup of G. The minimal pseudo-parabolic k-subgroups of G have the form Q = T U for a maximal k-torus T and a k-split smooth connected unipotent k-group U . Such a Q is a k-split solvable smooth connected k-group since k = ks , so by pseudo-completeness of G/H and Proposition C.1.5 there exists g ∈ G(k) such that gQg −1 ⊆ H . But gQg −1 is pseudo-parabolic, so H is also pseudo-parabolic (Proposition 3.5.8). In the reductive case (for which Cartan k-subgroups are always tori), we conclude that the quasi-parabolic k-subgroups are the pseudo-parabolic ksubgroups. These are precisely the parabolic k-subgroups, by Proposition 2.2.9. Now suppose that G is not reductive, so k is imperfect. In particular, p = char(k) > 0. If p = 2, then by Theorem 11.1.1 the Cartan k-subgroups of G are not tori. Hence, by what we have just proved above, G contains quasi-parabolic k-subgroups that are not pseudo-parabolic. Finally, assume k is imperfect with characteristic 2. In this case, Example 11.1.2 gives examples of non-reductive standard pseudo-reductive k-groups in which all Cartan k-subgroups are tori, and so in such k-groups the quasi-parabolic k-subgroups are precisely the pseudo-parabolic k-subgroups. 2

C.2 Conjugacy results in the smooth afﬁne case As an application of Proposition C.1.6, we can prove G(k)-conjugacy results that are well known in the connected reductive case but not so widely known in the general case. The proofs rest on the following lemma that generalizes a known result in the connected reductive case [Bo2, 20.5]. Lemma C.2.1 Let G be a smooth connected afﬁne group over a ﬁeld k, and P a pseudo-parabolic k-subgroup of G. The map G → G/P has sections Zariski-locally on G/P if k is inﬁnite, and in general G(k) → (G/P)(k) is surjective.

C.2 Conjugacy results in the smooth afﬁne case

505

Our proof is a variation on the argument in [Spr, 15.1.3]. The main difﬁculty, in comparison with the connected reductive case, is that G(k) may not be Zariski-dense in G when k is imperfect. Proof. If k is ﬁnite then the desired surjectivity is a well-known consequence of Lang’s theorem on the vanishing of degree-1 Galois cohomology for smooth connected k-groups (such as P); see [Bo2, 16.5(ii)]. Thus, we may assume k is inﬁnite. We shall argue by induction on dim G, the case of dimension 0 being clear. Now assume dim G > 0 and that the result is known for all smooth connected afﬁne k-groups of strictly smaller dimension. Let U = Rus,k (G) be the maximal k-split smooth connected unipotent normal k-subgroup of G, so U ⊆ P. The map G → G/U is a U -torsor over G/U , so it has sections over every afﬁne open subset of G/U because the pointed étale cohomology set H1 (X , U ) is trivial for every afﬁne k-scheme X (as we may verify by using the k-split property of U to reduce to the case U = Ga , for which we can apply the equality of Zariski and étale cohomology for quasi-coherent sheaves). The quotient group P/U is a pseudo-parabolic k-subgroup of G/U with (G/U )/(P/U ) = G/P, so if U = 1 then the settled case of G/U gives the result for G. We can therefore assume U = 1. By Proposition 2.2.4, the triviality of U implies that P = PG (λ) for some khomomorphism λ : GL1 → G. The multiplication map UG (−λ)×P → G is an open immersion, so UG (−λ) maps isomorphically onto a dense open subscheme V ⊆ G/P. If G(k) is Zariski-dense in G then its image in G/P is Zariski-dense, so G/P is covered by the translates g.V for g ∈ G(k). Since g.UG (−λ) maps isomorphically onto g.V , we get the desired sections Zariski-locally on G/P when G(k) is Zariski-dense in G. In general, let Gt be the smooth connected k-subgroup of G generated by the k-tori. By Proposition A.2.11, this is normal in G and unirational. In particular, Gt (k) is Zariski-dense in Gt since k is inﬁnite. Note that λ factors through Gt , so Pt := P ∩Gt = PGt (λ). The normality of Gt in G implies that Rus,k (Gt ) = 1 since Rus,k (G) = 1. In particular, Pt is a pseudo-parabolic k-subgroup of Gt . Since either dim Gt < dim G or Gt = G (in which case G(k) is Zariski-dense in G), the map Gt → Gt /Pt has Zariski-local sections. We shall prove that the natural map Gt /Pt → G/P is an isomorphism of k-schemes, so we will then be done. The quotient G/Gt makes sense as a k-group since Gt is normal in G, and since λ has trivial image in this quotient we see via Corollary 2.1.9 that the k-subgroup P = PG (λ) in G maps onto G/Gt . It follows that the natural khomomorphism θ : Gt P → G is surjective, and its kernel is isomorphic to Gt ∩ P = Pt (which is smooth). Since Pt P is a k-subgroup of Gt P containing ker θ, we get the composite isomorphism Gt /Pt (Gt P)/(Pt P) ((Gt P)/ ker θ )/((Pt P)/ ker θ) = G/P that is the natural map.

2

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Lemma C.2.2 Let G be a smooth connected afﬁne group over a ﬁeld k, and H a k-split solvable smooth connected k-subgroup. For any pseudo-parabolic k-subgroup P ⊆ G, there exists g ∈ G(k) such that g −1 Hg ⊆ P. Proof. Consider the left translation action of H on G/P. By Propositions C.1.6 and C.1.5, there exists x ∈ (G/P)(k) ﬁxed by H . This lifts to some g ∈ G(k) by Lemma C.2.1, so HgP ⊆ gP. In other words, g −1 Hg ⊆ P. 2 Theorem C.2.3 (Borel–Tits) Let G be a smooth connected afﬁne group over a ﬁeld k. All maximal k-split k-tori in G are G(k)-conjugate. See Proposition A.2.10 for a simpler proof when k = ks . Proof. We argue by induction on dim G, the case of dimension 0 being clear. Now assume dim G > 0 and that the result holds in all strictly smaller dimensions. Let U = Ru,k (G). By Lemma 2.2.3, there is a proper pseudo-parabolic k-subgroup P in G if and only if G/U contains a non-central k-split k-torus. First suppose that G does not contain any proper pseudo-parabolic ksubgroups, so all k-split tori in G map into the maximal central k-split k-torus S in G/U . Thus, the preimage H of S in G is a smooth connected solvable k-subgroup of G that contains all k-split tori of G. If H = G then we are done since dim H < dim G, so we can assume G is solvable. In this case, by [Bo2, 19.2] the maximal k-tori in G are G(k)-conjugate, so the maximal k-split k-tori in G are also G(k)-conjugate. Assume instead that there is a proper pseudo-parabolic k-subgroup P in G, and let S be a maximal k-split k-torus in G. Since S is k-split solvable, by Lemma C.2.2 there exists g ∈ G(k) such that g −1 Sg ⊆ P. Hence, we can again conclude by induction on dimension since dim P < dim G. 2 As an application of Theorem C.2.3, for any smooth connected afﬁne group G over a ﬁeld k and any pseudo-parabolic k-subgroup P of G, the maximal k-split tori S of P are maximal k-split tori in G and Proposition 2.2.11 is valid over any k when using maximal k-split tori. More importantly, if P is minimal for the property of containing S then it is minimal as a pseudo-parabolic k-subgroup of G. (In particular, every k-split k-torus in G lies in a minimal pseudo-parabolic k-subgroup, though this also follows from Lemma C.2.2.) Indeed, in the case that S is a maximal k-torus in G this was proved in the discussion preceding Proposition 3.3.15 by passing up to ks so as to apply the conjugacy result in Proposition A.2.10, and we can carry out the same argument using S directly over k by using Theorem C.2.3. Proposition C.2.4 Let P be a pseudo-parabolic k-subgroup of a smooth connected afﬁne group G over a ﬁeld k. Let S be a maximal k-split k-torus in P. Then ZG (S) ⊆ P, and P is a minimal pseudo-parabolic k-subgroup of G if and only if P = ZG (S) · Rus,k (P).

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Proof. By Remark 3.5.6 and Proposition 3.5.8, P is minimal in G if and only if P has no proper pseudo-parabolic k-subgroup of its own. Applying Lemma 2.2.3(1) to P, we conclude that P is minimal in G if and only if all k-split k-tori in P := P/Ru,k (P) are central. It is equivalent to say that one maximal k-split k-torus in P is central. Consider the image S of S in P. For any k-torus T in P containing S, the preimage H of T in P is a smooth connected solvable k-subgroup of P that is an extension of T by the unipotent group Ru,k (P). Hence, any maximal k-torus of H maps isomorphically onto T . Applying this to a maximal k-torus of H that contains S, it follows from the maximality of S as a k-split k-torus in P that S is maximal as a k-split k-torus in P. We conclude that P is minimal in G if and only if S is central in P. But ZP (S) → ZP (S) is surjective (Proposition A.2.5), so P is minimal in G if and only if P = ZP (S) · Ru,k (P). We will now prove that P = ZP (S) · Rus,k (P) when P is minimal. For this, we can replace P by P/Rus,k (P) to reduce to the case where (by Corollary B.3.5) Ru,k (P) is k-wound. But according to Corollary B.4.4 a k-wound smooth connected unipotent k-group does not admit a nontrivial action by a k-torus. Hence, Ru,k (P) ⊆ ZP (S). This proves that P = ZP (S) · Rus,k (P). Finally we will show that ZP (S) = ZG (S) for all pseudo-parabolic ksubgroups P containing S. Since P = PG (λ)Ru,k (G) for some λ : GL1 → G, by Theorem C.2.3 applied to P we can replace λ with a suitable P(k)-conjugate to arrange that λ takes values in S. Then ZG (S) ⊆ ZG (λ) ⊆ P, so indeed ZG (S) = ZP (S). 2 In the pseudo-reductive case, there is an alternative minimality criterion in the setting of Proposition C.2.4: by Theorem C.2.3 applied to P we can ﬁnd λ ∈ X∗ (S) such that P = PG (λ), so ZG (S) ⊆ ZG (λ), and then P is minimal in G if and only if ZG (S) = ZG (λ). Indeed, since P = ZG (λ) UG (λ) we have P/Ru,k (P) = ZG (λ), so the minimality criterion in Proposition C.2.4 yields the desired criterion in the pseudo-reductive case. Theorem C.2.5 (Borel–Tits) Let G be a smooth connected afﬁne group over a ﬁeld k. All minimal pseudo-parabolic k-subgroups of G are G(k)-conjugate. Proof. Let P and Q be minimal pseudo-parabolic k-subgroups of G. We noted above Proposition C.2.4 that maximal k-split k-tori in P are also maximal in G, and likewise for those in Q. In particular, by replacing P with a suitable G(k)-conjugate we can arrange that P and Q share a common maximal k-split k-torus S. By Proposition C.2.4, P = ZG (S) · Rus,k (P) and ZG (S) ⊆ Q. The smooth connected solvable k-subgroup H := S Rus,k (P) ⊆ P is k-split. By Lemma C.2.2 there exists g ∈ G(k) such that g −1 Hg ⊆ Q. Replacing P with g −1 Pg and S with g −1 Sg then brings us to the case Rus,k (P) ⊆ Q without affecting the property that P and Q share a maximal

