Representations of Algebraic Groups

Representations of Algebraic Groups

This is volume 131 in PURE AND APPLIED MATHEMATICS H. Bass, A. Borel, J. Moser, and S.-T. Yau, editors Paul A. Smith and Samuel Eilenberg, founding editors A list of titles in this series appears at the end of this volume.

Representations of Algebraic Groups Jens Carsten Jantzen Mathematisches Seminar der Universitat Hamburg Federal Republic of Germany

ACADEMIC PRESS, INC. Harcourt Brace Jooanovich, Publishers

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Copyright 01987 by Academic Press, Inc. All rights reserved. N o part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Academic Press, Inc. Orlando, Florida 32887 United Kingdom Edition published b y ACADEMIC PRESS INC. (LONDON) LTD. 24-28 Oval Road, London N W l 7DX

Library of Congress Cataloging-in-Publication Data Jantzen, Jens Carsten. Representations of algebraic groups.

(Pure and applied mathematics; v. 131) Bibliography: p. Includes index. 1. Representations of groups. 2. Linear algebraic groups. I. Title. 11. Series: Pure and applied mathematics (Academic Press); 131. QA3.P8 510 s C512.21 86-26619 LQA1711 ISBN 0-12-380245-8 (alk. paper)

87 88 89 90 9 8 7 6 5 4 3 2 1 Printed in the United States of America

Contents Introduction

vii

Part I: General Theory 1: Schemes 2: Group Schemes and Representations 3: Induction and Injective Modules 4: Cohomology 5 : Quotients and Associated Sheaves 6: Factor Groups 7: Algebras of Distributions 8: Representations of Finite Algebraic Groups 9: Representations of Frobenius Kernels 10: Reduction mod p

1 3 21 43 55 73 95 109 129 145 161

Part 11: Representationsof Reductive Groups 1: Reductive Groups 2: Simple G-Modules 3: Irreducible Representations of the Frobenius Kernels 4: Kempf’s Vanishing Theorem 5 : The Borel-Bott-Weil Theorem and Weyl’s Character Formula 6: The Linkage Principle 7: The Translation Functors 8: Filtrations of Weyl Modules 9: Representations of G, T and G, B 10: Geometric Reductivity and Other Applications of the Steinberg Modules 11: Injective G,-Modules

171 173 197 213 227 243 259 28 1 297 319 337 351 V

vi

Contents

12: Cohomology of the Frobenius Ke'rnels 13: Schubert Schemes 14: Line Bundles on Schubert Schemes

369 38 1 395

References

417

Index

435

Introduction

I This book is meant to give its reader an introduction to the representation theory of such groups as the general linear groups GL,(k), the special linear groups SL,(k), the special orthogonal groups SO,(k), and the symplectic groups Sp,,(k) over an algebraically closed field k. These groups are algebraic groups, and we shall look only at representations G + G L ( V ) that are homomorphisms of algebraic groups. So any G-module (vector space with a representation of G) will be a space over the same ground field k. Many different techniques have been introduced into the theory, especially during the last fifteen years. Therefore, it is necessary (in my opinion) to start with a general introduction to the representation theory of algebraic group schemes. This is the aim of Part I of this book, whereas Part I1 then deals with the representations of reductive groups. I1 The book begins with an introduction to schemes (Chapter 11) and to (affine) group schemes and their representations (Chapter 12). We adopt the “functorial” point of view for schemes. For example, the group scheme SL, over Z is the functor mapping each commutative ring A to the group SL,(A). Almost everything about these matters can also be found in the first two chapters of [DG]. I have tried to enable the reader to understand the basic definitions and constructions independently of [DG]. However, I refer to

vii

viii

Introduction

[DG] for some results that I feel the reader might be inclined to accept without going through the proof. Let me add that the reader (Df part I) is supposed to have a reasonably good knowledge of varieties and algebraic groups. For example, he should know [Bo] up to Chapter 111, or the first seventeen chapters of [HuZ], or the first six ones of [Sp2]. (There are additional prerequisites for Part I1 mentioned below.) In Chapter 13, induction functors are defined in the context of group schemes, their elementary properties are proved, and they are used in order to construct injective modules and injective resolutions. These in turn are applied in Chapter 14 to the construction of derived functors, especially to that of the Hochschild cohomology groups and of the derived functors of induction. In contrast to the situation for finite groups, the induction from a subgroup scheme H to the whole group scheme G is not exact, only left exact. The values of the derived functors of induction can also be interpreted (and are so in Chapter 15) as cohomology groups of certain associated bundles on the quotient G/H (at least for algebraic schemes over a field). Before doing that, we have to understand the construction of the quotient G/H. The situation gets simpler and has some additional features if His normal in G. This is discussed in Chapter 16. One can associate to any group scheme G an (associative) algebra Dist(G) of distributions on G (called the hyperalgebra of G by some authors). When working over a field of characteristic 0, it is just the universal enveloping algebra of the Lie algebra Lie(G) of G. In general, it reflects the properties of G much better than Lie(G) does. This is described in Chapter 17. A group scheme G (say over a field) is called finite if the algebra of regular functions on G is finite dimensional. For such G the representation theory is equivalent to that of a certain finite dimensional algebra and has additional features (Chapter 18). For us, the most important cases of finite group schemes arise as Frobenius kernels (Chapter 19) of algebraic groups over an algebraically closed field k of characteristic p # 0. For example, for G = GL,(k) the map F : G -,G sending any matrix (a,) to (a;) is a Frobenius endomorphism. The kernel of F' (in the sense of group schemes) is the rth Frobenius kernel G, of G. The representation theory of G , (for any G) is equivalent to that of Lie(G) regarded as a p-Lie algebra. In order to apply our rather extensive knowledge of the representation theory of groups like SL,(C) to that of SL,(k), where k is a field of prime characteristic, one uses the group scheme SL, over Z. One chooses SL,stable lattices in SL,(C)-modules and tensors with k in order to get SL,(k)-modules. Some general properties of this procedure are proved in Chapter 110.

Introduction

ix

From Part I, the contents of Chapters 1 (until 1.6), 2, 3, 4 (until 4.18), 5 (mainly 5.8-5.13), and 6 (until 6.9) are fundamental for everything to follow. The other sections are used less often. In Part 11, the reader is supposed to know the structure theory of reductive algebraic groups (over an algebraically closed field) as to be found in [Bo], [ H u ~ ] ,[Sp2]. The reader is invited (in Chapter 111) to believe that there is for all possible root data a (unique) group scheme over Z that yields for each field k (by extension of the base ring) a split reductive algebraic group defined over k having the prescribed root data. Furthermore, he has to accept that all “standard” constructions (like root subgroups, parabolic subgroups, etc.) can be carried out over Z. (The sceptical reader should turn to [SGA 31 for proofs.) I have included a proof (following Takeuchi) of the uniqueness of an algebraic group with a given root datum (over an algebraically closed field) that does not use case-by-case considerations. 111 Let me describe a selection of the contents of the remaining chapters in more detail. Assume from now on (in this introduction) that k is an algebraically closed field and that G is a (connected) reductive algebraic group over k with a Bore1 subgroup B c G and a maximal torus T c B. Let X ( T )be the group of characters of T. In case char(k) = 0 the representation theory of G is well understood. Each G-module is semi-simple. The simple G-modules are classified (as in the case of compact Lie groups or of complex semi-simple Lie algebras) by their highest weights. Furthermore, one has a character formula for these simple modules. In fact, Weyl’s formula for the compact groups holds when interpreted in the right way. (For us, the character of a finite dimensional G-module will always be the family of the dimensions of its weight spaces with respect to T. As the semi-simple elements in G are dense in G and as each semi-simple element is conjugate to one in T, the character determines the trace of any g E G on the Gmodule.) The situation in prime characteristic is much worse. Except for the case of a torus, there are non-semi-simple G-modules. Except for a few low rank cases, we do not know a character formula for the simple modules, and Weyl’s formula will certainly not carry over. Only one property survives: The simple modules are still classified by their highest weights, and the possible highest weights are the “dominant” weights in X ( T ) . (The notion of dominant depends on the choice of an ordering of X ( T ) .We shall always work with an ordering for which the weights of T in Lie(B) are negative.) This classification is due to Chevalley, cf. [SC]. Let L(A) denote the simple module with the highest weight A.

Introduction

X

The difference of the situations in zero and in prime characteristic can be illustrated already in the case G = SL,(k). Let H ( n ) be the nth symmetric power of the natural representation of G on k 2 . If char&) = 0, then H(n) = L(n) for all n E N. (For SL, we can identify X ( T ) N Z in such a way that the dominant weights correspond to N.)If char(k) = p # 0, obviously not all H ( n ) can be simple: For all r, n E N, > 0 the map f H f p r maps H ( n ) to a proper submodule of H(p'n), so H(p'n) is not simple. It is not too difficult to show for any n that H ( n ) contains L(n) as its unique simple submodule, and that H ( n ) = L(n) if and only if n = up' - 1 for some a, r E N with 0 < a I p . So for all other n the module H ( n ) is not semi-simple. For arbitrary G one gets L(A) as the unique simple submodule of an induced module Ho(L): One extends 1, E X ( T ) to a one dimensional representation of B such that the unipotent radical of B acts trivially. Then Ho(A) is the G-module induced by this B-module. It is nonzero if and only if A is dominant. (In the case G = SL,(k) the H o ( L ) are just the H ( n ) from above.) This is the main content of Chapter 112. The case G = SL,(k) with char&) = p # 0 can serve to illustrate other general results also. For any vector space I/ over k let be the vector space that is equal to I/ as an additive group and where any a E k acts as ap-' does on I/. Then the map f H f p r is linear when regarded as a map H(n)(')--t H ( p r n ) , hence a homomorphism of G-modules. It is not difficult to show: If n= aipi with 0 I ai < p for all i, thenf, 0 f i 0 . .$ 0f , H is an isomorphism

XI=,

nr=,fpi

H(a,) 0 H(a1)(')0 . . . 0 H(a,)(') 2 L(n). This result was generalized in [Steinberg 21 to all G: A suitable p-adic expansion of the highest weight 1 leads to a decomposition of L(L) into a tensor product of the form L(1,) 0 L(A1)(')0 . . . 8 L(Ar)(r).This tensor product theorem reduces the problem of calculating the characters of all simple G-modules to a finite problem (for each G). Steinberg's proof relied on a theorem from [Curtis 13 on the representations of Lie(G).In the special case of G = SL,(k), it can be proved in a quite elementary way. It says then: Each L(n) with n < p remains simple for Lie(G), and each simple module of Lie(G) regarded as a p-Lie algebra is isomorphic to exactly one L(n)with n < p . More generally, each L(n) with n < p' is simple for the rth Frobenius kernel of SL,(k), and we get thus each simple module for this infinitesimal group scheme. This result again has an extension to all G and then leads to a rather simple proof of Steinberg's tensor product theorem, discovered by Cline, Parshall, and Scott. (All this is done in Chapter 113.)