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k-split k-torus S. Hence, P = ZG (S) · Rus,k (P) ⊆ Q, so P = Q by minimality of Q. 2 Corollary C.2.6 Let G be a smooth connected afﬁne k-group, and let P and Q be pseudo-parabolic k-subgroups of G. Let K be a separable extension of k. If PK and QK are G(K)-conjugate then P and Q are G(k)-conjugate. Proof. Choose minimal pseudo-parabolic k-subgroups P and Q of G with P ⊆ P and Q ⊆ Q. The G(k)-conjugacy of P and Q (Theorem C.2.5) allows us to replace Q with a suitable G(k)-conjugate so that P ∩ Q contains the minimal pseudo-parabolic subgroup P . By Proposition 3.5.12(1), the ksubgroup P ∩ Q in G is smooth and connected, so it is a pseudo-parabolic k-subgroup of G (Proposition 3.5.8). Thus, if PK and QK are G(K)-conjugate 2 then Proposition 3.5.12(3) implies that PK = QK , so P = Q. Proposition C.2.7 Let G be a smooth connected afﬁne group over a ﬁeld k and P and Q be two pseudo-parabolic k-subgroups. Then there exists a maximal k-split k-torus S of G such that P ∩ Q contains the centralizer ZG (S) of S. Proof. We may and do assume that P is a minimal pseudo-parabolic ksubgroup. By Proposition 3.5.12, P ∩ Q is a smooth connected k-subgroup and (P ∩ Q)ks contains a maximal ks -torus of Gks . Hence, any maximal k-torus T of P ∩ Q is maximal in G. Fix such a T . Under the natural surjective homomorphism π : P → P/Ru,k (P) =: H , the maximal k-torus T maps isomorphically onto a maximal k-torus of H . In particular, π(T ) contains the maximal central k-torus of H . Since P is a minimal pseudo-parabolic k-subgroup of G, by Proposition C.2.4 the center of H contains a k-torus isomorphic to a maximal k-split k-torus of G. We conclude from this that the maximal k-split k-subtorus S of T is isomorphic to a maximal k-split k-torus of G. Hence, S is a maximal k-split k-torus of G (Theorem C.2.3). Now, by conjugacy of maximal k-split k-tori of P (resp. Q) under P(k) (resp. Q(k)), we see that there exist 1-parameter k-subgroups λ, μ : GL1 ⇒ S such that P contains PG (λ) and Q contains PG (μ). But since both PG (λ) and 2 PG (μ) clearly contain ZG (S), so does P ∩ Q. The following gives the Bruhat decomposition of G(k) for an arbitrary smooth connected afﬁne group G over any ﬁeld k. Theorem C.2.8 (Borel–Tits) Let G be a smooth connected afﬁne group over a ﬁeld k, S a maximal k-split k-torus, and P a pseudo-parabolic k-subgroup of G containing S. Let N = NG (S) and Z = ZG (S) ⊆ P. Then G(k) = P(k)N (k)P(k) and if P is minimal then the surjective map N (k)/Z(k) → P(k)\G(k)/P(k) is bijective. Proof. Let g ∈ G(k). We consider the pseudo-parabolic k-subgroup gPg −1 . According to Proposition C.2.7, P ∩ gPg −1 contains a maximal k-split k-torus

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T of G. Thus, P contains the maximal k-split k-tori S, T , and g −1 Tg. By conjugacy of such tori under P(k) (Theorem C.2.3) we deduce that there exist p1 , p2 ∈ P(k) such that p1−1 Tp1 = S = p2 g −1 Tgp2−1 . Then n := p1−1 gp2−1 lies in NG (S)(k). Thus, g = p1 np2 ∈ P(k)N (k)P(k). Now we prove that N (k)/Z(k) → P(k)\G(k)/P(k) is bijective when P is minimal. Let P = P/Ru,k (P), so minimality of P says exactly that the map Z → P is surjective (Proposition C.2.4). The natural k-homomorphism S → P has trivial kernel (as S ∩ Ru,k (P) is both unipotent and of multiplicative type), so S embeds as a central k-torus in P. In particular, any g ∈ P(k) ∩ N (k) must centralize Sk , so P(k) ∩ N (k) = Z(k). Thus for any n ∈ N (k) − Z(k) we must have Pk ∩ (Pk nPk ) = ∅. In particular, P(k)nP(k) is disjoint from P(k) for any n ∈ N (k) − Z(k). This is a special case of what we wish to prove, and we will use it to deduce the general case. Consider any n, n ∈ N (k) such that P(k)nP(k) = P(k)n P(k). We will prove n−1 n ∈ Z(k). The equality of double cosets implies that nq = pn for some p, q ∈ P(k), so pSp−1 = (pn )S(pn )−1 = n(qSq−1 )n−1 ⊆ nPn−1 . Letting H be the smooth connected k-subgroup of P and nPn−1 generated by S and pSp−1 , it follows that S and pSp−1 are maximal k-split k-tori in H . Thus, by Theorem C.2.3 (applied to H ), there exists h ∈ H (k) such that pSp−1 = hSh−1 , so p = hz for some z ∈ N (k). But z = h−1 p ∈ P(k), so z ∈ P(k) ∩ N (k) = Z(k). Since H ⊆ nPn−1 we can write h = np n−1 for some p ∈ P(k), so nqn

−1

= p = hz = np n−1 z.

It follows that p −1 q = n−1 zn = n−1 n (n −1 zn ). This implies that n−1 n Z(k) 2 meets P(k), so P(k)n−1 n P(k) meets P(k). Hence, n−1 n ∈ Z(k). Remark C.2.9 In the setting of Theorem C.2.8, when P is minimal in G we can use the second assertion in Proposition 3.5.7 to prove that if n, n ∈ N (k) are distinct modulo Z(k) then PnP and Pn P are disjoint as locally closed subvarieties of G; this reﬁnes the fact that n and n have distinct images in P(k)\G(k)/P(k). To prove this reﬁnement, we ﬁrst reduce to the pseudo-reductive case as follows. Let π : G → G = G/Ru,k (G) be the maximal pseudo-reductive quotient, so S := π(S) is a maximal k-split k-torus in G and Z := ZG (S ) = π(Z). Let N = NG (S ), so π(N ) ⊆ N . Since π : S → S is an isomorphism, the map N (k)/Z(k) → N (k)/Z (k) is injective. The image P = π(P) = P/Ru,k (G) is a pseudo-parabolic k-subgroup of G which is minimal (by Proposition 2.2.10 and the minimality hypothesis for P), and PnP = Pn P if and only if P π(n)P = P π(n )P , so we can replace (G, S, P) with (G , S , P ) and replace

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n and n with their images in N (k) to reduce to the pseudo-reductive case. In particular, P = PG (λ) and by Theorem C.2.3 (applied to P) we can arrange that λ ∈ X∗ (S). Note that ZG (λ) = ZG (S) due to the minimality of P in G (and the minimality criterion in the pseudo-reductive case, as noted following the proof of Proposition C.2.4). This equality of centralizers for P := PG (λ) is what will be used; neither the pseudo-reductivity of G nor the pseudo-parabolicity of P will be needed below. Consider the following general assertion for an arbitrary smooth connected afﬁne k-group G, k-torus S in G, and k-homomorphism λ : GL1 → S such that ZG (S) = ZG (λ): for N = NG (S), Z = ZG (S), P = PG (λ), U = UG (λ), and distinct elements n, n ∈ N (k), UnU is disjoint from Un U . Once this is proved, we apply it to the preceding setup after scalar extension to k (noting that if n, n ∈ N (k) are distinct modulo Z(k) then they are distinct modulo Z(k)) to deduce that (PnP)(k) is disjoint from (Pn P)(k) since P = Z U . So now we may assume k = k and just have to show that if nu = u n for some u, u ∈ U (k) then n = n . Since n normalizes S it is easy to check that the solvable smooth connected group V := n −1 Un is normalized by S (since U is normalized by S ⊆ Z), so by Proposition 2.1.12(2) we have (V ∩ U − ) × (V ∩ Z) × (V ∩ U ) = V via multiplication, with U − := UG (−λ). But V ∩ Z = n −1 (U ∩ Z)n = 1. Hence, n −1 u n ∈ V (k) = (V ∩ U − )(k) × (V ∩ U )(k), so n −1 nu = n −1 u n = vu with v ∈ U − (k) and u ∈ U (k). Thus, n −1 n = vu u−1 ∈ U − (k)×U (k). This element lies in N (k), and we claim that this forces v, u u−1 ∈ Z(k) = ZG (λ)(k), from which it follows that v = u u−1 = 1, so n = n as desired. To prove the above claim it will sufﬁce to show that if H is any smooth subgroup scheme of Z (such as S) then NG (H ) ∩ (U − × Z × U ) = (U − ∩ ZG (H )) × NZ (H ) × (U ∩ ZG (H )). Working functorially, consider a k-algebra R and points u ∈ U − (R), z ∈ Z(R), and u ∈ U (R) such that u zu normalizes HR in GR . We aim to show that u and u centralize HR and z normalizes HR . To check this we may work with points of H valued in an R-algebra R , so by renaming R as R we aim to show that u and u centralize H (R) and z normalizes H (R). Let φ be the automorphism of HR deﬁned by conjugation by u zu, so for any h ∈ H (R) we have u · zh · h−1 uh = (φ(h)u φ(h)−1 ) · φ(h)z · u. But U − ×Z × U is a subfunctor of G via multiplication, so u = φ(h)u φ(h)−1 , zh = φ(h)z, and h−1 uh = u. Thus, u and u indeed centralize H (R) and z normalizes H (R). This completes the proof that UnU and Un U are disjoint if n = n.

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Proposition C.2.10 Let G be a smooth connected afﬁne group over a ﬁeld k, S a k-split torus in G, and W (G, S) = NG (S)/ZG (S) the associated ﬁnite étale quotient group. The k-group W (G, S) is constant, and if S is a maximal k-split torus then the natural homomorphism NG (S)(k) → W (G, S)(k) is surjective. In particular, NG (S)(k)/ZG (S)(k) → W (G, S)(k) is an isomorphism if S is a maximal k-split torus in G. Proof. To prove that W (G, S) is a constant k-group, it sufﬁces to prove that all of its ks -points are invariant under = Gal(ks /k); i.e., the natural action of on W (G, S)(ks ) = NG (S)(ks )/ZG (S)(ks ) is trivial. Choose any g ∈ NG (S)(ks ), so x → gxg −1 is a ks -automorphism of Sks . But any ks -automorphism of Sks is deﬁned over k since S is k-split, so for any γ ∈ we have γ (gxg −1 ) = gγ (x)g −1 for x ∈ S(ks ). This says that γ (g)−1 g ∈ ZG (S)(ks ), so the image of g in W (G, S)(ks ) coincides with that of γ (g). Hence, the -action is indeed trivial. Now assume that S is a maximal k-split torus, and let W = W (G, S). There is a natural action of W (k) on the set P of minimal pseudo-parabolic k-subgroups of G containing S, described as follows. For a given P ∈ P and w ∈ W (k) let us choose an n ∈ NG (S)(ks ) which maps onto w. Then nPks n−1 is a pseudoparabolic ks -subgroup of Gks which by the Galois-criterion is deﬁned over k and hence descends to a k-subgroup of G containing ZG (S). By Proposition 3.5.2(1) this k-subgroup of G is a pseudo-parabolic subgroup, and as its dimension equals that of P, it is in fact a minimal pseudo-parabolic k-subgroup (by Theorem C.2.5); we will denote this minimal pseudo-parabolic k-subgroup by w ◦ P since it is easily seen to depend only on w and not on a particular choice of n. The action of W (k) on P is then (w, P) → w ◦ P. It readily follows from Proposition 3.5.7 and the fact that NG (S) ∩ P = ZG (S) for any P ∈ P (see the proof of Theorem C.2.8) that W (k) acts freely on P. On the other hand, given P, P ∈ P, by applying Theorems C.2.5 and C.2.3 we can ﬁnd an n ∈ NG (S)(k) such that nPn−1 = P . Thus, under the natural action of W (k) on P described above, the image W of NG (S)(k) in W (k) acts transitively on P. As the action of W (k) on P is free, we conclude that W = W (k); i.e., NG (S)(k) maps onto W (k). 2

C.3 Split unipotent subgroups of pseudo-reductive groups Let G be a smooth connected afﬁne group over a ﬁeld k. If there is a non-central k-split k-torus T in G then there is a non-central 1-parameter k-subgroup λ : GL1 → G. Hence, ZG (λ) is a proper subgroup of G, yet UG (−λ)·ZG (λ)·UG (λ) is open in G (Proposition 2.1.8(3)), so at least one of the subgroups UG (−λ) and UG (λ) is nontrivial. Since both UG (−λ) and UG (λ) are k-split smooth

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connected unipotent k-groups (Proposition 2.1.10), if G contains a non-central GL1 then G contains Ga . In this section we will prove a converse to the preceding conclusion for pseudo-reductive groups: if a pseudo-reductive G contains Ga then it contains a non-central GL1 . In fact, we will go further and show that any such Ga must occur inside of Rus,k (P) for a proper pseudo-parabolic k-subgroup P of G. (Recall from Lemma 2.2.3 that the existence of such a P is equivalent to the existence of a non-central GL1 in G.) We will also prove a unipotent analogue of Theorem C.2.3: all maximal k-split smooth connected unipotent k-subgroups U in a pseudo-reductive k-group G are G(k)-conjugate. This is closely related to the G(k)-conjugacy of minimal pseudo-parabolic k-subgroups P of G (Theorem C.2.5) because we will prove that all such U have the form U = Rus,k (P) for some minimal P. The preceding facts were announced by Borel and Tits without proof. In [Ti3], Tits sketched proofs of some of them. We will provide proofs over a general ground ﬁeld, based on a variant of an argument of Kempf in [Kem]. The key fact to be shown is that if G is pseudo-reductive over k then any nontrivial k-split smooth connected unipotent k-subgroup U ⊆ G is contained in Rus,k (P) for some pseudo-parabolic k-subgroup P of G. (By pseudo-reductivity of G, such a P must be a proper pseudo-parabolic k-subgroup of G.) We now ﬁx a pseudo-reductive group G over a separably closed ﬁeld k. (This restriction on k will be eliminated later by using uniqueness results to carry out Galois descent from ks to k.) We also ﬁx a faithful representation ρ : G → GLn over k, and let Mat n denote the k-algebra scheme of n × n matrices, in which GLn is the open unit group. Endow Mat n with a structure of ﬁnite-dimensional G-module over k via the conjugation action g.x = gxg −1 for g ∈ G and x ∈ Mat n . Finally, deﬁne X∗ (G) := Homk (GL1 , G) to be the set of all 1-parameter k-subgroups λ : GL1 → G. There is a natural left action of G(k) on the set X∗ (G) via conjugation: for g ∈ G(k) and λ ∈ X∗ (G), deﬁne (g.λ)(t) = gλ(t)g −1 . Let U be a nontrivial k-split smooth connected unipotent k-subgroup of G. For each λ ∈ X∗ (G), consider the set Sλ of integers m such that for all u ∈ U (k), the k-scheme map GL1 → Mat n deﬁned by t → t −m (λ(t).(u − 1)) (using the linear structure on Mat n to deﬁne the scaling action by t −m ∈ GL1 ) extends to a k-map Ak1 → Mat n . By considering the weights for λ(GL1 ) acting on Mat n , we see that all sufﬁciently negative m lie in Sλ . Likewise, by choosing u ∈ U (k) − {1} and considering the nontrivial components of u − 1 along some weight spaces for λ(GL1 ) in Mat n we see that Sλ is bounded above. Hence, it is legitimate to deﬁne f (λ) = max Sλ ∈ Z.

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Note that f (λ) depends on U (as well as on the ﬁxed faithful representation ρ of G), but U will be ﬁxed throughout our discussion. Lemma C.3.1 For λ ∈ X∗ (G), let P = PG (λ) be the associated pseudoparabolic k-subgroup of G. For all p ∈ P(k), we have f (p.λ) = f (λ). Proof. Let p ∈ P(k). For all u ∈ U (k) and m ∈ Z, since the G-action on Mat n is k-linear we see that t −m ((pλ(t)p−1 ).(u − 1)) = (pλ(t)p−1 λ(t)−1 ).(t −m (λ(t).(u − 1))). By deﬁnition of P = PG (λ), the k-scheme map GL1 → G deﬁned by t → λ(t)p−1 λ(t)−1 extends to a k-scheme map Ak1 → G, so it is obvious from the above equation that Sλ ⊆ Sp.λ . Since PG (p.λ) = PG (λ) and p−1 .(p.λ) = λ, we conclude that Sλ = Sp.λ . Hence, f (p.λ) = f (λ). 2 Now let S be any maximal k-torus of G. Any character χ of S gives rise to a Z-linear functional X∗ (S) → Z via λ → χ , λ. Deﬁne VS = R ⊗Z X∗ (S). Extending scalars to R deﬁnes an R-linear functional Sχ : VS → R that carries the lattice X∗ (S) into Z via χ, · . Using the left S-action on Mat n via conjugation, consider the S-submodule S.(U − 1) of Mat n generated by the points u − 1 for u ∈ U (k). Deﬁnition C.3.2 Let !S be the set of S-weights occurring in the linear representation space S.(U − 1) for S. For v ∈ VS , deﬁne S (v) = minχ∈!S Sχ (v). Since each Sχ is a linear form on VS , S is a continuous function. It is easy to check that it satisﬁes the following three properties: (i) for λ ∈ X∗ (S), we have f (λ) = S (λ), (ii) S (sv) = sS (v) for any nonnegative real number s, and (iii) S is a concave function; i.e., for all 0 s 1 and v, v ∈ VS , we have S (sv + (1 − s)v ) sS (v) + (1 − s)S (v ). By (iii), for any c ∈ R the set {v ∈ VS | S (v) c} is convex in V S . Its boundary is easily checked to be (S )−1 (c), which in turn is polyhedral since S is a minimum over a set of ﬁnitely many linear forms (but we do not need this fact). Deﬁnition C.3.3 A maximal k-torus S in G is special with respect to the nontrivial k-split smooth connected unipotent k-subgroup U ⊆ G if the set {v ∈ VS | S (v) 1} is non-empty, or equivalently there exists λ ∈ X∗ (S) such that f (λ) = S (λ) > 0. (This deﬁnition depends on the choice of faithful representation ρ of G.) Lemma C.3.4 There exist maximal k-tori in G that are special with respect to U .