Introduction

xi

The choice of the notation H o ( i ) for the induced module has been influenced by the fact that Ho(%)is the zeroth cohomology group of a line bundle on G/B associated to %.Let Hi(%)denote the ithcohomology group (for not only for dominant ones). We could have constructed Hi(%) any 1 E X(T), also by applying the i derived functor of induction from B to G to the onedimensional B-module defined by A. Another result from characteristic zero that does not carry over to prime characteristic is the Borel-Bott-Weil theorem. It describes explicitly all H i ( p ) with i E N and p E X ( T ) :For each p there is at most one i with H i ( p ) # 0, and this H i ( p )can then be identified with a specific L ( i ) .We know already that we cannot expect the simplicity of the H i ( p )in prime characteristic. But, even worse, there can be more than one i for a given p with H i @ ) # 0, and the character of H i ( p ) will depend on the field. (This was first discovered by Mumford.) It is crucial for the representation theory that one special case of the Borel-Bott-Weil theorem holds for any k: If % is dominant, then Hi(%) = 0 for all i > 0. This is Kempf’s vanishing theorem from [Kempf 11. The proof here in Chapter 114 is due to Haboush and Andersen (independently). In Chapter 115, we give Demazure’s proof of the Borel-Bott-Weil theorem in case char&) = 0. Furthermore, we prove (following Donkin) that Weyl’s character formula yields the alternating sum (over i) of the characters of all Hi(%). Assume from now on char(k) = p # 0. Kempf’s vanishing theorem implies that one can construct for any k the Ho(%)with A dominant by starting with the similar object over C,taking a suitable lattice stable under a Z-form of G, and then tensoring with k. To construct representations in this way has the advantage that one can carry out specific computations more easily. Several examples computed especially by Braden then led Verma in the late sixties to several conjectures (cf. [Verma]) that had a great influence on the further development of the theory. One conjecture is the linkage principle (Chapter 116): If L ( p ) is a composition factor of Ho(A) (or, more generally, if L ( p ) and L(A) are both composition factors of a given indecomposable Gmodule), then p E W,.%. Here W, is the group generated by the Weyl group W and by all translations by pa with a a root. The dot is to indicate a shift in the operation by p, the half sum of the positive roots (i.e., w.A = w(A + p ) - p). For large p this principle was proved in [Humphreys 13. The result was then extended by several people to almost all cases, but a general proof appeared only in 1980 (in [Andersen 41). It relies on an analysis of the failure of Demazure’s proof (of the Borel-Bott-Weil theorem) in prime characteristic. Another conjecture of Verma was a special case of the translation p belong to the same principle (Chapter 117): If two dominant weights i,

xii

Introduction

"facet" with respect to the affine reflection group W,, then the multiplicity of any L(w.l) with w E W, as a composition factor of Ho(A) should be equal to that of L ( w . p ) in H o ( p ) . This was proved (modulo the linkage principle) in [Jantzen 21. The approach to the Ho(A) via representations over Z also has the advantage that it allows the construction of a certain filtration (Chapter 118)of Ho(A).One can compute the sum of the characters of the terms in the filtration ([Jantzen 31 for large p , [Andersen 121 in general) and use this information to get information about composition factors. For example, for G = SL,(k) or for G of type G2 this leads to a computation of the characters of all simple modules. If I and 1+ pv for some v E X(T)are weights that are "small" with respect to p 2 and are "sufficiently dominant" (confer 11.9.14-9.15 for a more precise condition), then one gets the composition factors of H o ( I + pv) from those of Ho(l)by adding p v to the highest weights. This was proved first in [Jantzen 4) using involved computations. Later on it was realized that it follows rather easily if one develops the representation theory of the group scheme G,T. For 1 as above experimental evidence (cf. [Humphreys lo]) indicated that the H'(w.1) with w E W satisfy a weak version of the Borel-Bott-Weil theorem ( W ( w . 2 ) # 0 for at most one i). This was then proved in [Cline, Parshall, and Scott lo] using the representation theory of the group scheme G,B. All this is described in Chapter 119. Let us assume that G is semi-simple and simply connected. There is for each r 2 1 a unique simple G-module that is simple and injective for G,. It is called the rth Steinberg module and was first discovered by Steinberg within the representation theory of finite Chevalley groups. We do not look at its great importance there but discuss some applications to the representation theory of G (Chapter 1110). It plays a crucial role in Haboush's proof that G is geometrically reductive. One may wonder whether any injective G,-module can be extended to a G-module. For large p this was proved by Ballard. We discuss this (with some applications to the representation theory of G) in Chapter 1111. One can write down the character of a simple G-module L(A) if one knows all extension groups Ext",(L(A), Ho(p)),cf. 11.6.21. Unfortunately, rather little is known about these groups. There has been a considerable amount of work (especially by Cline, Parshall, and Scott) to understand better especially Hochschild cohomology groups H"(G, M ) 1: Ext",k, M ) . One has H " ( G , M )N l&H"(G,,M) if dimM c 00, so one may hope to get information on G-cohomology from information on G,-cohomology. Here the most remarkable theorem known is due to Friedlander and

Introduction

xiii

Parshall: For large p the cohomology ring H’(G,, k ) is isomorphic to the ring of regular functions on the nilpotent cone in Lie(G). This result can be found in Chapter 1112. The orbits of B on G / B are isomorphic to affine spaces. They are called Bruhat cells, their closures are called Schubert varieties. For example, G / B itself is a Schubert variety. One can extend Kempf’s vanishing theorem to any Schubert variety Y c G/B:If one restricts to Y the line bundle on G / B corresponding to a dominant weight A, then all higher cohomology groups vanish. As an application one can prove the normality of Y and a character formula for the space of global sections. These results have been proved only recently (1984) using old ideas of Kempf and Demazure by Mehta, Ramanathan, Seshadri, Ramanan, and Andersen. One can find this in Chapter 1114, whereas I1 13 provides the necessary background on the Schubert varieties. The last seven chapters of Part I1 can be divided into three groups (118, 119-12, 1113-14), which are independent of each other. Also, the logical interdependence of chapters 1110-12 is rather weak.

IV Suppose that Fq is a finite field and that k is an algebraically closed extension of Fq.The representation theory of groups like GL,(k) or Sp,,(k) has always been developed in close interaction with that of groups like GL,(Fq) or Sp,,(F,). It would therefore have been desirable to have a third part of the book dealing with representations of finite Chevalley groups (say over fields of the same characteristic as that over which they are defined). In fact, I promised to write such a part in a preliminary foreword to a preprint version of Part I. However, I hope to be forgiven if I break this promise, as otherwise the book would have grown to an unreasonable size. Furthermore, I feel that people most interested in these finite groups would prefer another book where they would not have to devour at first all of Parts I and 11. V In the summer of 1984, I gave a series of lectures on some topics discussed in this book at the East China Normal University in Shanghai. I had been asked in advance to provide the audience with some notes. When doing so I was still undecided about the precise contents of my lectures. I therefore included more material than I could possibly cover in my lectures. This book has grown out of those notes. I should like to use this opportunity to thank the mathematicians I met in Shanghai, especially Professor Cao Xihua, for their hospitality during my stay and for the patience with which they listened to my lectures. Thanks are also due to Henning Haahr Andersen, Burkhard Haastert, and Jim Humphreys for useful comments on my manuscript and for providing lists of misprints.

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Part 1

General Theory

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1

Schemes

It is the purpose of this first chapter to give the necessary introduction to schemes following the functorial approach of [DG]. This approach appears to be most suitable when dealing with group schemes later on. After trying to motivate the definitions in 1.1, we discuss affine schemes in 1.2-1.6. What is done there is fundamental for the understanding of everything to follow. As far as arbitrary schemes are concerned, it is most of the time enough to know that they are functors with some properties so that all affine schemes are functors and so that over an algebraically closed field any variety gives rise to a scheme in a canonical way. Sometimes, e.g., when dealing with quotients, it is useful to know more. So we give the appropriate definitions in 1.7-1.9 and mention the comparison with other approaches to schemes and with varieties in 1.11. The elementary discussion of a base change in 1.10 is again necessary for many parts later on. There is also a discussion of closed subfunctors and of closures (1.12-1.14). Finally, we describe the functor of morphisms between two functors (1.15)and prove some of its properties. Again, these results are used only in few places. A ring or an associative algebra will always be assumed to have a 1, and homomorphisms are assumed to respect this 1. Let k be a fixed commutative ring. Notations of linear algebra (like Hom, 0 )without special reference to a ground ring always refer to structures as k-modules. A k-algebra is always

3

4

Representations of Algebraic Groups

assumed to be commutative and associative. (For noncommutative algebras we shall use the terminology: algebras over k.) 1.1 Before giving the definitions, I want to point out how functors arise naturally in algebraic geometry. Assume for the moment that k is an algebraically closed field. Consider a Zariski closed subset X of some k" and denote by I the ideal of all polynomials f E k[T,, T,, . . . , T,,] with f(X)= 0. Instead of looking at the zeroes of I only over k, we can look also at the zeroes over any k-algebra A, i.e., at %(A) = {x E A" ( f ( x ) = 0 for all f E I } . The map A H % ( A ) from {k-algebras} to {sets} is a functor: Any homomorphism cp: A -+ A' of k-algebras induces a map cp": A" -+ (A')", (a,, a , , . . * a,) H (cp(a,), cp(a,), ' . * with f(cp"X)) = cp(f(x)) for all x E A" and f E k[T,,. . . , T,,]. Therefore cp" maps %(A) to %(A'). Denote its restriction by %(cp): % ( A ) -+ %(A'). For another homomorphism cp': A' -+ A" of k-algebras, one has obviously %(cp')%(cp) = %(cp' o cp), proving that % is indeed a functor. A regular map from X to a Zariski closed subset Y of some k" is given by rn polynomials f 1 , f 2 , . . .,f, E k[T,, T,, . . ., T,] as f : X Y, x H ( f , ( x ) , f 2 ( x ) ,. . .,fm(x)). The fi define for each k-algebra A a map f(A):A" + A", x H ( f , ( x ) , . .. , f m ( x ) ) .The comorphism f ( k ) * :k [ T , , . .., T,] -+ k[Tl,. . . ,T,] maps the ideal defining Y into the ideal I defining X . This implies that any f(A) maps %(A) into Y(A). The family of all f(A) defines a morphism f : % - + % of functors, i.e., a natural transformation. The more general discussion in 1.3 (cf. 1.3(2)) shows that the map f H f is bijective (from {regular maps X --+ Y } to {natural transformations % -+ Y}). Taking this for granted, we have embedded the category of all affine algebraic varieties over k into the category of all functors from {k-algebras}to {sets}as a full subcategory. This embedding can be extended to the category of all algebraic varieties, see 1.11. One advantage of working with functors instead of varieties (i.e., of working with % instead of X ) will be that it gives a natural way to work with "varieties" over other fields and over rings. Furthermore, we get new objects over k (algebraically closed) in a natural way. Instead of working with I , we might have taken any ideal I' c k[T,, . . .,T,,] defining X , i.e., with X = {x E k" I f ( x )= O for all f~ 1') or, equivalently by Hilbert's Nullstellensatz, with f i = I. Replacing I by I' in the definition of X,we get a functor, say X’, with %'(A) = %(A) for each field extension A 2 k (or even each integral domain), but with %'(A) # %(A) for some A if I # 1'. Such functors arise in a 7