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Proof. By Lemma C.2.2, there exists a minimal pseudo-parabolic k-subgroup P of G that contains U . Since k = ks by our hypothesis, the minimality of P forces the pseudo-reductive k-group P/Rus,k (P) to be commutative (Corollary 2.2.5 and Proposition 3.3.15(3)), so it contains no nontrivial smooth connected unipotent k-subgroup. In particular, the image of U in this quotient is trivial, so U ⊆ Rus,k (P). Let S be a maximal k-torus of P, so S is also a maximal ktorus of G. We may and do choose λ ∈ X∗ (S) such that P = PG (λ). Since U ⊆ Rus,k (P) = UG (λ), we have S (λ) = f (λ) 1. Thus, S is special with respect to U . 2 It will be convenient to introduce a Z-valued positive-deﬁnite inner product on X∗ (S) for each maximal k-torus S of G (though only special S will be of interest to us) such that the isomorphism X∗ (S) X∗ (gSg −1 ) deﬁned by gconjugation for any g ∈ G(k) is an isometry. To make such a construction, ﬁx one maximal k-torus T in G and let W = NG (T )(k)/ZG (T )(k) be the associated ﬁnite Weyl group. There is a natural left W -action on X∗ (T ), akin to the left G(k)-action on X∗ (G) via conjugation. Lemma C.3.5 The natural map W \ X∗ (T ) → G(k)\ X∗ (G) is bijective. Proof. In view of the G(k)-conjugacy of maximal k-tori of G (Proposition A.2.10), for any λ ∈ X∗ (G) there is a g ∈ G(k) such that g.λ ∈ X∗ (T ). This proves surjectivity. To prove injectivity, consider g ∈ G(k) and λ ∈ X∗ (T ) such that g.λ also belongs to X∗ (T ). Hence, the smooth connected afﬁne k-group ZG (λ) contains the maximal k-tori g −1 Tg and T , so by ZG (λ)(k)-conjugacy of maximal k-tori of ZG (λ) there exists z ∈ ZG (λ)(k) such that z −1 g −1 Tgz = T . This implies gz ∈ NG (T )(k), so gz represents a class w ∈ W such that g.λ = w.λ. 2 Now ﬁx a W -invariant Z-valued positive-deﬁnite inner product on X∗ (T ). For any maximal k-torus S of G, there exists g ∈ G(k) such that gTg −1 = S. The identiﬁcation of S with T via g-conjugation is independent of g up to the W -action on T . Hence, if we identify X∗ (S) with X∗ (T ) via g-conjugation and thereby transfer the W -invariant positive-deﬁnite inner product on X∗ (T ) over to a Z-valued inner product on X∗ (S) then this latter inner product is independent of g. In this way, for every S we get a Z-valued positive-deﬁnite inner product on the lattice X∗ (S) such that for every g ∈ G(k) the identiﬁcation of X∗ (S) with X∗ (gSg −1 ) via g-conjugation is compatible with the inner product. Applying extension of scalars to R gives an inner product on each VS = R ⊗Z X∗ (S) with an analogous compatibility property. We will denote the norm of v ∈ VS under this inner product by ||v||S . Consider any maximal k-torus S in G that is special with respect to U (Lemma C.3.4). The non-empty convex closed set {v ∈ VS | S (v) 1} contains a point vS with minimal distance to the origin relative to the inner product we have constructed on VS . By convexity, vS is unique. We must have S (vS ) = 1, for if

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S (vS ) > 1 then by the homogeneity of S we could scale vS by some c ∈ (0, 1) such that S (cvS ) 1, contradicting the minimality of the distance from vS to the origin. The points c · vS with c > 0 can obviously be characterized as the nonzero elements of VS where the function v → S (v)/||v||S takes the maximum value. Lemma C.3.6 There exists c > 0 such that c · vS lies in the lattice X∗ (S) in VS . Proof. Since S (vS ) = 1, from the deﬁnition of S we see that the set !S := {χ ∈ !S | Sχ (vS ) = 1} is non-empty. Consider the saturated Z-submodule L = {λ ∈ X∗ (S) | Sχ (λ) = Sχ (λ) for all χ , χ ∈ !S } of X∗ (S). Since each Sχ is Z-valued on X∗ (S), the corresponding R-subspace V = {v ∈ VS | Sχ (v) = Sχ (v) for all χ , χ ∈ !S } of VS is identiﬁed with R ⊗Z L , so L = V ∩ X∗ (S) and L is a lattice in V . By deﬁnition of !S we have vS ∈ V , so V = 0. The functions Sχ for χ ∈ !S restrict to a common linear form φ on V , and this linear form is Z-valued on the lattice L . Since φ(vS ) = 1 by deﬁnition of !S , we see that φ = 0. Hence, H := ker φ is a hyperplane in V . We claim that the line RvS in V through 0 is the orthogonal complement H ⊥ of H (in V ) with respect to the inner product we have constructed on VS at the outset, or in other words that vS is perpendicular to H . Near vS in VS , the linear forms Sχ for χ ∈ !S − !S are all strictly larger than 1, so near vS in V the function φ coincides with S . Hence, the point vS in the half-space {φ 1} is a local minimum for distance to the origin, and so it is the unique global minimum. It follows by elementary Euclidean geometry that vS is perpendicular to the hyperplane H = ker φ parallel to the half-space boundary φ −1 (1), as claimed. Since φ and the inner product on VS = R ⊗Z X∗ (S) respectively arise from a Z-valued linear form on X∗ (S) and a Z-valued inner product on X∗ (S), orthogonal projection from H = ker φ is deﬁned over Q. Hence, the orthogonal complement H ⊥ = RvS meets X∗ (S)Q in a Q-line and so meets X∗ (S) in an inﬁnite cyclic group. 2 By Lemma C.3.6, there is a unique positive multiple λS of vS that lies in X∗ (S) and generates X∗ (S) ∩ RvS . We call it the optimal 1-parameter k-subgroup of S. Of course, λS depends on U . Observe that λS also depends on the Z-valued inner product on X∗ (S) (constructed from a choice of W -invariant Z-valued inner product on X∗ (T ) for an initial choice of maximal k-torus T in G). For λ ∈ X∗ (S) we have S (λ) = f (λ), so the earlier characterization of the point c · vS with c > 0 (as the nonzero elements of VS where the function v →

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S (v)/||v||S takes the maximum value) leads to the following useful alternative description of λS : (∗) λS is the unique non-divisible element of the lattice X∗ (S) where the function λ → f (λ)/||λ||S takes the maximum value. Our interest in working with special maximal k-tori (with respect to U ) is the following remarkable property of their associated optimal 1-parameter k-subgroups: Lemma C.3.7 For each special maximal k-torus S in G with respect to U , consider the pseudo-parabolic k-subgroup PS = PG (λS ) associated to the optimal 1-parameter k-subgroup λS . These k-subgroups all coincide; i.e., they are independent of S. Proof. Let S and S be two such maximal k-tori. Consider the intersection PS ∩ PS . This intersection contains a maximal k-torus S of G (Proposition 3.5.12(1)), and PS (k)-conjugacy of maximal k-tori in PS provides p ∈ PS (k) such that p S p−1 = S. Likewise, there exists p ∈ PS (k) such that p S p−1 = S. Since f (λ ) = f (p .λ ) and ||λ ||S = ||p .λ ||S for all λ ∈ X∗ (S ), by Lemma C.3.1 and the characterization (∗) of optimal 1-parameter subgroups given above, it follows that the maximal k-torus S in G is special and that its optimal 1-parameter k-subgroup λS is equal to p .λS . Hence, as p ∈ PS (k), we conclude that P := PG (λS ) is equal to PS . In a similar manner, λS = p .λS and P = PS . Thus, PS = PS . 2 Now we are in position to prove the ﬁrst main result, dropping the preceding hypothesis that the ground ﬁeld is separably closed. Theorem C.3.8 Let G be a pseudo-reductive group over a ﬁeld k, and let U be a k-split smooth connected unipotent k-subgroup of G. There exists a pseudo-parabolic k-subgroup P such that U ⊆ Rus,k (P). Proof. We may assume that U is nontrivial, so the discussion above is applicable to Uks inside of Gks . Fix a faithful representation ρ : G → GLn over k. Let T be a maximal k-torus in G, so W := W (G, T )(ks ) and X∗ (Tks ) are equipped with natural actions by := Gal(ks /k) factoring through a ﬁnite discrete quotient. Hence, we can construct a W -invariant Z-valued inner product on X∗ (Tks ) that is also -invariant. Consider the natural left action by on the set of closed subschemes Z of Gks by using the k-structure G on Gks . This action is deﬁned via extension of scalars with respect to the natural action of on ks over k via automorphisms: γ Z := ks ⊗γ ,ks Z ⊆ γ ∗ (Gks ) = Gks . For any maximal ks -torus S in Gks and γ ∈ we can also apply such extension of scalars to 1-parameter subgroups of S, viewed as ks -homomorphisms from the ks -group GL1 (where GL1 is equipped with its natural k-structure) to S, and we can do similarly for characters. This identiﬁes X∗ (S) with X∗ (γ S) and identiﬁes X(S) with X(γ S) as Z-modules. We claim