9

-+

5

Schemes

natural way even when we deal with varieties, and they play an important role in representation theory. Before giving the proper definitions, let us describe the functor % without using the embedding of X into k”. For each k-algebra A , we have a bijection . . . ,a(T,)). Homk-,,,(k[Tl, T,,. . . ,T,], A ) --f A“, sending any a to (a(T1),a(T2), The inverse image of %(A) consists of those a with 0 = f(a(T1),. . .,a(T,,)) = a ( f ) for all f E I, hence can be identified with Horn,.,,,(k[ Tl, T,, . . . , T,,]/Z, A ) . As k[Tl, T,, . . .,T,,]/Z is the algebra k[X] of regular functions on X, we have thus a bijection .%-(A) N Homk.,,,(k[X], A ) . If cp: A + A’ is a homomorphism of k-algebras, then %(q) corresponds to the map Homk~,l,(k[X],A ) + Hom,,,,(k[X], A ’ ) with u H cp a. A morphism f: X + Y is given by its comorphism f*:k [ Y ] + k[X]. Then the morphism p: 3+ g is given by / ( A ) : Hom,_,,,(k[X], A ) Hom,_,,,(k[ Y ] ,A ) , ct H ct f* for any k-algebra A . 0

0

--f

1.2 (&-functors)Let us assume k to be arbitrary again. In the definitions to

follow, we shall be rather careless about the foundations of mathematics. Instead of working with ‘‘all’’ k-algebras, we should (as in [DG]) take only those in some universe. We leave the appropriate modifications to the interested reader. A k-functor is a functor from the category of k-algebras to the category of sets. Let X be a k-functor. A subfunctor of X is a k-functor Y with Y ( A )c X(A) ( ~ )all k-algebras A, A’ and all cp E Homk-,&i, A’). and Y(cp) = X ( V ) / ~ for Obviously, a map Y that associates to each k-algebra A a subset Y ( A )c X(A) is a subfunctor if and only if X(cp)Y(A)c Y(A’) for each homomorphism cp: A + A’ of k-algebras. For any family ( Q oof r subfunctors of X, we define their intersection 5 through (fliery ) ( A )= q ( A )for each k-algebra A. This is again a subfunctor. The obvious definition of a union is not the useful one, so we shall not denote it by UiElY;:. For any two k-functors X,X’, we denote by Mor(X,X’) the set of all morphisms (i.e., natural transformations) from X to X . For any f E Mor(X,X’) and any subfunctor Y’ of X’, we define the inverse image f - ‘ ( Y ) of Y under f through f - ’ ( Y ) ( A ) = f ( A ) - ’ ( Y ’ ( A ) )for each kalgebra A . Clearly f - ’ ( Y ’ ) is a subfunctor of X. (The obvious definition of an image of a subfunctor is not the useful one.) Obviously, f- commutes with intersections. For two k-functors X,, X,, the direct product XI x X, is defined through (X, x X,)(A) = X,(A) x X,(A) for all A. The projections pi:X, x X, 4 Xi

niEr

niEI

6

Representations of Algebraic Groups

are morphisms and ( X , x X,,p,,p,) has the usual universal property of a direct product. For three k-functors X , , X,, S and two morphisms f,:X , + S, f, : X 2 + S , the jibre product X , x s X , (relative to f,, f2) is defined through

The projections from X , x s X , to X , and X , are morphisms, and X , x s X,, together with these projections, has the usual universal property of a fibre product. Of course, we may also regard X , x s X 2 as the inverse image of the diagonal subfunctor D, c S x S (with Ds(A) = {(s,s) I s E S ( A ) }for all A ) under the (obvious) morphism ( f 1 , f 2 ) : X ,x X , -+ S x S. (On the other hand, inverse images and intersections can also be regarded as special cases of fibre products.) 1.3 (Affine Schemes) For any n E N,the functor A" with A"(A) = A" for all A and A"(q)= q " : ( a , , . . .,an)H ( q ( a l ).,. . ,q(a,)) for all cp: A + A' is called the afJine n-space over k. (We also sometimes use the notation A; when it may be doubtful which k we consider.) Note that A' is the functor with Ao(A) = (0) for all A . Hence there is for each k-functor X exactly one morphism from X to Ao (i-e., Ao is a final object in the category of k-functors), and we can regard any direct product as a fibre product over A'. For any k-algebra R, we can define a k-functor Sp,R through (Sp,R)(A) = HOmk-alg(R, A ) for all A and (Sp,R)(q): HOmk.alg(R,A ) + HOmk-alg(R,A'), a H q 0 a for all homomorphisms q : A + A'. We call Sp,R the spectrum of R. Any k-functor isomorphic to some Sp,R is called an afJine scheme over k. (Note that the Sp,R generalize the functors 9"considered in 1.1.) For example, the affine n-space A" is isomorphic to Sp,k[T,, . . . , T,] (and will usually be identified with it), where k[T,, ..., T,] is the polynomial ring over k in n variables T,, ...,T,. We can recover R from Sp,R. This follows from:

Yoneda's Lemma: For any k-algebra R and any k-functor X , the map

f H f(R)(id,) is a bijection

Indeed, take any k-algebra A and any a E Homk-alg(R, A ) = (Sp,R)(A). As f is a natural transformation, we have X ( a ) 0 f(R) = f(A)0 (Sp,R)(a). Let us ab-

7

Schemes

breviate xs = f(R)(id,). As (SpkR)(a)(idR)= a 0 id,

= a, we

get

f(A)(a) = X ( a ) ( x s ) .

(1)

This shows that f is uniquely determined by x f and indicates how to construct an inverse map. For each x E X ( R ) and any k-algebra A , let f,(A): ( S p k R ) + X ( A ) be the map Z H X(r)(x). Then one may check that f , ~ Mor(SpkR,X ) and that x H f, is inverse to f H x f . An immediate consequence of Yoneda’s lemma is MOr(SpkR,SpkR’) 1 HOmk-a,g(R’,R )

(2)

for any k-algebras R , R ‘ . We denote this bijection by f ~ f and * call f * the comorphism corresponding to f. As Homk-,,,(k[T1], R ) r R under a H a( T,) we get especially

(3)

Mor(SpkR,A ’ ) 3 R .

For any k-functor X , we denote M o r ( X , A ’ ) by k [ X ] . This set has a natural structure as a k-algebra and (3) is an isomorphism k [ S p k R ] 7 R of k-algebras. (For f i , f 2 E k [ X ] , define f, f2 through (f, f , ) ( A ) ( x ) = f , ( A ) ( x ) f , ( A ) ( x ) for all A and all x E X ( A ) . Similarly, f1f2 and af, for a E k are defined.) We shall usually write f ( x ) = f ( A ) ( x ) for x E X ( A ) and f E k [ X ] . Note that for X = SpkR and f E R N k [ X ] we have f ( x ) = x ( f ) for X E (SPkR)(A)= HOmk-alg(R,A ) . The universal property of the tensor product implies immediately that a direct product X, x X , of affine schemes over k is again an affine scheme over k with k [ X , x X , ] N k [ X , ] 0 k [ X , ] . More generally, a fibre product X , x s X , with X , , X , , S affine schemes is an affine scheme with

+

+

+

1.4 (Closed Subfunctors of AffineSchemes) Let X be an affine scheme over k.

For any subset I c k [ X ] , we define a subfunctor V ( I )of X through (1)

V ( I ) ( A )= {x E X ( A )I f ( x ) = 0 N

for all f

{ a E Homk-,,,(k[X],A)Ia(I) =

E

I}

o}

for all A . (One can check immediately that this is indeed a subfunctor, i.e., that X ( ( p ) V ( I ) ( Ac ) V ( I ) ( A ’ for ) any homomorphism q : A + A’.) Of course, V(1)depends only on the ideal generated by I in k [ X ] . We claim: (2) The map I H V ( I )from {ideals in k [ X ] } to {subfunctors of X } is injective.

8

Representations of Algebraic Groups

More precisely, we claim for two ideals I , I ’ of k [ X ] : I c I ’ 0 V ( I ) 3 V(Z’). (3) Of course, the direction “+-” is trivial. O n the other hand, consider the canonical map a : k [ X ] + k [ X ] / I ’ , which we regard as an element of X ( k [ X ] / I ’ ) .As a(Z’) = 0, it belongs to V(Z’)(k[X]/Z’).If V(Z’) c V ( I ) , then a E V ( Z ) ( k [ X ] / I ’and ) a ( I ) = 0, hence I c I’. We call a subfunctor Y of X closed if it is of the form Y = V ( I )for some ideal I c k [ X ] . Obviously, any closed subfunctor is again an affine scheme over k as

(4) V ( I ) = SPk(kCXI/I). For any family ( I j ) j E of ideals in k [ X ] , one checks easily

Thus the intersection of closed subfunctors is closed again. For each subfunctor Y of X , there is a smallest closed subfunctor F of X with Y ( A ) c P ( A ) for all A . (Take the intersection of all closed subfunctors with the last property.) This subfunctor 7 is called the closure of Y. We really do not have to assume here that Y is a subfunctor: Any map Y will do that associates to each A a subset Y ( A ) c X ( A ) .We can, for example, fix an A and consider a subset M c X ( A ) . Then the closure M of M is the smallest closed subfunctor of X with M c M ( A ) . Let I , , I , be ideals in k [ X ] . Because of (3), the closure of the subfunctor A H V ( I , ) ( A )u V ( I , ) ( A )is equal to V ( I , n I,). If A is an integral domain, then one checks easily that V ( I , ) ( A )u V(Z,)(A)= V(I1 n I,)(A). For arbitrary A, this equality can be false. Still, we dejne the union as V ( I , )u V ( I , ) = VUl n 121. Let f:X ’ --t X be a morphism of affine schemes over k. One easily checks for any ideal I of k [ X ] that

(6)

f-’V(I) = V(k[X’]f*(I)).

Thus the inverse image of a closed subfunctor is again a closed subfunctor. For ) any ideal I ’ c k [ X ’ ] , the closure of the subfunctor A H f ( A ) ( V ( I ‘ ) ( A )is V ( ( f * ) - ’ I ’ ) . This functor is also denoted as f(V(Z’)), but we do not want to define f (V(I’))here. For two affine schemes X , , X , over k and ideals I , c k [ X , ] , I , c k [ X , ] , one checks easily (7)

V(I1) x V(I2) N V(I1 0 k [ X , l

+k[Xil@

9

Schemes

If S is another affine scheme and if morphisms X, one gets

(8)

v(zl) x S v ( z 2 )

21 V(zl

@k[S]k[x21

-, S,

X,

--+

S are fixed, then

+ k[xll @k[S]I 2 ) *

(Use, e.g., that V ( I , ) xs V(Z2)= p i ’ V(Il) n p i ’ V(I,) together with (5), (6), where pi:XI xs X, -,Xi for i = 1,2 are the canonical projections.)

1.5 (Open Subfunctors of Affine Schemes) Let X be an affine scheme over k. A subfunctor Y of X is called open if there is a subset I c k[X] with Y = D ( I ) where we set for all k-algebras A : (1)

D ( I ) ( A )= {x E X(A) I

1Af(x)= A )

ier

= { a E HOmk-,,,(k[X],A)I A a ( l ) = A }

Note that (1) defines for each ideal I a subfunctor: For each cp E Homk.alg(A, A ’ ) and each x E D(I)(A), one has A’f(X(cp)x) = A’cp(f(x)) = A ’ q ( x , , , Af(x)) = A’cp(A) = A’. Obviously:

xfEr

ErBr

(2) If A is a j e l d , then D ( I ) ( A )= U f B r ( xE X(A) (f(x) # 0). Of course, the right hand side in (2) would be the obvious choice for something open. But it does not define a subfunctor, as homomorphisms between kalgebras are not injective in general. Therefore, we have to take (1) as the appropriate generalization to rings. For I of the form I = {f} for some f E k[X], one writes X, = D ( f ) = D({f}) and gets (3)

Xf(A) = {.

Homk-alg(k[X1, A ) 1 @ ( f ) E A x 1,

hence

(4)

xf

Spk(k[XIf)

where k[X], = k[X][f-’] is the localization of k[X] at f. So the open subfunctors of the form X, are again affine schemes. For arbitrary I, however, D ( I ) may be no longer an affine scheme. Obviously, D ( I ) depends only on the ideal of k[X] generated by I . As any proper ideal in any ring is contained in a maximal ideal, we have for any A D ( I ) ( A )= { a E Homk-,,,(k[X], A ) I a ( ] ) $ m

I

for any M E Max(A)}

= { a E Homk-,,,(k[X], A ) a, E D ( I ) ( A / ~ )for any

rn E Max(A)}

where Max(A) is the set of all maximal ideals of A and a, is the composed map k[X] 4 A % A/rn. This shows that D ( I ) is uniquely determined by its

10

Representations of Algebraic Groups

values over fields and especially that D ( I ) = D ( J I ) for any ideal I c k [ X ] . Denote for each prime ideal P c k [ X ] the quotient field of k [ X ] / P by QPand the canonical homomorphism k [ X ] + k [ X ] / P + Q Pby a p . Then up 4 D ( I ) ( Q p )0 a p ( I )= 0 0 P

As &is the intersection of all prime ideals P ideals I , I‘ of k [ X ]

I> I

3

I.

of k [ X ] , we get for any two

D ( I ) c D(I’) o 8 c f i

(5)

Thus I H D ( I ) is a bijection (ideals I of k[X] with I = J I } subfunctors of X } . For two ideals I , I ’ in k [ X ] , one checks easily (6)

+ {open

.

D ( I ) n D(I’) = D(I n 1 ’ ) = D ( I 1 ’ )

and gets especially for any f,f ’E k [ X ] X, n X,. = X,,,.

(7)

For any ideal I in k [ X ] one has (8) If A is a field, then X ( A ) is the disjoint union of D ( I ) ( A )and V ( I ) ( A ) , For arbitrary A , the union may be smaller. Also, the next statement may be false for arbitrary A : Consider a family ( I j ) j e J of ideals in k [ X ] . Then obviously (9) If A is a field, then

ujEJ D ( I j ) ( A )= D ( c j E JIj)(A).

For any morphism f:X ’ + X of affine schemes over k, one has (10)

f-’D(I) = D(k[X’]f*(I))

for any ideal I c k [ X ] , as one may check easily. We get especially for any

f ’E “ 1 (11)

f - ’ ( x , . ) = X&.

For any fibre product X , xs X , of affine schemes over k (with respect to suitable morphisms) and any ideals I, c k[X,], I, c k [ X , ] , one has

(12)

DV,) xs W

Z )

= D(I1 Ok(S1IZ).

(Argue as for 1.4(8).) 1.6 (Affine Varieties and Affine Schemes) An affine scheme X is called algebraic if k [ X ] is isomorphic to a k-algebra of the form k[T,, . . . , % ] / I for some n E N

Schemes

11

and a finitely generated ideal I in the polynomial ring k[Tl,. . . , T I .It is called reduced if k [ X ] does not contain any nilpotent element other than 0. Assume until the end of Section 1.6 that k is an algebraically closed field. Any affine variety X over k defines as in 1.1 a k-functor 3 which we may identify with S p , [ X ] . One gets in this way exactly all reduced algebraic affine schemes over k. For two affine varieties X , X ‘ , one has M o r ( X , X ‘ )= Hom,-,,,(k[X’], k [ X ] ) ‘v Mor(X, 3’). So we have indeed embedded the category of affine varieties as a full subcategory into the category of affine schemes. When doing this, one has to be aware of several points. Any closed subset Y of an affine variety X is itself an affine variety. The functor CiY is the closed subfunctor V ( I )c 3,where I = { f E k [ X ] I f(Y ) = O}. In this way one gets an embedding {closed subsets of X } + {closed subfunctors of X}. On the level of ideals (cf. 1.4(2)), it corresponds to the inclusion {ideals I of k [ X ] with I = f i }c {ideals of k [ X ] } . The embedding is certainly compatible with inclusions (i.e., Y c Y’ c> Y c CiY ), but in general not with intersections: It may happen that Y n Y’ is strictly larger than the functor corresponding to Y n Y’. Take for example in X = k 2 (where k [ X ] = k[Tl, T,]) the line Y = {(a,O) 1 a E k } and the parabola Y’ = {(a,a’) I a E k } . Then Y n Y’ = {(O,O)]. The ideals I,I of Y , Y’ are I = (T,) and I‘ = ( T t - T,), hence I + I’ = (Tf, T,) # (TI, T2) and Y n CiY = V ( I )n V(I’)= V(I + 1’)differs from the subfunctor V(Tl, T2)corresponding to Y n Y’. So, when regarding affine varieties as (special) affine schemes, we have to be careful, whether intersections are taken as varieties or as schemes. The same is true for inverse images and (more generally) for fibre products. Similar problems do not arise with open subsets. To any open Y c X we can associate the ideal I = { f k~[ X ] I f ( X - Y ) = 0) and then the open subfunctor D ( I ) , which we denote by Y. Because of 1.5(5), the map Y H Y is a bijection from {open subsets of X > to {open subfunctors of z } that is compatible with intersections. It follows from 1.5(10), (12) that this bijection is also compatible with inverse images and fibre products. (In case Y is affine, the notation C?l is compatible with the earlier one.)

1.7 (Open Subfunctors) (Let k again be arbitrary.) Let X be a k-functor. A subfunctor Y c X is called open if for any affine scheme X ‘ over k and any morphism f:X ’ + X there is an ideal I c k[X ] with f -‘(Y) = D ( I ) . Note that this definition is compatible with the one at the beginning of 1.5 because of 1.5(10). From 1.5(6) one gets

(1) I f Y, Y’ are open subfunctors of X , then so is Y n Y’.

12

Representations of Algebraic Groups

Let f:X’

+X

be a morphism of k-functors. Then one has, obviously,

(2) If Y is an open subfunctor of X , then f

-I(

Y ) is an open subfunctor of X’.

Let X , , X,, S be k-functors and suppose X , x s X , is defined with respect to some morphisms. Then one gets (using Y, xs Y, = p;’(Y,) n p i 1 (Y,)) (3) If Y, c X , and Y, c X, are open subfunctors, then Y, xs Y2 is an open subfunctor of X , xs X , .

Let Y, Y’ be open subfunctors of X . Then

(4) Y

Y o Y ( A ) = Y ’ ( A )for each k-algebra A that is a jield. (Of course ‘‘3” is trivial. In order to show e suppose , Y # Y . Then there is some k-algebra A with Y ( A )# Y’(A).Assume that there is x E Y ( A ) with x # Y’(A). Via Y ( A ) N Mor(SpkA,Y ) c Mor(SpkA,X), regard x as a morphism SpkA + X . Then idA E x - ’ ( A ) ( A ) , # x - ’ ( Y ) ( A ) , hence x - ’ ( Y ) # x-’(Y‘). Now the result follows from the discussion preceding 1.5(5).) A family ( y j ) j E of open subfunctors of X is called an open covering of X , if X(A)= ? ( A ) for each k-algebra A which is a jield. If X is affine and if 5 = D ( I j )for some ideal Ij c k [ X ] , then formula 1.5(9) implies that the D ( I j )form an open covering of X if and only if k [ X ] = Ij. A comparison with the case of an affine variety shows that this is the appropriate generalization of the notion of an open covering. Note that especially =

uje

cj,

( 5 ) Let X be afine and f i , f 2 ,...,f,E k [ X ] . Then the X f i form an open covering of X if and only if k [ X ] = k[Xlfi.

ci=,

Let Y c X be an open subfunctor, and let cp: A + A‘ be a homomorphism of k-algebras. Then

( 6 ) If A’ is a faithfully flat A-module via cp, then Y ( A )= X ( q ) - ’ Y ( A ’ ) . We have to prove only ‘‘3”. Suppose at first that X is affine. Then Y = D(Z) for some ideal Z c k [ X ] . Consider some a E X ( A ) = Hom,,,,,(k[X],A) with cp 0 a = X(cp)(.) E Y(A’), i.e., with A’ = A’cp(a(I)). The isomorphism A OAA’ % A’, a 0 a’ H cp(a)a’ induces an isomorphism Aa(I)0, A’ r A’cp(a(1)).Therefore A’ = A’cp(a(1))together with the flatness of A‘ implies ( A / A a ( I ) )0, A’ = 0, hence Aa(I) = A by the faithful flatness. For arbitrary X , we regard x E X ( A ) as a morphism x: SpkA + X with x(A)(id,) = x , hence with X(cp)x = x(A’)Sp,(cp)(id,). So, if x E X(cp)-’ Y(A’), then id, E Spk(cp)-’(x-’(Y)(A‘)),hence id, E x - ’ ( Y ) ( A ) ,as x - ’ ( Y ) is an open subfunctor of the affine scheme Spk(A).Now x = x(A)(id,) E Y ( A )as desired.

13

Schemes

Of course, ( 6 )implies that we can restrict to algebraically closed fields in (4). Also, a family ( I.;)je of open subfunctors of X is an open covering of X if and only if X ( A ) = U j , ? ( A ) for all k-algebras A that are algebraically closed fields. 1.8 (Local Functors) As the notion of an affine scheme generalizes the notion of an affine variety, we want to define the notion of a scheme generalizing the notion of a variety. Certainly a scheme should (by analogy) be a k-functor admitting an open covering by affine schemes. This is, however, not enough. Consider two k-functors X , Y and an open covering ( I.;)j, of Y. If X , Y correspond to geometric objects, then a morphism f: Y + X ought to be determined by its restrictions jirJ to all I.;. Furthermore, given for each j a morphism fj: I.; -+ X such that filrJ r,, = I y, y J , for all j, j’ E J , then there ought to be a (unique) morphism f: Y -+ X with fir, = fi for all j. In other words, the sequence

ought to be exact where a( f ) = (fir,)jcJ and p((fj)j,J) resp. ~ ( ( f j ) ~has ,~) (j,j’)-component fil YJnY,. resp. f j ’ , Y , n Y , . . For arbitrary X , Y,(I.;),the sequence (1) will not be exact. So we define a kfunctor X to be local if the sequence (1) is exact for all k-functors Y and all open coverings ( I.;)jc (One can express this as saying that the functor Mor(?,X ) is a sheaf in a suitable sense.) For any k-algebra R and any fl,. . . ,f, E R with RJ;.= R, the Sp,(Rf,) form an open covering of the affine scheme SpkR.In this case the sequence (1) takes (because of Yoneda’s lemma) the form J .

where the maps have components of the form X ( a )with a one of the canonical maps R + Rji or Rj, + R f i f j Now . one can prove (cf. [DG], I, 51, 4.13). Proposition: A k-functor X is local if and only if for any k-algebra R and any RJ;.= R the sequence (2) is exact.

f l ,. . . ,f , E R with

(Note that in [DG] the second property is taken as the definition of “local”.) For R and f l,. . . ,f, as in (2) the sequence (3)

14

Representations of Algebraic Groups

(induced by the natural maps R --+ R f i and R f i + R f i f j is ) exact. (This is really the description of the structural sheaf on Spec R, e.g., in [Ha], II,2.2.) For an affine scheme X over k the exactness property of Hom,,,,(k[X],?) = X ( ? ) shows that the exactness of (3) implies the exactness of (2). Thus we get

nl=, Ai)r n;=,

(4) Any ajine scheme over k is a local k-functor.

nl=,

Consider k-algebras A , , A , , . . , ,A , and the projections p j : Ai + A j . If we apply (2) to R = A iand the f i = (0,. . . ,0,1,0,. . . ,O),then we get

( 5 ) If X is a local functor, then X ( n l = , algebras A , , A , , . . . , A , .