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that this identiﬁcation of cocharacter groups respects the inner products deﬁned by transport from X∗ (Tks ). To check the compatibility with inner products, pick g ∈ G(ks ) such that S = gTks g −1 . The inner product on X∗ (S) is obtained from that on X∗ (Tks ) via g-conjugation. Clearly γ S = γ (g)Tks γ (g)−1 , so the inner product on X∗ (γ S) is obtained from that on X∗ (Tks ) via γ (g)-conjugation. But the composite of g-conjugation X∗ (Tks ) X∗ (S) with γ -twisting X∗ (S) X∗ (γ S) is the same as the composite of the γ -action on X∗ (Tks ) and γ (g)-conjugation X∗ (Tks ) X∗ (γ S). Indeed, for any λ ∈ X∗ (Tks ) the γ -twist of the map ks× → S(ks ) deﬁned by t → gλ(t)g −1 is t → γ (g) · γ ∗ (λ)(t) · γ (g)−1 into γ S(ks ), so we are reduced to checking the -invariance of the inner product chosen on X∗ (Tks ). We arranged this invariance property when constructing the inner product on X∗ (Tks ). In addition to the γ -twisting identiﬁcation of X∗ (S) and X∗ (γ S) respecting inner products for γ ∈ , the action of γ on the set of closed subschemes of (Mat n )ks takes S.(Uks − 1) to γ S.(Uks − 1). Hence, the isomorphism X(S) → X(γ S) obtained via γ -twisting carries !S to !γ S . It follows that the isomorphism VS Vγ S deﬁned by scalar extension to R carries Sχ to γ γ γ S∗ (χ) and S to S , so by deﬁnition we see that S is special if and only if γ S is special, with γ ∗ (vS ) ∈ Vγ S necessarily equal to vγ S . We conclude from the deﬁnitions that when S is special, γ ∗ (λS ) = λγ S for any γ ∈ . Consider the common pseudo-parabolic ks -subgroup Q = PGks (λS ) for all special maximal ks -tori S in Gks (Lemma C.3.7). For any γ ∈ , we have γ ∗ (Q) = PGks (γ ∗ (λS )) = PGks (λγ S ) = Q. Thus, by Galois descent, Q descends to a k-subgroup P in G; this is a pseudo-parabolic k-subgroup by Proposition 3.5.2(1). The containment Uks ⊆ Rus,ks (Q) = Rus,k (P)ks then implies that U is contained in Rus,k (P). 2 Corollary C.3.9 Let G be a smooth connected afﬁne group over a ﬁeld k, and assume that the maximal pseudo-reductive quotient G = G/Ru,k (G) does not contain a non-central k-split k-torus. Then every k-split smooth connected k-subgroup of G is contained in Ru,k (G) (and hence in Rus,k (G)). Proof. The formation of images of smooth connected k-subgroups under the quotient map G G preserves the property of being k-split unipotent, so we can replace G with G so that G is pseudo-reductive. Assume that G contains a nontrivial k-split smooth connected unipotent k-subgroup U . By Theorem C.3.8, G contains a pseudo-parabolic k-subgroup P such that U ⊆ Rus,k (P). This forces P = G (as U = 1), so G contains a proper pseudo-parabolic ksubgroup. Since G is pseudo-reductive over k, it then follows from Lemma 2.2.3 that G contains a non-central k-split k-torus, contrary to the hypotheses. 2

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Remark C.3.10 For a pseudo-reductive k-group G, the condition that it contains no non-central k-split k-torus is equivalent to the condition that D (G) is k-anisotropic (i.e., contains no nontrivial k-split k-torus). This follows from Lemma 1.2.5(iii). Proposition C.3.11 Let G be a smooth connected afﬁne group over a ﬁeld k, and P a minimal pseudo-parabolic k-subgroup of G. Then Rus,k (P) contains every k-split smooth connected unipotent k-subgroup of P. Proof. By Corollary B.3.5, it sufﬁces to prove that Ru,k (P) contains every k-split smooth connected unipotent k-subgroup of P. By deﬁnition of pseudoparabolicity we have Ru,k (G) ⊆ Ru,k (P), so we can replace G with the pseudoreductive group G/Ru,k (G) and replace P with P/Ru,k (G) to reduce to the case that G is pseudo-reductive. Let λ : GL1 → G be a 1-parameter k-subgroup such that P = PG (λ) = ZG (λ) UG (λ). Since UG (λ) = Ru,k (P) = Rus,k (P) (Corollary 2.2.5), we see that ZG (λ) P/Rus,k (P) is pseudo-reductive over k, and by minimality of P in G the k-group ZG (λ) contains no proper pseudo-parabolic k-subgroup (due to Proposition 2.2.10 and Remark 3.5.6). Hence, by Lemma 2.2.3, ZG (λ) does not contain any non-central k-split tori. It then follows from Corollary C.3.9 that ZG (λ) contains no nontrivial k-split smooth connected unipotent k-subgroup. Consider any k-split smooth connected unipotent k-subgroup U of P. The image of U in P/Rus,k (P) is a k-split smooth connected unipotent k-subgroup, but we have just seen that the only such k-subgroup is the trivial one. Hence, U ⊆ Rus,k (P). 2 Theorem C.3.12 Let G be a smooth connected afﬁne group over a ﬁeld k. All maximal k-split unipotent smooth connected k-subgroups (resp. all maximal k-split solvable smooth connected k-subgroups) of G are G(k)-conjugate to each other. Proof. By Lemma C.2.2 and Proposition C.3.11, any maximal k-split unipotent smooth connected k-subgroup U of G has the form U = Rus,k (P) for some minimal pseudo-parabolic k-subgroup P of G. The G(k)-conjugacy of minimal pseudo-parabolic k-subgroups (Theorem C.2.5) then gives the desired G(k)conjugacy of all such U . Now let H be a maximal k-split solvable smooth connected k-subgroup of G. Then by Lemma C.2.2 there is a minimal pseudo-parabolic k-subgroup P which contains H . Since H · Rus,k (P) is a k-split solvable smooth connected k-subgroup, maximality of H implies that it contains Rus,k (P). The structure of k-split solvable smooth connected afﬁne k-groups [Bo2, 15.4(i)] implies that H = S Rus,k (H ) for a k-split k-torus S. According to Proposition C.3.11, Rus,k (P) contains every k-split unipotent smooth connected k-subgroup of P. Hence, Rus,k (H ) ⊆ Rus,k (P), so H = S Rus,k (P).

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Since H is maximal among k-split solvable smooth connected k-subgroups, S is a maximal k-split torus of P. Proposition C.2.4 implies that P = ZG (S) Rus,k (P), and as S is central in ZG (S) we conclude that the k-subgroup H = S Rus,k (P) is the unique maximal k-split solvable smooth connected k-subgroup of P. The G(k)-conjugacy of minimal pseudo-parabolic k-subgroups (Theorem C.2.5) now implies the G(k)-conjugacy of the maximal k-split solvable smooth connected k-subgroups of G. 2

C.4 Beyond the smooth afﬁne case The rational conjugacy of maximal split tori in smooth connected afﬁne groups is valid more generally in all connected groups of ﬁnite type over a ﬁeld: it is not necessary to assume smoothness or afﬁneness. To bypass afﬁneness we will use general structure theorems as recorded in Theorem A.3.7 and Theorem A.3.9. The following lemma is useful for avoiding smoothness hypotheses more generally in situations that involve properties of rational points over separable extensions of the ground ﬁeld, as we illustrate in Example C.4.3. Lemma C.4.1 Let X be a scheme locally of ﬁnite type over a ﬁeld k. There is a unique geometrically reduced closed subscheme X ⊆ X such that X (k ) = X (k ) for all separable extension ﬁelds k /k. The formation of X is functorial in X and commutes with the formation of products over k and separable extension of the ground ﬁeld. In particular, if X is a k-group scheme then X is a smooth k-subgroup scheme. In this lemma we permit non-algebraic separable extensions. This is important for applications with k a global function ﬁeld (over a ﬁnite ﬁeld) and k its completion at some place. Proof. First we check uniqueness. Let ks /k be a separable closure. If X is such a closed subscheme then X (ks ) = X (ks ) is Zariski-dense in X by geometric reducedness (which implies generic smoothness), so the closed subscheme Xks ⊆ Xks has to be the Zariski closure of the subset X (ks ). This settles uniqueness, and for existence consider the Zariski closure Z of X (ks ) in Xks . This is a closed subscheme, and Z is geometrically reduced over ks (as we may see by reducing case of afﬁne X and using injectivity of the natural map to the K ⊗K Vi → (K ⊗K Vi ) for any ﬁeld extension K /K and any indexed collection of K-vector spaces Vi ). By Galois descent, Z descends uniquely to a closed subscheme X ⊆ X that is necessarily geometrically reduced over k. Thus, we just have to check that X (k ) = X (k ) for all separable extensions k /k. If ks /k denotes a separable closure then X (k ) ⊆ X (ks ) is the subset of Gal(ks /k )-ﬁxed points, and likewise for X (k ) ⊆ X (ks ). Thus, if the functorial

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inclusion X (ks ) ⊆ X (ks ) is an equality then the same holds for k -points. We may therefore restrict our attention to the case when k is separably closed. Fixing a k-embedding of ks into k thereby reduces us to the case when k is separably closed (in view of how X was deﬁned via Galois descent). Hence, we now have that X is (by deﬁnition) the Zariski closure X (k) of X (k) in X . Our aim is to prove X (k ) ⊆ X (k)(k ) for all separable extensions k /k (when k is separably closed). Since k is the direct limit of its k-smooth subalgebras, if I is the coherent (radical) ideal sheaf of X (k) in X then it sufﬁces to show that for all geometrically reduced k-algebras A and x ∈ X (A) the x-pullback of I to an ideal in A vanishes. Since A is reduced it sufﬁces to check such vanishing in A/p for a Zariski-dense set of primes p of A. But A is a generically smooth k-algebra and k is separably closed, so such a Zariski-dense set is given by the k-rational maximal ideals m of A. That is, it sufﬁces to treat the case A = k, and this case is a tautology by the deﬁnition of I in terms of X (k). This settles the existence aspect, and functoriality is clear by the construction. Uniqueness and the deﬁning property of X imply that its formation commutes with any separable extension on k. It remains to prove that if X and Y are two k-schemes locally of ﬁnite type then the geometrically reduced closed subscheme X × Y ⊆ X × Y is (X × Y ) . By functoriality with respect to the projections, we have (X × Y ) ⊆ X × Y . Likewise, by functoriality the inclusion X × Y → X × Y factors through (X × Y ) because (X × Y ) = X × Y . Hence, the desired equality holds. 2 Remark C.4.2 When using Lemma C.4.1, it is difﬁcult to control connectedness properties. More speciﬁcally, for any imperfect ﬁeld k with char(k) = p > 0 there exists a connected afﬁne k-group scheme H of ﬁnite type such that the maximal smooth k-subgroup H as in Lemma C.4.1 is disconnected. In fact, p−1 but H = Z/pZ. For example, if we there are H such that (Hk )red Ga consider G as in Example 1.6.3 with [k : k] = p there then we can take H to be the preimage under the Frobenius isogeny FG/k : G → G (p) of the k-unipotent (p) radical Ru,k (G (p) ) (which descends Ru (G ) in such cases). k