X(Ai) for all k-

(The bijection maps any x to ( X ( p i ) x ) l s i s , , . ) 1.9 (Schemes) A k-functor is called a scheme (over k) if it is local and if it admits an open covering by affine schemes. Obviously, 1.8(4)implies (1) Any ajine scheme over k is a scheme over k.

The category of schemes over k (a full subcategory of {k-functors})is closed under important operations:

( 2 ) If X is a local k-functor (resp. a scheme over k) and i f X‘ is an open subfunctor of X, then X ‘ is local (resp. a scheme). In the situation of 1.8(1), the injectivity of tl for X implies its injectivity for X ’ . In order to show the exactness for X ’ , one has to show then for any f~ M o r ( Y , X ) such that each f i Y j factors through X ’ , that also f factors through X ’ . The assumption implies r j c f - ’ ( X ’ ) for each j ’ , hence by the definition of an open covering that f - ’ ( X ’ ) ( A )= Y ( A )for each k-algebra A that is a field. Then 1.7(4) implies Y c f - ’ ( X ’ ) and f factors through X ’ . In order to get the affine covering of X ’ in case X is a scheme, one can restrict to the case where X is affine, hence X ’ = D ( I )for some ideal. Then the ( X f ) f E I form an open affine covering. Let X , , X , , S be k-functors and form X , x s X , with respect to suitable morphisms. Then:

(3) If X , , X,,S are local (resp. schemes), then so is X , xs X,, The proof may be left to the reader. The most important non-affine schemes are the projective spaces and, more generally, the Grassmann schemes gr,,for each r, n E N. For any k-algebra A , one sets gren(A) equal to the set of direct summands of the A-module A ‘ + ,

15

Schemes

having rank r. (Then P = 3’,,”is the projective space of dimension n.) In [DG], 1.1.3.9/13, there is a proof that all gr,”are schemes. 1.10 (Base Change) Let k‘ be a k-algebra. Any k’-algebra A is in a natural way also a k-algebra, just by combining the structural homomorphisms k -,k‘ and k’ A . We can therefore associate to each k-functor X a k’-functor Xk! by Xk.(A)= X(A) for any k’-algebra A. For any morphism f:X -, X’ of kfunctors, we get a morphism fk‘: Xk, -+ Xi, of k‘-functors simply byfk.(A)= f(A) for any k’-algebra A . In this way we get a functor X H &,, f H fk, from {k-functors} to {k’-functors}, which we call base change from k to k‘. For any subfunctor Y of a k-functor X, the k’-functor G. is a subfunctor of xkf.Furthermore, the base change commutes with taking inverse images under morphisms, with taking intersections of subfunctors, and with forming fibre products. The universal property of the tensor product implies that (SpkR),. = Spk.(R0 k’) for any k-algebra R. In other words, if X is an affine scheme over k, then XkT is an affine scheme over k‘ with k’[Xk.] N k[X] 0 k‘. For any ideal I of k[X], one gets then V(& = V ( I 0 k’) and D(I)kr = D(I 0 k’). ( w e really ought to replace 1 0 k‘ in these formulas by its canonical image in k[X] 0 k‘, but for once we shall indulge in some abuse of notation.) For any k’-algebras A, R one has

(Spk,R)(A)= Homk,-,,g(R,A) C HOmk-alg(R,A ) = (SPkR)k‘(A). Thus we have embedded SpkfR as a subfunctor into (SpkR),,. For any ideal I of R, denote the corresponding subfunctors as in 1.4/5 by V ( I ) ,D ( I ) c SpkRand l$(I), Dkf(I)c Spk,R.Then one sees immediately Dkf(I)= (Sp,.R) n D(I)k,and h,(I)= (Spk.R) V(I)k,. Using the last results, it is easy to show for any open subfunctor Y of a kfunctor X that G. is open in Xk,.If X is a local k-functor, then obviously Xk. is a local k’-functor. Now it is easy to show that Xk! is a scheme over k‘ if X is one over k. Let k, be a subring of k. We say that a k-functor X is dejinedouer k, if there is a fixed k,-functor X, with X = (X1)k. 1.11 (“Schemes”) In text books on algebraic geometry (like that by Hartshorne to which I shall usually refer in such matters) another notion of scheme is introduced that I shall denote by “schemes” in case a distinction is useful. A “scheme” is a topological space together with a sheaf of k-algebras and an open covering by “affine schemes”. The “affine schemes” are the prime spectra Spec(R) of the k-algebras R endowed with the Zariski topology and a sheaf having sections R, on each Spec(R,) c Spec(R) for all f E R. To each

16

Representations of Algebraic Groups

such “scheme” X one can associate a k-functor 95 via %(A) = Mor(Spec A , X ) for all A . O n the other hand, one can associate in a functorial way to each k-functor X a topological space (XI together with a sheaf such that ISp,R( = Spec(R) for each k-algebra R. It turns out that 1x1is a “scheme” if and only if X is a scheme and that X H 1x1and X’ H %’ are quasi-inverse equivalences of categories. (This is the content of the comparison theorem [DG], I, §1,4.4.) One property of this construction is that the open subfunctors of any kfunctor X correspond bijectively to the open subsets of 1x1, cf. [DG], I, 41, 4.12. More precisely, if Y is an open subfunctor of X, then I Y I can be identified with an open subset of and the k-algebra of sections in I Y I of the structural sheaf of 1x1is isomorphic to Mor(Y,A’), ibid. 4. 14/15. Suppose that k is an algebraically closed field. Consider a scheme X over k that has an open covering by algebraic affine schemes. We can define on X(k) a topology such that the open subsets are the Y(k) for open subfunctors Y c X. The map Y H Y(k) turns out to be injective ([DG], I, §3,6.8). We can define a sheaf ox(,) on X(k) through ox(,)( Y(k)) = Mor( Y, A’), Then X H (X(k), Ox,,,) is a faithful functor and its image contains all varieties over kin the usual sense. There are some fundamental notions of algebraic geometry (like smoothness and dimension) that we shall have to consider only in a few places. The necessary definitions and the main properties from the point of view of kfunctors are contained in [DG]. I do not want to repeat what is done there in order to keep the length of this book down. Any reader who is familiar with these notions in the context of “schemes” (e.g., from [Ha]) can use the correspondence of X and 1x1as above to translate. For example, a scheme Xis smooth if and only if 1x1is so.

1x1

1.12 (Closed Subfunctors) A subfunctor Y of a k-functor X is called closed if and only if for each affine functor X’ and any morphism f:X’ X of kfunctors the subfunctor f -‘(Y) of X’ is closed in the old sense (as in 1.4). Because of 1.4(6), this is compatible with the old definition in case X is affine. The following statements are clear from 1.4(5) or by definition: --f

nie,

(1) If ( y i ) i e , is a family of closed subfunctors of a k-functor X, then yi is closed in X. (2) Let fi X X ‘ be a morphism of k-functors. If Y’ c X‘ is a closed subfunctor, then f -‘(Y’) c X is closed. (3) Let X , , X,, S be k-functors with fixed morphisms X, -+ S and X, -+ S . If Y, c X, and Y, c X2 are closed subfunctors, then Y, xs Y2 c X, xs X, is closed.

17

Schemes

Because of (l), we can define the closure of any subfunctor Y of X as the intersection of all closed subfunctors containing Y. In order to get some deeper results we need (4) Let X be an afJine scheme and ( X j ) j , J an open covering of X . If Y, Y' are local subfunctors of X with Y n Xj= Y' n X j for all j E J , then Y = Y'. If X j = D(lj) for some ideal 'j c k [ X ] , then cjsJlj = k [ X ] , cf. 1.7. We can choose a finite subset J, c J and & E lj for all j E Jo such that k [ X ] = J o k [ X ] f , . Then the D(4.) c 5.with j E J, form also an open covering of X (refining the original one). We have also Y n D(&) = Y ' n D(&)for all j E J,, so we may as well assume that J = {I,2,. . . ,r> and X j = X f j for some f j E k [ X ] with k [ X ] = k[X]fj. Consider now x E X ( A ) = Hom,-,,,(k[X], A ) for some k-algebra A ; set f;= x(h) E A and x i E X ( A f ; )corresponding to the composed homomork [ X ] L = k [ X ] implies A = phism k [ X ] A A 2 A J i . Now A f i , so the local property of Y and Y' yields

cis

Xi=,

x E Y ( A )0 xi E Y ( A f ; )

for all

i

o x i E ( Y n Xi)(AfJ

for all i

o xi E

(Y' n X i ) ( A f i )

for all i

ox E

Y'(A).

In the affine case, any closed subfunctor is again an affine scheme, cf. 1.4(4), hence local, so we can apply (4) to it.

(5) Let be an open covering of some k-functor X . If Y c X is a closed subfunctor with Y 3 Xjfor all j , then Y = X . Indeed, consider x E X ( R ) for some k-algebra R and let f:Sp,R + X be the morphism with f (R)(idR)= x, cf. 1.3. We can apply (4) to the closed, hence local, subfunctors f - I ( Y ) and f - ' ( X ) = Sp,(R) of Sp,(R) and the open covering (f-’(xj))j,J. We get f-'(Y) = f -’(X),hence idR E f - ' ( Y ) ( R ) and x E Y(R).

( 6 ) Any closed subfunctor Y of a local functor X (resp. a scheme X ) is again local (resp. a scheme). Indeed, consider a morphism f:X ' + X and an open covering ( X j ) j , , of X ' such that each fix; factors through Y, i.e., with X j c f - '( Y ) . As f - I ( Y ) is closed, ( 5 ) yields f - ' ( Y ) = X ' , hence f factors through Y. This together with the local property of X implies easily that Y is local. If X is a scheme and if

18

Representations of Algebraic Groups

(Xj)js is an open covering by affine schemes, then ( Y n Xj)js is an open covering by closed subschemes of affine schemes, hence by affine schemes. The proof of the following statement is left as an exercise: (7) For any closed subfunctor Y of a k-functor X and any k-algebra k' the subfunctor &, of X,. is closed. 1.13 Lemma: Let X be a local functor and Y c X a local subfunctor of X. Let (Xj)je be an open covering of X. Then Y is closed in X if and only if each Y n Xj is closed in Xj.

Proof: One direction being obvious, let us suppose that each Y n Xj is closed in Xj. For any morphism J X ' -+ X with X' affine also, f -'(Y) z X'x,Y is local, the f - ' ( X j )are an open covering of X ' , and each f - '( Y) n f -'(Xj) = f - '( Y n Xj) is closed in f - ’(4.). So we may as well assume that X is affine. As in the proof of 1.12(4),we can assume J = { 1,2,. , . ,r } and 4. = X,, for somefi E k[X]. Let I resp. Ij be the kernel of the restriction map k[X] --* k[ Y] resp. k[X] -+ k [ X j n Y]. Then F = V ( I ) . As Y n X j is closed, we have Y n Xj = V(Zj),j. The Y n Xj form an open covering of Y. So the restriction induces an injective map k[Y] -+ k[Y n X,], hence I = 4. We have for all i,j

ns=

ns=l

hence (Ij),8f, = ( I J J i f j . So for any a E Ii,there is some n with (fih)"a E Ijfor all j , hence with f :a E 4 for all j and thus f :a E I = Zj. This implies ZSi = (Zi)si for all i, hence r n X i = Y n X , . Now apply 1.12(4) to the local subfunctors Y and F of and get Y = F.

r

1.14 (Closures and Direct Products) Let us assume in this section that k is noetherian (in order to simplify the following definition). A scheme X over k is called algebraic if it admits a finite open covering by affine subschemes which are algebraic in the sense of 1.6. (One can check that this yields the old definition in the affine case.) Let Y,Z be schemes and X a subscheme of Y such that X and Y are algebraic and such that Z is flat. (This means that Z admits open and affine covering ( Z i ) isuch that each k[Zi] is a flat k-module.) Then we have in Y x Z (1)

x x z = 5? x 2.

This follows, e.g., by applying [DG], I, $2, 4.14 to Y' = Y x Z and the projection Y x Z + Y.

19

Schemes

1.15 (Functors of Morphisms) For any k-functors X , Y , we can define a kfunctor A o + ( X ,Y ) through

&oz(X, Y ) ( A )= M o r ( X , , Y,)

(1)

for any k-algebra A . For any homomorphism cp: A -+A' of k-algebras A G ~ ( XY)(cp) , maps any morphism f : X A + Y, to the morphism f A , : X A'v , ( X A ) A-+ f ( YA)As = Y,. using the structure of A' as an A-algebra via cp. The construction of A o 2 ( X , Y ) is clearly functorial: To each morphism X ' -+ X resp. Y' -+ Y of k-functors there corresponds an obvious morphism A o . c ( X , Y ) -+ A u t ( X ' , Y ) resp. A G 2 ( X , Y ' ) -+ A o h ( X , Y ) . If Y' is a subfunctor of Y , then we shall always regard &02(X, Y') as a subfunctor of &Oh(X, Y ) . Consider an open covering ( X j ) j , of X and a closed subfunctor Y' of Y. Let p j : A G ~ ( XY, ) -+ &&(Xj, Y ) be the obvious restriction map. We claim

Of course, one inclusion (’’c") is trivial. Consider on the other hand

f E A o Q ( X ,Y ) ( A )= M o r ( X , , YA) for some k-algebra A with p j ( A ) f E M o r ( X j A ,Y a ) for all j E J, i.e., with X j A c f-'(YA) for all j. Now the ( X j A ) j E J are an open covering of X, and f-’( Y a )is a closed subfunctor of X,, so 1.12(5) yields f - ' ( Y > ) = X,, hence f E A G ~ ( XY)(A). , If X is an affine scheme, i.e., if there is a k-algebra R with SpkR = X , then A u h ( X , Y ) can also be described as follows: One has for any k-algebra A Auh(SpkR, Y ) ( A ) Mor((SpkR)A, YA) 2:

Mor(Sp,(R 0 A ) , Y ) N Y ( R 0 A ) .

Any Y E Y ( R 0 A ) defines by Yoneda's Lemma (1.3) a morphism f,: Spk(R 0 A ) Y mapping any y E Homk-AI,(R 0 A , B ) = Spk(R 0 A ) ( B ) to Y(y)(y).Using the identification as above, we get also a morphism fk: SpkA -+ A&h(SpkR, Y ) . For any k-algebra B, we can regard f ; ( B ) as the map Homk-AI,(A,B) -+ Y ( R 0 B ) with p H Y(idR0 p)(y). So f b ( B ) is the COmpOSitiOn Of p H id^ 0 p, Homk-A],(A, B ) + Homk-~],(R0 A , R 0 B ) with fJR 0 B). We claim: If R is free as a k-module and if Y' c Y is a closed subfunctor, then fF'Ao@pkR, Y ' )is closed in SpkA. To start with, we know that f ; ' ( Y ' ) is closed in Spk(R0 A ) , so there is an ideal I' c R 0 A with f;’( Y ' ) = V(1'). The description of f; as above implies (for any k-algebra B ) -+

f;'&oh(SpkR,

Y ' ) ( B )= { p E HOmk.Alg(A, B ) I I' c ker(idR 0 p)} =

( p E Homk-~],(A,B ) 11' c R 0 ker(p))

20

Representatiolrs of Algebraic Groups

using the freeness of R for the last equality. This freeness implies also R0 li = R 0 li for any family (li)iof ideals in A . If we take as the li all ideals with R 0 li 3 I’, then I = li is the smallest idea of A with I' c R Q I. Then

n

n

n

I' c R Q ker(p) o I SO

t

ker(p) o p E V(I)(B),

f;'.,doh(Sp,R, Y ' ) = V ( I )is closed. As we can apply this to all A and all y , this implies that Jtio.t(Sp,R, Y ' ) is

closed in .,dob(SpkR,Y ) .Using ( 2 ) we get now ( 3 ) Let X , Y be k-functors and Y' c Y a closed subfunctor. If X admits an open covering ( X j ) j E with J afine schemes such that each k [ X j ] is free as a kmodule, then .,doh(X, Y ' ) is closed in J t i u ~ ( XY, ) .

If X is a scheme, then X is called locally free if and only if there is an open covering as above.

L

Group Schemes and Representations

In this section we define group schemes and modules over these objects and discuss their fundamental properties. As in Chapter 1 we more or less follow CDGI. After making the definitions of k-group functors and k-group schemes in 2.1, we describe some examples in 2.2. The relationship between algebraic groups and Hopf algebras generalizes to group schemes. This is done in 2.32.4. (We always assume our group schemes to be affine.) We then discuss the class of diagonalizable group schemes in 2.5 and group operations in 2.6. We then go on to define representations (2.7) and to discuss the relationship between G-modules and k[G]-comodules (2.8).We generalize standard notions of representation theory to G-modules: submodules (2.9),fixed points (2.10), centralizers and stabilizers (2.12), and simple modules (2.14). The definition of a submodule has an unpleasant aspect that disappears only when G is a flat group scheme (ie., a group scheme such that k[G] is a flat k-module). This is the reason why we shall restrict ourselves to such groups later on. We show that representations of group schemes are locally finite (2.13). Furthermore, we describe representations of diagonalizable group schemes (2.1 1) and mention results about modules for trigonalizable and unipotent groups over fields (in 2.14).One can twist a given representation of an abstract group over k by composing with a group endomorphism. One can also

21

22

Representations of Algebraic Groups

construct a new representation by changing the operation of k on the module by a ring endomorphism. This is generalized to group schemes in 2.15-2.16. 2.1 (Definitions) A k-group functor is a functor from the category of all k-

algebras to the category of groups. We can regard any k-group functor also as a k-functor by composing it with the forgetful functor from {groups} to {sets}. In this way we can and shall apply all ideas and notions from section 1 also to k-group functors. For two group functors G, H, we shall denote by Mor(G, H) the set of all morphisms ( = natural transformations) from G to H considered as k-functors and by Hom(G, H) the set of all morphisms from G to H considered as k-group functors. So Hom(G,H) consists of those f E Mor(G, H) with f(A)a group homomorphism for each k-algebra A . These elements are called homomorphisms from G to H. Let Aut(G) be the group of automorphisms of the k-group functor G. A k-group scheme is a k-group functor that is an afJine scheme over k when considered as a k-functor. (Of course, we really ought to call such an object an affine k-group scheme and drop the word “affine” in the definition of a k-group scheme. But we shall consider only affine group schemes, and then it is more economical to call them group schemes.) An algebraic k-group is a k-group scheme that is algebraic as an affine scheme. A k-group scheme is called reduced if it is so as an affine scheme. Over an algebraically closed field, the category of algebraic groups as in [Hu2] or [Sp2] can be identified with the subcategory of all reduced algebraic k-groups. Let G be a k-group functor. A subgroup functor of G is a subfunctor H of G such that each H(A) is a subgroup of G ( A ) . The intersection of subgroup functors is again a subgroup functor. The inverse image of a subgroup functor under a homomorphism is again one. A direct product of k-group functors is again a k-group functor; so is a fibre product if the morphisms used in its definition are homomorphisms of k-group functors. A subgroup functor H of G is called normal (resp. central) if each H(A) is a normal (resp. central) subgroup of G ( A ) .Again, normality is preserved under taking intersections and inverse images under homomorphisms. The kernel ker cp of a homomorphism cp: G -+ G’ is always a normal subgroup scheme. A closed subgroup scheme of a k-group scheme G is a subgroup functor H which is closed if considered as a subfunctor of the affine scheme G over k. If G and H are algebraic k-groups, we simply call H a closed subgroup of G. A k-group functor G is called commutative if all G ( A )are commutative. 2.2 (Examples) The notations introduced here for special group functors G and their algebras k[G] will be used always, The additive group over k is