Example C.4.3 Let G be a group scheme of ﬁnite type over a global ﬁeld k, and let S be a ﬁnite set of places of k. Let X 1S (k, G) denote the pointed set of isomorphism classes of right G-torsors E for the fppf topology over Spec k that are locally trivial away from S (i.e., admit a kv -point for all places v ∈ S). By using Lemma C.4.1, the study of such torsors can be carried out with a smooth k-group of ﬁnite type in place of G. To explain how this goes, we ﬁrst make some preliminary observations. Any separable extension of k (such as kv /k for any v ∈ S) is a directed union of fraction ﬁelds of smooth k-algebras, so the specialization method as in Proposition 1.1.9(1) shows that such E are automatically torsors for the étale topology on Spec k. In other words, E can be described as a descent over k of the trivial

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GK -torsor GK for some ﬁnite Galois extension K/k. Since G is quasi-projective over k (Proposition A.3.5), this description via Galois descent implies that E is quasi-projective over k. Let G be the maximal smooth closed k-subgroup, as in Lemma C.4.1; this is afﬁne when G is afﬁne, but it may be disconnected even if G is connected. The pointed set X 1S (k, G ) is more accessible than X 1S (k, G) in general because it can be studied in the framework of Galois cohomology (since G is smooth). Thus, it is a useful fact that the natural map X 1S (k, G ) → X 1S (k, G) deﬁned via pushout of torsors along G → G is bijective. (Such a pushout is the quotient of E × G by the equivalence relation (xg , g) ∼ (x, g g) for points g of G . This quotient exists as a trivial GK -torsor for a ﬁnite Galois extension K/k trivializing the G -torsor E , so it exists over k due to the effectivity of Galois descent for quasi-projective schemes over a ﬁeld.) To prove bijectivity of the pushout map X 1S (k, G ) → X 1S (k, G), we now describe an inverse in terms of a functor on torsors. Let E be a right G-torsor over k such that E(kv ) is non-empty for all places v ∈ S. Deﬁne E in terms of E as in Lemma C.4.1 (i.e., it is the k-descent of the Zariski closure of E(ks ) in Eks ). The compatibility of the formation of E with respect to any separable extension of the ground ﬁeld can be applied to the extension k → kv0 for a single v0 ∈ S, so E is a right G -torsor within the right G-torsor E because the analogous claim is obvious over kv0 . Also, E (kv ) = E(kv ) for all v since kv /k is separable, and this is non-empty for v ∈ S. Hence, E represents a class in X 1S (k, G ). The inclusion E → E is equivariant with respect to the inclusion of G into G, so pushout along G → G carries the class [E ] of E in X 1S (k, G ) to the class [E] of E in X 1S (k, G). A map X 1S (k, G) → X 1S (k, G ) in the other direction is induced by [E] → [E ], and we have just shown that it is an inverse to the map X 1S (k, G ) → X 1S (k, G). As an application of Lemma C.4.1 over general ﬁelds, we can analyze conjugacy of maximal split tori in possibly non-smooth groups and prove some other related facts without hypotheses of smoothness or afﬁneness. We ﬁrst check that the notion of “maximal k-torus” interacts well with extension on k in many situations going beyond the familiar smooth afﬁne case. Lemma C.4.4 Let G be a group scheme locally of ﬁnite type over a ﬁeld k, T a maximal k-torus in G, and K/k an extension ﬁeld. If G is smooth or commutative, or if K/k is separable, then TK is a maximal K-torus in GK . Proof. Let G ⊆ G be as in Lemma C.4.1. This is a smooth k-subgroup of G whose formation commutes with any separable extension on k, and it contains every k-torus of G, so the case when K/k is separable is reduced to the case when G is smooth. In both the smooth and commutative cases we can replace G with G 0 so that G is connected and hence of ﬁnite type [SGA3, VIA , 2.4].

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Suppose G is connected and commutative. Any ks -torus in Gks has a Gal(ks /k)-orbit consisting of ﬁnitely many ks -tori, so by commutativity they collectively generate a Gal(ks /k)-stable ks -subtorus. Thus, by Galois descent, Tks is maximal in Gks . In general we can replace K with Ks , so we may assume k and K are separably closed. For any integer n not divisible by char(k), G[n] is ﬁnite étale, hence a constant group. Every K-torus S in GK contains as a Zariski-dense subset the union of the S[n](K) for n not divisible by char(K). But S[n](K) ⊆ G[n](K) = G[n](k) and so each S[n] uniquely descends to a k-subgroup of G. The Zariski-closure of these descents is a smooth k-subgroup H in G such that HK = S, so H is a k-torus. In particular, if S contains TK then H contains T . By maximality of T , this forces S = TK and so settles the commutative case. Finally, assume G is connected and smooth. In this case, the result for afﬁne G is well known; see Corollary A.2.6. In general, by Proposition A.8.10(1) we can replace G with ZG (T )0 so that T is central, and then pass to G/T so that T = 1. In this case we need to show that GK has no nontrivial K-tori. By Theorem A.3.9 there is a short exact sequence 1→Z →G→G→1 with G a smooth connected afﬁne k-group and Z a smooth connected central k-subgroup. Since Z contains no nontrivial k-tori, the settled commutative case shows that this property persists after any ground ﬁeld extension. By replacing G with any maximal k-torus T and replacing G with the smooth connected preimage G of T in G, we may assume that G is a k-torus. With G a k-torus, G is necessarily commutative since its commutator deﬁnes a pairing G × G → Z that is easily checked to be bi-additive and hence trivial (since a biadditive pairing between k-tori valued in any commutative group of ﬁnite type must be valued in a torus and hence vanish, due to the rigidity of tori). Thus, the settled commutative case implies that GK contains no nontrivial K-tori. 2 Proposition C.4.5 Let G be a group scheme locally of ﬁnite type over an arbitrary ﬁeld k. (1) All maximal k-split k-tori in G are G(k)-conjugate. (2) Let f : G H be a surjective homomorphism onto a k-group H locally of ﬁnite type. If f is smooth then it carries maximal k-tori onto maximal k-tori, and likewise for maximal k-split k-tori. Moreover, if T is a maximal k-torus (resp. maximal k-split k-torus) in H then it lifts to one in G provided that f −1 (T )0red is smooth (e.g., f is smooth). Proof. To prove (1), the same argument as at the start of the proof of Lemma C.4.4 allows us to reduce to the case when G is smooth and connected. By

C.4 Beyond the smooth afﬁne case

523

[BLR, 9.2/1], G is an extension of an abelian variety A by a possibly nonsmooth connected afﬁne k-group H . Every k-torus in G is contained in H , so we may replace G with H to reduce to the case when G is afﬁne but possibly not smooth. Now going through the reduction to the smooth connected case again brings us to the smooth connected afﬁne case. This case is Theorem C.2.3. To prove (2), we ﬁrst treat the assertion concerning images of maximal k-tori, so we may assume f is smooth. The case when G is afﬁne (so H is also afﬁne) is Proposition A.2.8. In general, let S be a maximal k-torus in G and let T be a maximal k-torus in H containing f (S). To show T = f (S) we may replace H with T and G with f −1 (T ) so that H = T is a k-torus and G is smooth. In this case we want f (S) = T . We note for later purposes that smoothness of f will no longer be used. Since G is smooth, by Lemma C.4.4 we may assume that k is algebraically closed. If the subtorus f (G 0 ) in T is distinct from T then each connected component of G maps onto a translate of this subtorus and so f would not hit the generic point of T , contrary to the surjectivity hypothesis. Hence, f (G 0 ) = T , so we can replace G with G 0 so that G is smooth and connected. By Theorem A.3.9, there is a central extension 1→Z →G→G→1 with Z a smooth connected commutative k-group and G a smooth connected afﬁne k-group. The natural map G = G/Z → T /f (Z) induced by f between smooth connected afﬁne k-groups is surjective, so by the known case of smooth connected afﬁne groups there is a k-torus S in G containing the image of S such that S maps onto T /f (Z). We may replace G with the preimage of S in G (as this maps onto T under f ), so G is a central extension of the torus S by the smooth connected Z. This forces G to be commutative, as we saw in the proof of Lemma C.4.4 (using rigidity of tori and the commutativity of Z). Thus, the quotient G/S makes sense and the map G/S → T /f (S) surjective. By commutativity of G, the smooth connected quotient G/S contains no nontrivial torus. Hence, by Chevalley’s structure theorem (Theorem A.3.7), G/S is an extension of an abelian variety by a smooth connected unipotent group. In particular, G has no nontrivial torus quotient, so T /f (S) is trivial. That is, T = f (S) as desired. Having settled (2) for the formation of images of maximal k-tori over any k, now consider the image under f of a maximal k-split k-torus S0 in G. The k-split k-torus T0 = f (S0 ) in H lies in a maximal one, say Z0 , and we want T0 = Z0 . We may replace H with Z0 and G with f −1 (Z0 ) to reduce to the case where H is a k-split torus. In this case if S is a maximal k-torus in G containing S0 then f (S) = H , so f (S0 ) = H since H is a k-split torus. Consider the problem of lifting a maximal k-torus T in H to one in G, assuming that G := f −1 (T )0red is smooth. If a maximal k-torus S in G maps

524

Rational conjugacy in connected groups

onto T then any k-torus S in G containing S also maps onto T , so S ⊆ G and hence S = S . That is, such an S would be maximal in G and so would do the job. We can therefore replace G with G so that H = T is a k-torus and G is smooth and connected. The same argument works if we replace “maximal k-torus” with “maximal k-split k-torus” throughout. The preceding argument for images of k-tori may now be applied at the point beyond which smoothness of f was never used, so we get the required lift. 2 Remark C.4.6 The conjugacy in Proposition C.4.5(1) allows us to generalize the surjectivity result in Lemma 3.2.1 as follows. Let f : H → H be a surjective homomorphism between group schemes locally of ﬁnite type over a ﬁeld F, and let T be an F-torus in H , with T := f (T ) its F-torus image in H . Assume that the F-torus (T ∩ ker f )0red is maximal in ker f and remains maximal after scalar extension to F (e.g., (ker fF )0red is unipotent, as in Lemma 3.2.1). In such cases we claim that the map NH (T ) → NH (T ) between scheme-theoretic normalizers is surjective. To prove this we can assume F = F, and then it sufﬁces to prove surjectivity on F-points. By conjugacy of maximal F-tori in ker f (Proposition C.4.5(1) applied to ker f over F = F), the hypothesis on T ∩ ker f implies that T is a maximal torus in f −1 ( f (T )). Consider any h ∈ NH (T )(F), and choose h ∈ H (F) such that f (h) = h . Then hTh−1 is contained in f −1 ( f (T )). By dimension considerations and the conjugacy of maximal F-tori in f −1 ( f (T )) (Proposition C.4.5(1)), it follows that hTh−1 = gTg −1 for some F-point g in f −1 ( f (T )). In other words, h−1 g ∈ NH (T )(F), so h −1 f (g) = f (h−1 g) ∈ NH (T )(F). Replacing h with h · h −1 f (g) = f (g) is therefore harmless. But f (g) = f (t) for some t ∈ T (F), so we are done.

References

[Bo1] [Bo2] [BoSe] [BoSp] [BoTi1] [BoTi2] [BLR] [Bou1] [Bou2] [Bri] [BrTi] [Chev] [Chow]

[Con1] [Con2] [DG] [SGA3]

A. Borel, On afﬁne algebraic homogeneous spaces, Arch. Math. 45(1985), 74–78. A. Borel, Linear algebraic groups (2nd ed.), Springer-Verlag, New York, 1991. A. Borel, J-P. Serre, Théorèmes de ﬁnitude en cohomologie galoisienne, Comm. Math. Helv. 39(1964), 111–164. A. Borel, T. Springer, Rationality properties of linear algebraic groups II, Tohoku Math. Journal (2) 20(1968), 443–497. A. Borel, J. Tits, Groupes réductifs, Publ. Math. IHES 27(1965), 55–151. A. Borel, J. Tits, Théorèmes de structure et de conjugasion pour les groupes algébriques linéaires, C. R. Acad. Sci. Paris 287(1978), 55–57. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Springer-Verlag, New York, 1990. N. Bourbaki, Lie groups and Lie algebras (Ch. 1–3), Springer-Verlag, New York, 1989. N. Bourbaki, Lie groups and Lie algebras (Ch. 4–6), Springer-Verlag, New York, 2002. M. Brion, Anti-afﬁne algebraic groups, Journal of Algebra 321(2009), 934–952. F. Bruhat, J. Tits, Groupes réductifs sur un corps local I, Publ. Math. IHES 41(1972), 5–251. C. Chevalley, Une démonstration d’un théorème sur les groupes algébriques, J. Mathématiques Pures et Appliqées, 39(1960), 307–317. W.-L. Chow, “On the projective embedding of homogeneous varieties” in Algebraic geometry and topology. A symposium in honor of S. Lefschetz, 122– 128, Princeton Univ. Press, Princeton, 1957. B. Conrad, A modern proof of Chevalley’s theorem on algebraic groups, Journal of the Ramanujan Math. Society, 17(2002), 1–18. B. Conrad, Finiteness theorems for algebraic groups over function ﬁelds, in preparation (2010). M. Demazure, P. Gabriel, Groupes algébriques, Masson, Paris, 1970. M. Demazure, A. Grothendieck, Schémas en groupes I, II, III, Lecture Notes in Math. 151, 152, 153, Springer-Verlag, New York (1970).

525

526

[Gre] [EGA] [SGA1] [Hum1] [Hum2] [KMT] [Kem] [Mat] [Mum] [Oes] [Pink] [PY1] [PY2] [RG] [Ri] [Ru] [SS]

[Ser1] [Ser2] [Ser3] [Spr] [Ti0]

[Ti1] [Ti2] [Ti3]

References

M. Greenberg, Rational points in henselian discrete valuation rings, Publ. Math. IHES 31(1966), 59–64. A. Grothendieck, Eléments de Géométrie Algébrique, Publ. Math. IHES 4, 8, 11, 17, 20, 24, 28, 32, 1960–7. A. Grothendieck, Séminaire de géométrie algébrique 1, Paris, 1961. J. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972. J. Humphreys, Linear algebraic groups (2nd ed.), Springer-Verlag, New York, 1987. T. Kambayashi, M. Miyanishi, M. Takeuchi, Unipotent algebraic groups, Lecture Notes in Math. 414, Springer-Verlag, New York (1974). G. Kempf, Instability in invariant theory, Ann. Math. 108(1978), 299–316. H. Matsumura, Commutative ring theory, Cambridge Univ. Press, 1990. D. Mumford, Abelian varieties, Oxford Univ. Press, 1970. J. Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique p, Inv. Math. 78(1984), 13–88. R. Pink, On Weil restriction of reductive groups and a theorem of Prasad, Math. Z. 248(2004), 449–457. G. Prasad, J.-K. Yu, On ﬁnite group actions on reductive groups and buildings, Inv. Math. 147(2002), 545–560. G. Prasad, J.-K. Yu, On quasi-reductive group schemes, Journal of Algebraic Geometry 15(2006), 507–549. M. Raynaud, L. Gruson, Critères de platitude et de projectivité. Techniques de “platiﬁcation” d’un module, Inv. Math. 13(1971), 1–89. R. W. Richardson, Afﬁne coset spaces of reductive algebraic groups, Bull. London Math. Soc. 9(1977), 38–41. P. Russell, Forms of the afﬁne line and its additive group, Paciﬁc Journal of Math. 32(1970), 527–539. C. Sancho de Salas, F. Sancho de Salas, Principal bundles, quasi-abelian varieties, and structure of algebraic groups, Journal of Algebra 322 (2009), 2751–2772. J.-P. Serre, Algebraic groups and class ﬁelds, Springer-Verlag, New York, 1988. J.-P. Serre, Lie groups and Lie algebras, Lecture Notes in Math. 1500, SpringerVerlag, New York, 1992. J-P. Serre, Galois cohomology, Springer-Verlag, New York, 1997. T. A. Springer, Linear algebraic groups (2nd ed.), Birkhäuser, New York, 1998. J.Tits, “Classiﬁcation of algebraic semisimple groups” in Algebraic groups and discontinuous groups, Proc. Symp. Pure Math., vol. 9, AMS, Providence, 1966. J. Tits, Lectures on algebraic groups, Yale Univ., New Haven, 1967. J. Tits, Théorie des groupes, Annuaire du Collège de France, 1991–92. J. Tits, Théorie des groupes, Annuaire du Collège de France, 1992–93.

Index

a∨ , 95 Aut(G), xx Aut G,C representability, 23–25, 79–82 Aut G,C , 23, 79 and exotic pseudo-reductive groups, 269 Aut G ,C and Weil restriction, 348 Aut G,T representability, 82 Aut G,T , 82 Bn± , 301 , 100 CG/k , 155 C[n], xx > , < , 218 D (G), xx D (G), 396 Dk (), 460 Dn , 301 fG , 159 FG/k , FX /S , 396, 404, 405 FX /S,n , 404 > , < , 218 (G, T ), 68 J , 88 λ0 , 112 λ<0 , 128 n , + n , 302 (G, S), 100 nd , 109

Ga , 115, 411 Ga , 392 Gα,n , 313 properties, 314–321 G> , G< , 226 g< , g> , 220 G/H , 395 GJ , 88 (G , k /k, T , C), 29 G M , 464 G pred , 325 G red , G ss , 397 k k Gt , 82, 401 HA (G), 103 (H , H ), 396 HV ,K/k , 286 and Weil restriction, 288 iG , 16 isomorphism criterion, 362–363 a , 291 Lie(G), 51 m(u), 117 μ< , 218 μn , xx NG (V ), 394 !G , 329, 334 ξG , 183 properties in basic exotic case, 261–265 properties in standard case, 183 properties of kernel, 184, 186–190

527

528

ξG (contd.) properties with non-reduced root system, 321–324, 329–331, 333–339 PG (λ), 48 for abstract action, 58 for quotients, 56 (λ), 61 PG qa , 292 qX , 429 ra , ra∨ , 95, 412 R∨ , 95 R(Gk ), 397 Rk (G), 3 R k /k , xiv Ru (Gk ), 397 Ru,k (G), xii, 3 Rus,k (G), 61, 484 X 1S (k, G), xii Symn , 301 t< , 218 2-linear, 292 Ua , 75 UA (G), 103 UG (λ), 49 for abstract action, 58 for quotients, 56 Un± , 301 U(a) , 69 for quotients and subgroups, 70 Usplit , 484 ∗ , 284 VK/k and Weil restriction, 288 W (G, T ), 97 wt(V ), 101 X ∨ , 94 X(M ), 393 n X (p ) , 403 Xred , xix Y H , 464 ZG , 395 ZG,C , 23, 78 and basic exotic pseudo-reductive groups, 270–271 and exotic pseudo-reductive groups, 269

Index

ZG ,C and Weil restriction, 348 ZG (λ), 49 for abstract action, 58 for quotients, 56 ZG (V ), 394

absolutely pseudo-simple, 86 adelic points, 243, 339 arises from a 4-tuple, 29 arithmetic properties basic exotic case, 243–247 non-reduced root system case, 339–341 automorphism functor, 23, 78–85 automorphisms representability results, 78–85 standard case, 23–25, 168–170 basic exotic case, 247–250, 268–271 generalized standard case, 350 totally non-reduced case, 321, 329, 334, 343–348 automorphisms standard case, 25

basic exotic pseudo-reductive group, 231–240 ZG,C , 270–271 arithmetic properties, 243–247 automorphisms, 268–271 center, 236 isomorphisms, 247–250 maximal geometric reductive quotient, 238 non-standardness, 255 pseudo-parabolic subgroup, 381–387 root groups, 236 root system, 238 separable ground ﬁeld extension, 238 simplicity, 257 Weil restriction, 256 basic non-reduced pseudo-simple group, 343 birational group law, 297–299