Group Schemes and Represeotations

23

the k-group functor G, with G,(A) = ( A , +) for all k-algebras A . It is an algebraic k-group with k[G,] isomorphic to (and usually identified with) the polynomial ring k [ T ] in one variable. Any k-module M defines a k-group functor Ma with Ma(A)= ( M 0A, + ) for all A . (So we have G, N ka). If M is finitely generated and projective as a kmodule, then Ma is an algebraic k-group with k[M,] = S ( M * ) , the symmetric algebra over the dual k-module M *. In case M = k" for some n E N, we may identify Ma with G, x G, x . . . x G, ( n factors) and k[M,] with the polynomial ring k[T,, T',. . . ,T,]. The multiplicatioe group over k is the k-group functor G, with G,(A) = A" = {units of A ) for all A. It is an algebraic k-group with k[G,] = k[T, 7-11. Any k-module M defines a k-group functor G L ( M ) with G L ( M ) ( A )= (End,(M 0A))" called the general linear group of M. In case M = k", we may identify G L ( M ) with GL, where G L , ( A ) is the group of all invertible (n x n)-matrices over A . Obviously, GL, is an algebraic k-group with k[GL,] isomorphic to the localization of the polynomial ring k [ q j , 1 Ii, j In] with respect to {(det)"I n E N}. More generally, if M is a finitely generated and projective k-module, then the k-functor A H End,(M 0A ) can be identified with the affine scheme ( M * @ M ) , from above and G L ( M )with the open subfunctor D(det). For such M (projective of finite rank), the determinant defines a homomorphism of algebraic k-groups G L ( M )+ G,. Its kernel is denoted by S L ( M )and is called the special linear group of M. It is an algebraic k-group. Similarly, we define SL, c GL,. Note that GL1 = G, and S L , = 1 = the group functor associating to each A the trivial group (1). For each n E N, let T, be the algebraic k-group such that T,(A)is the group of all invertible upper-triangular (n x +matrices of A , i.e., of all uppertriangular matrices such that all diagonal entries belong to A " . One may identify k [ T , ] 2: k [ q j l 1 I i I j I n, T i ' 11 I i I n]. Furthermore, let U, be the algebraic k-group such that each U,(A) consists of all g E T,(A) having all diagonal entries equal to 1. We may identify k[U,] = k [ T j I 1 I i < j I n]. For any n E N we denote by pc, the group functor with p(,)(A)= { a E A la" = l } for all A . It is an algebraic k-functor with k[p(,,] = k[T]/(T" - 1) and a closed subgroup of G,. Let p be a prime number and assume pl = 0 in k. Then we can define for each r E N a closed subgroup G,,, of G, through G,,,(A) = { a E A I apr = O}. 2.3 (Group Schemes and Hopf Algebras) Let G be a k-group functor. The group structures on the G ( A ) define morphisms of k-functors m G : G x G + G (such that each mG(A):G ( A ) x G ( A )+ G ( A ) is the multiplication), and

24

Representations of Algebraic Groups

l,:Sp,k + G (such that l,(A) maps the unique element of (Spkk)(A)to the 1 of G(A)),and i G : G + G (inducing on each G(A)the map g - g - ' ) . Now assume G to be a k-group scheme. Then these morphisms correspond uniquely to their comorphisms AG = m;: k[G] -+ k[G] Q k[G] (called comultiplication), E~ = 12: k[G] + k (called counit or augmentation), and aG= i a : k [ G ] -P k[G] (called coinverse or antipode). So, if AG(f) = fi Q f; for Some f E kCG1, then f ( g 1 s J = I;=, f i ( s l ) f : ( g J for each 91, g 2 E G(A) and any A . Furthermore, we have ~ ~ (=f f(1) ) and a G ( f ) ( g )= f ( g - ' ) for any g E G ( A )and any A. We shall drop in our notations the index G whenever no confusion is possible. As in the case of algebraic groups (cf. [Bo], 1.5 or [ H u ~ ] ,7.6 or [Sp2], 2.1.2), the group axioms imply that A, E, a satisfy (id Q A) A = (A Q id) A,

(1)

0

0

@ i d ) A = id = (id @ E ) A,

(2)

(E

(3)

( a @ i d ) o A= B = ( i d @ o ) o A .

0

0

(Here we denote by cp@ $ the map ~ Q u ' Hcp(a)$(a') in contrast to cp Q $ : a Q a' H cp(a)Q $(a') and by Bthe endomorphism a H ~ ( a )of l kCG].) A morphism cp: G -P G' between two k-group schemes is a homomorphism if and only if its comorphism cp*: k[G'] -+ k[G] satisfies (4)

AG q* = ( ~ p * Q ~ p * ) AGg. 0

0

If so, then one has automatically (5)

&G

cp* = &G'

and (6)

bG

q* = q*

QG'.

A Hopf algebra over k is an associative (not necessarily commutative) algebra R over k together with homomorphisms of algebras A: R --i R Q R, E : R -P k, and a: R -P R satisfying (1)-(3). A homomorphism between two Hopf algebras is a homomorphism of algebras satisfying additionally (4)-(6) (with the appropriate changes in the notation.) We call R commutative if it is so as an algebra, and cocommutative if s A = A, where s: R Q R + R Q R is the map a Q b H b @ a. Let R be a commutative Hopf algebra over k. Then we can define on each (Sp,R)(A)= Hom,(R, A ) a multiplication via ab = (a @ p) A. In this way we get on SpkR a structure as a k-group scheme. It is elementary to see that we 0

0

25

Group Schemes and Representations

get in this way a functor {commutative Hopf algebras over k } -, {k-group schemes} that is quasi-inverse to G H k [ G ] . Thus these categories are antiequivalent. Note that G is commutative if and only if k [ G ] is cocommutative. 2.4 (Properties of the Hopf Algebra Structure) Let us look at the Hopf algebra structures on k [ G ] in our examples in 2.2. In the case of G,, one has A(T) = 1 0 T + T 0 1, E ( T )= 0 and o ( T )= - T. Similar formulas hold for the Ga,r.In the case of G,, one has A(T) = T @ T, E ( T )= 1, and a(T) = T-'. In G L , , one has A(Tj) = q,,,0 Tmjand c ( q j ) = 6 , (the Kronecker delta). The formula for o ( T j ) is more complicated. Furthermore, one has A(det) = det 0 det, e(det) = 1, and o(det) = det-'. Let G be a k-group scheme and set I , = k e r ~ ,the augmentation ideal in k [ G ] . One has k [ G ] = k l @ I,, and a H al, k -,k l is bijective. This implies k [ G ] 0 k [ G ] = k(1 0 1) @ (k 0 I , ) @ ( I , 60 k) 0 ( I , 0 1'). The formula 2.3(2) implies

,

A ( f ) ~ f O+ l 10f+I,OZ,

(1)

forall f e z ,

and then the formula 2.3(3) implies o ( f )E -f

(2)

+ 1:

for all f

E

I,.

Set

X ( G ) = Hom(G, G,,,).

(3)

This is a commutative group in a natural way. The embedding of affine schemes G , c G, = A' yields an embedding

X ( G ) c Mor(G, G,) c M o r ( G , G,) N k [ G ] that is compatible with the multiplication. Take f E k [ G ] . One has f * ( T )= f. Therefore 2.3(4) implies easily

(4)

X(G)

{f kcGI I f(l)= 1, A G ( f )

=f 0f>*

Of course A G ( f ) = f 0 f implies f(1)' = f(1). If f(1) = 0, then f(g) = f ( g 1) = f(g)f(l) = 0 for all g E G ( A ) and all A , hence

-

I

(4') If k is an integral domain, then X ( G ) N (f E k [ G ] A&)

f ZO}.

Let me refer to [ D G ] , II,§l, 2.9 for the proof of ( 5 ) If k is a jeld, then X ( G ) is linearly independent.

=f

0 f,

26

Representations of Algebraic Groups

(This is just another variation on the theme “ h e a r independence of characters”.) Usually we shall write the group law in X ( G ) additiuely. Let I be an ideal in k[C]. Using 1.4(6), (7) one checks easily that V ( I ) is a subgroup functor if and only if (61

A(I) c I 0 k[G]

+ k[G]

@ I , &(I) = 0,a(I)c I.

If so, it will be a normal subgroup if and only if (7)

c * ( I ) c k[G] 0 I ,

where c* is the comorphism of the conjugation map c: G x G -, G with c(A)(g1,g2)= g1g2gi1for all A and gl, g2 E G ( A ) .One may check that

2.5 (Diagonalizable Groups) Let A be a commutative group (written multiplicatively) and let us identify A with the canonical basis of the group algebra k[A]. We make k[A] into a commutative and cocommutative Hopf algebra via A(i) = i0 iand &(A) = 1 and ~(i) = A- for all iE A. In this way we associate to A a k-group scheme which we denote by Diag(A). If A is finitely generated, then Diag(A) is an algebraic k-group. So Diag(A)(A) consists (for any k-algebra A ) of all group homomorphisms cp: A -, A ‘. The multiplication is the obvious one: (cpcp’)(A) = cp(A)cp’(A) for all I. E A. We call a k-group scheme diagonalizable if it is isomorphic to Diag(A) for some commutative group A. For example G, Y Diag(Z) and p(”, N Diag(Z/(n)) are diagonalizable. We get also direct products of these groups as Diag(Al x A2) Y Diag(A,) x Diag(A,) for all commutative groups Al, A 2 . Any group homomorphism a: A1 -, A2 induces a homomorphism of group algebras a*: k[AI] + k[A2] which is a homomorphism of Hopf algebras, hence we get a homomorphism Diag(a): Diag(A,) + Diag(A2) of k-group schemes. Thus A H Diag(A) is a functor from {commutative groups} to {kgroup schemes} that maps {finitely generated commutative groups) into {algebraic k-groups}. If LX is surjective, then Diag(cc)(A)is injective for each A, and we can regard Diag(A2) as a subfunctor of Diag(A,). If a is not surjective, then there a nontrivial homomorphism A 1 / a ( A 2 + ) A x for some k-algebra A , hence Diag(a)(A) is not injective.

Group Schemes and Representations

27

Suppose that k is an integral domain. Then an easy computation shows (cf. [DG], 11, $1, 2.1 1) for all A, A‘ (1)

X(Diag(A)) N A

( k integral)

and (2)

Hom,,(A, A’) r Hom(Diag(A’),Diag(A))

(k integral)

Thus in this case Diag(?) is an anti-equivalence of categories from {commutative groups} to {diagonalizable k-group schemes}. Furthermore, A is finitely generated if and only if Diag(A) is an algebraic k-group. We get from (1) that a k-group scheme G is diagonalizable if and only if X(G) is a basis of k [ G ] (for k integral). 2.6 (Operations) Let G be a k-group functor. A left operation of G on a kfunctor X is a morphism a: G x X + X such that for each k-algebra A the map a ( A ) :G ( A ) x X(A) + X(A) is a (left) operation of the group G ( A ) on the set X(A). We usually write gx instead of a(A)(g,x ) for g E G ( A )and x E X(A). We can similarly define right operations. For example, the conjugation map c in 2.4 is an operation of G on itself. Other operations of G on itself are by left ( a ( A ) ( g , g ‘ )= gg’) and right ( a ( A ) ( g , g ’ )= g ’ g - ’ ) multiplication. Let k‘ be a k-algebra. Then any operation of G on a k-functor X defines in a natural way an operation of Gkeon Xkn. For any operation a as above we set

(1)

XG(k)= { x E X(k) Igx = x

for all g E G ( A ) and all A } .

(This is done by some abuse of notation. The x in gx = x is really the image of x under the map X ( k ) + X ( A ) corresponding to the structural morphism k + A. We shall stick to this abuse.) We can define a subfunctor X G of X, the Jixed point functor via XG(A) = (X,)GqA)

(2)

= {x E X(A)(gx= x

for all g E G ( A ’ ) and all A-algebras A ’ } .

If Y is a subfunctor of X, then its stabilizer in G is the subgroup functor StabG(Y) with

(3)

Stab,(Y)(A) = { g E G ( A )1 gY(A’) c Y(A’)

for all A-algebras A ’ }

28

Representatiolls of Algebraic Groups

for all A, and its centralizer is the subgroup functor CentG(Y) with

(4)

Cent,(Y)(A) = { g E G ( A )I gy

=y

for all y E Y(A’) and all A-algebras A ’ }

for all A . These constructions can also be described using the k-functors Aur(X1,X,) as in 1.15. The operation of G defines a morphism y:G-+Au+(X,X) associating to each g E G ( A ) the morphism X, + X, defined by the action of 9. For any subfunctor Y of X we get by restriction a morphism y y : G + Aur( Y, X). Now obviously (5)

StabG(Y) =

J ;)

’ A u h (Y,

Y).