Index

Bruhat decomposition smooth connected afﬁne groups, 508 split pseudo-reductive groups, 119

Cartan subgroup and Galois cohomology, 119 and quotient groups, 400 Lie algebra, 399 pseudo-reductive group, xv, 13 rational expression in root groups, 163 reductivity criterion, 358 standard pseudo-reductive group, 175 Weil restriction, 438–441 cckp-kernel, 482 center (scheme-theoretic), 395 central extension, 163, 182, 256, 321, 328, 334, 346 central quotient, 78 central subgroup, 395 centralizer (scheme-theoretic), 394 character group, 393 character lattice, 97 Chevalley’s theorem, 406–407 closed set of roots, 63, 109 cocharacter lattice, 97 commutator subgroup, 396 conjugacy for maximal split tori, 506 for maximal split unipotent subgroups, 518 for minimal pseudo-parabolic subgroups, 507 coroot lattice, 97 coroots, 95

dense in ﬁbers, 298 derived group, xx, 396 in pseudo-reductive group, 15 direct spanning, 99, 107–110 of Ru,k (P), 111 divisible root, 63

étale morphism, xx exceptional pseudo-reductive group, 210

529

and basic exotic pseudo-reductive groups, 261–265 Existence Theorem, 412–414 exotic pseudo-reductive group, 265 Aut G,C and ZG,C , 274 automorphism scheme, 269 characterization of invariants, 257–260 direct factor of pseudo-reductive group, 275–277 isomorphisms, 247–250, 266–268 parameterization of maximal tori, 359–361 root system, 265 separable ground ﬁeld extension, 271–274 simple factors, 257 faithfully ﬂat descent, xx, 182 ﬁeld of deﬁnition for closed subgroup, 452 for closed subscheme, 7 for linear subspace, 7, 451 separable base change, 8 Frobenius morphism, 396, 403–405 functorial criterion for smoothness, xx functorial linear representation, 461 Galois cohomology and Cartan subgroup, 119 commutative pseudo-reductive case, 371 pseudo-semisimple case, 372 generalized standard construction, xvi, 347–349 generalized standard presentation, 349, 351 pseudo-parabolic subgroup, 376–379 root group, 374 generated by tori, 82 geometric radical, 397 geometric unipotent radical, 397 Greenberg functor, 442 height 1, 456

530

Index

imaginary part, 300 Isogeny Theorem, 418–419 Isomorphism Theorem, 412–416 isomorphisms criterion for iG , 362–363 exotic pseudo-reductive group, 247–250, 266, 268 generalized standard pseudo-reductive group, 353 standard pseudo-reductive group, 172–174

kernel (scheme-theoretic), 394 k-radical, 3 k-split unipotent radical, 61 k-subgroup, 394 k-unipotent radical, xii, 3 normal subgroups and torus centralizers, 417 k-wound, 480

Levi subgroup, 115, 121, 126, 250–254, 398, 441–446 Weil restriction, 441 Lie algebra, 51, 447–450 and Weil restriction, 450 invariants under linearly reductive group, 463–467 p-Lie algebra structure, 455 linear homomorphism, 392 linear structure, 75, 392 uniqueness, 71 linearly reductive, 463 action on reductive group, 468–470 scheme of invariants, 463–467

maximal central torus, 14, 399 maximal tori, 392 and ground ﬁeld extension, 399 and normal subgroups, 400 and quotient groups, 400 conjugacy, 401 exotic pseudo-reductive groups, 359–361

generalized standard pseudo-reductive groups, 352 in non-afﬁne groups, 521 standard pseudo-reductive groups, 152 Weil restriction, 438–441 multipliable root, 63 multiplicative type, 393 complete reducibility, 461 equivalence with Galois lattices, 457 split, 460 non-divisible root, 63 non-multipliable root, 63 normal subgroup, 394 generalized standard pseudo-reductive group, 366–368 pseudo-reductive group, 15, 89 standard pseudo-reductive group, 177 normalizer (scheme-theoretic), 394 open Bruhat cell, 114, 316 parabolic set of roots, 63–64 perfect group, 397 pinning, 420–422 p-Lie algebra, 455–456 p-morphism, 409 positive system of roots, 98 p-polynomial, 474 principal part, 475 pseudo-complete, 494–498 quotient by pseudo-parabolic subgroup, 498 pseudo-parabolic subgroup, 59 and basic exotic pseudo-reductive group, 381–387 and basic non-reduced pseudo-reductive group, 379–381 and quasi-parabolic subgroup, 503 classiﬁcation, 130–146 completeness for cosets, 127 conjugacy, 114, 143 conjugacy in minimal case, 507 connectedness, 139 containing a ﬁxed torus, 133 in reductive groups, 64

Index

intersection, 143 normalizer, 136 parameterization via generalized standard presentation, 376–379 parameterization via G red , 142 k parameterization via roots, 130 Weyl group, 140 pseudo-reductive group, xiii, 3 basic exotic, 231–240 basic non-reduced, 343 Bruhat decomposition, 119 Cartan subgroup, xv, 13 central quotient, 31 commutative, xv, 13–14, 18–19, 39–40, 368–372 derived subgroup, 15 exceptional, 209, 210 exceptional and basic exotic, 261–265 exotic, 265 exotic direct factor, 275–277 generalized standard, xvi, 349, 351–354 Levi subgroup, 121–126 non-reduced root system, 291–297, 314–317, 321–324 non-reduced root system construction, 302–313 non-unirational, 368–371 normal subgroup, 15, 89 perfect normal subgroup, 91 pseudo-simple, xviii pseudo-split, 43 reductive group, 126 root datum, 96 standard, xv, 25–29 standardness criterion, 186–190 totally non-reduced, 342 unirationality, 372 Weyl group, 97, 118 pseudo-reductivity derived subgroup, 15 for Cartan subgroup, 13 ground ﬁeld extension, 6 invariants by linearly reductive group, 470

531

p-torsion, 39 quotients, 6, 41 solvability, 13 Weil restriction, 10 pseudo-semisimple, 364–368 pseudo-semisimplicity, 22 pseudo-simple group, xviii, 86 pseudo-split pair, 67 pseudo-reductive group, 43 smooth connected afﬁne group, 67 quasi-parabolic subgroup, 501 and pseudo-parabolic subgroup, 503 quotient comparison with classical case, 395 comparison with quotient by p-Lie subalgebra, 457 maximal adjoint semisimple, 397 maximal adjoint semisimple, exactness properties, 417 maximal geometric reductive/semisimple, 397 maximal simple adjoint semisimple, 397 maximal simple semisimple, 397 universal maximal absolutely simple adjoint, 155 quotient (scheme-theoretic), 395 real part, 300 reductive group, 397 reﬂection, 95 root datum, 96 adjoint, 97 semisimple, 97 simply connected, 97 root group, 75, 77 and generalized standard presentation, 374 of basic exotic pseudo-reductive group, 236 of central quotient, 78 of pseudo-split pseudo-reductive group, 373 short, long, 218

532

Index

root lattice, 97 root system basic exotic pseudo-reductive group, 238 dual, 219, 224 exotic pseudo-reductive group, 265 for Weil restriction, 68 generalized standard pseudo-reductive group, 354 invariants by linearly reductive group, 471 inverse, 219 of central quotient, 78 standard pseudo-reductive group, 381 roots, 94

semigroups, 100 associated subgroups, 102–107 semisimple group, 397 separable ﬁeld extension, xix polynomial, 474 simply connected central cover, 420 simply connected datum, 159 smooth morphism, xx solvable pseudo-reductive groups, 13 special isogeny, 223 split central extension, 163, 182, 256, 321, 328, 334, 346 conjugacy for maximal split tori, 506 conjugacy for maximal split unipotent subgroups, 518 reductive groups, 407–422 reductive pair, 407 smooth connected solvable group, 392, 473 standard construction, 25–29 for semisimple group, 35 standard presentation, 151 compatibility with a maximal torus, 153 relation with Cartan subgroup, 175 relation with maximal torus, 172

standard pseudo-reductive group, xv, 25 and Cartan subgroup, 175 central quotient, 31 derived subgroup, 170 invariance under separable extension, 174, 189 normal subgroup, 177 pseudo-simple normal subgroup, 179 root system, 381 standardness criterion, 186–190 in characteristic 2, 290 in characteristic 3, 278 strict birational group law, 299 totally non-reduced pseudo-reductive group, 342 arithmetic properties, 339–341 general properties, 343–347 2-linear, 292 type of an exotic pseudo-reductive group, 265 unipotent group, 393 maximal split normal subgroup, 484 maximal split, conjugacy, 518 torus action, 485–493 wound, 480 universal maximal absolutely simple adjoint quotient, 155 vector group, 392 separable ground ﬁeld extension, 481 torus action and linear structure, 491 very special isogeny, 222 very special quotient, 222 weight lattice, 97 Weil restriction, xiv, 4–5, 422–437 automorphism scheme, 23–25, 348 basic exotic pseudo-reductive group, 256 centralizer, 438–441 ﬁeld of deﬁnition, 36 for simply connected groups, 437

Index ∗ , 288 HV ,K/k and VK/k isogenies, 17 Levi subgroup, 441 Lie algebra, 450 maximal tori, 438–441 perfectness criterion, 453 pseudo-reductivity, 10 quasi-projectivity, 432 root system, 68, 98 simply connected group, 454 standard pseudo-reductive group, 31

533

Weil’s theorem on birational group laws, 299 Weyl group ﬁniteness, 401 for maximal geometric reductive quotient, 97 for pseudo-parabolic subgroup, 140 for pseudo-split pseudo-reductive group, 118 for smooth connected afﬁne group, 97 wound group, 480–485 torus action, 493