Let cp: G -+ A u r ( Y, X x X) be the morphism associating to each g the morphism y H (gy,y). Using the notation D, for the diagonal subfunctor of X as in 1.2 we see (6)

Cent,(Y) = cp-’Aur(Y, D,).

Let I): X -,Aur(G,X x X) be the morphism associating to each x the morphism g H (gx,x). Then (7)

xG= I ) - ~ A ~ C ( GD,). ,

Now 1.15(3) implies: ( 8 ) If Y is a closed subfunctor of X and a locally free k-scheme, then StabG(Y ) is closed in G. (9) If Y is a locally free k-scheme and if D, is closed in X x X, then CentG(Y) is closed in G. (10) If G is locally free and if D, is closed in X x X, then X Gis closed in X. (One calls X separate if D, is closed in X x X. Any affine scheme is separate.) In case X = G with G acting via conjugation one usually calls StabG(Y) the normalizer of Y and denotes it by NG(Y). Furthermore, we then usually write CG(Y)instead of Cent,(Y), and Z ( G )instead of CG(G).Of course, Z ( G )is just the centre of G . Consider a k-algebra k‘ which is an algebraically closed field. Suppose that Xk, is algebraic and separate. The map Z H Z(k’) is a bijection from {closed and reduced subfunctors of Xkr>to {closed subsets of X(k’)),cf. 1.6. We claim for any closed subfunctor Y of X such that yk. is reduced: (11)

StabG(Y)(k’) = StabGog,( Y(k’)),

(12)

CentG(Y)(k’) = Cent,(,,,( Y(k‘)).

Group Schemes and Representatiom

29

Indeed, if g E G(k’),then g & and 5.are two closed and reduced subfunctors of xk,, hence g & = yk. if and only if (gY,.)(k’)= gY(k’)is equal to Y(k’),i.e., if g E StabG(kt)(Y(k’)).This yields (11). In order to get (12), we embed 5. via y H ( y ,y ) and via y H ( y ,gy) into & x Xk,. The images Y, and Y2 are closed subfunctors of Xkpx &, both isomorphic to &, hence reduced. Therefore Y, = Y2 if and only if Y,(k’) = Y,(k’). On the other hand, g E Cent,(Y)(k’) resp. g E CentG(kr)(Y(k’))if and only if Y, = Y2 resp. Yl(k’) = Y,(k’). This implies (12). Suppose G acts on another k-group functor H such that each G ( A )acts on H ( A ) through group automorphisms. Then we can define the semi-direct product G DC H where each (G PC H ) ( A ) is the usual semi-direct product G ( A ) DC H ( A ) . As a k-functor G DC H is of course the direct product of G and H . Let H , N be subgroup functors of G such that H normalizes N , i.e., that each H ( A ) normalizes N ( A ) . We can then construct H cx N as above and get a homomorphism cp: H DC N + G via (h, n) H hn for all h E H ( A ) , n E N ( A ) and all A . Its kernel is isomorphic to H n N under h H ( h , h - ’ ) for all h E H ( A ) n N ( A ) and all A . If cp is an isomorphism, then we say that G is the semi-direct product of H and N and write G = H LX N . (If G is a k-group scheme and G = H DC N , then necessarily H and N are closed subgroup schemes.) 2.7 (Representations) Let G be a k-group factor and M a k-module. A representation of G on M (or: a G-module structure on M ) is an operation of G on the k-functor Ma (as in 2.2) such that each G ( A ) operates on Ma@) = M 63 A through A-linear maps. Such a representation gives for each A a group homomorphism G ( A )+ End,(M @I A ) ” ,leading to a homomorphism G + GL(M) of group functors. Vice versa, any such homomorphism defines a representation of G on M. There is an obvious notion of a G-module homomorphism (or G-equivariant map) between two G-modules M and M ‘ . The k-module of all such homomorphisms is denoted by HomG(M,M’). The representations of G on the k-module k, for example, correspond bijectively to the group homomorphisms from G to G L , = G,, ie., to the elements of X(G). For each 3, E X(G) we denote k considered as a G-module via 3, by k , . In case 3, = 1 we simply write k. Given one or several G-modules we can construct in a natural way other G-modules. For example

(1) Any direct sum of G-modules is a G-module in a natural way. (2) The tensor product of two G-modules is a G-module in a natural way.

30

Representations of Algebraic Groups

( 3 ) Any symmetric and exterior power of a G-module is a G-module in a natural way. In (3), for example, we consider for each commutative ring R the functor FR from R-modules to itself with & ( M ) = S”M. We have for each R-algebra R‘ canonical isomorphisms FR(M) ORR’ N FR,(M ORR’) for all R-modules M , i.e., the functors M H FR(M) ORR’ and M H FR,(M ORR’) are isomorphic. If M is a G-module, then G operates on the functor A H F,(M 0 A ) , each g E G ( A ) via FA(g). By our assumption this functor is isomorphic to Fk(M),, hence we get a G-module structure on F,(M). The functor M H A”M has the same property, hence we can argue as above. Our reasoning can easily be extended to functors in several variables and then yields (l), (2). If we deal with contravariant functors (FR)Rin our situation above, we ought to let g E G(A) act via F,(g-’), This applies to the functor M H M * which will, however, “commute with ring extensions” only when restricted to finitely generated and projective modules. Thus we get (4) Let M be a G-module which is finitely generated and projective over k. Then

M * is a G-module in a natural way.

For M as in (4)one has canonically M * 0 M ‘ k-module M ‘ . Combining (2) and (4),we get

N

Hom(M, M ’ ) for any

( 5 ) Let M , M ’ be G-modules with M finitely generated and projective over k. Then Hom(M, M ‘ ) is a G-module in a natural way. The following result is obvious from the definitions:

( 6 ) Let k‘ be a k-algebra and M a G-module. Then M @ k’ is a Gk.-mOdUk in a natural way. Another way, how representations arise, is from an operation of G on an affine scheme X. Then we get a G-module structure on k[X]: If g E G(A) and f E k[X] 0 A = A[X,] for some k-algebra A, then gf E A[X,] is defined through ( g f ) ( x )= f ( g - ’ x ) resp. = f ( x g )(for a left resp. right operation) for all x E X ( A ’ ) = X,(A’) and all A-algebras A’. (Again, the g in g - ’ x or xg is really the image of g under G ( A ) -+ G(A’). . .). In case G is a k-group scheme, we get thus the left and right regular representations of G on k[G] derived from the action of G on itself by left and right multiplications. We shall always denote the corresponding homomorphisms G -+ GL(k[G]) by pI and pr. The coinverse 0, is an isomorphism of G-modules from k[G] with pr to k[G] with p l . Furthermore, the

31

Group Schemes and Representations

conjugation action of G on itself gives rise to the conjugation representation of G on k[G]. 2.8 (The Comodule Map) Let G be a k-group scheme. If M is a G-module, then

id,[,] E G(k[G]) = End,.,,,(k[G]) acts on M 0 k[G], so we get a k-linear map A M : M + M 0 k[G] with AM(m)= idktGl(m 0 1) for all m E M . We call A M the comodule map of the G-module M . It determines the representation of G on M completely: For any k-algebra A and any g E G ( A ) = Homk_,,,(k[G],A) we have a commutative diagram

1

1

’ MOkCGl

G(kCG1) x ( M O kCG1)

8 9)

G(g)

G ( A ) x ( M 0A )

idM89

> M O A

by the functorial property of an operation. As G(g)cp = g cp for any cp E G ( k [ G ] ) ,we have g = G(g)id,[,,, hence g(m 6 1) = (id, 0 g) 0 AM(m) for all m E M . More explicitly, if AM(m)= m i0 fi, then 0

The fact that each G ( A ) operates on M O A (ie., g(g’m) = (gg’)m and l m = m)yields easily the following formulas: (2)

(AM

8 id,[,,)

O

AM

= (idM 8 A G )

O

AM

and

(3)

(idM Q E G )

0

AM

= id,.

If M ’ is another G-module and if cp: M + M ’ is a linear map, then cp is a homomorphism of G-modules if and only if (4)

AM,

Cp

= (P 0id,,,,)

AM.

A comodule over the Hopf algebra k[G] is a k-module M together with a linear map A M : M + M 0 k[G] such that (2) and (3) are satisfied. A homomorphism between two comodules is a linear map satisfying (4).So we have defined a faithful functor from {G-modules} to {k[G]-comodules). On the other hand, any k[G]-comodule gives rise to a G-module: Just take (1) as a definition. In this way we can see that the two categories of G-modules and of k [G] -comodules are equivalent. Let a: X x G + X be an action of G on an affine scheme X over k . Then k[X] is a G-module in a natural way (see 2.7) and the comodule map

32

Representatiom of Algebraic Croup

Akfx1:k[X] + k[X] @ k[G] is easily checked to be the comorphism a*. If we take X = G and the action by right multiplication, we get Apr = A G .

(5)

(We write APr and also Ap, below instead of AkIG] in order to indicate which representation is considered.) For the left regular action we get (6)

Ap, = s O

(‘JG

@ id,[,])

O

AG

with s(f @ f’) = f ’@ f for all f,f’. For the conjugation representation on k[G] the comodule map is equal to (7)

where

t’

O

(&[GI

@ AG) O AG

t’(fi@ f 2 @ f 3 ) = f 2 @ aG(fi)f3.

Remark: Suppose for the moment that k is an algebraically closed field and that G is a reduced algebraic k-group. There is a natural notion of representations of G(k) as an algebraic group (or of a rational G(k)-module),cf. [ H u ~ ] ,p. 60. One can show, as above, that the category of G(k)-modules is equivalent to the category of comodules over k[G(k)] = k [ G ] , hence to that of G-modules. (To a G-module M we associate the operation of G(k)on M given by the definition of a G-module.)Similarly one can show that the notions of Gsubmodules (to be defined in 2.9) and of G(k)-submodules coincide, using ) M G (to be defined in 2.10), using 2.10(2).Furthermore, 2.9(1),and that M G ( k = one has HomG(M,M ’ ) = Hom,(,,(M, M ’ ) for any two G-modules M , M ‘ (using (4) above). 2.9 (Submodules) Let G be a k-group functor. If

k is a field, we can define a

submodule of a G-module M as a subspace N c M such that N @ A is a G(A)stable submodule of M @ A for each k-algebra A. Then N itself is a G-module in a natural way. For arbitrary k this works out well as long as the natural map N @ A + M @ A is injective for each A, e.g., if N is a direct summand of M . Taking only such “pure” submodules (as in [DG], 11, 1.3-4) would be too restrictive and not allow kernels and images of all homomorphisms. So let us define a submodule of a G-module M to be a k-submodule N of M that has itself a G-module structure such that the inclusion of N into M is a homomorphism of G-modules. If so, then M I N has a natural structure as G-module: We have for each A an exact sequence of G(A)-modules N @I A -+ M @I A + ( M / N )@ A + 0. We call M I N the factor module of M by N . It has the usual property of a factor module.

Group Schemes and Representations

33

Still, our definition of a submodule has one disadvantage. A given ksubmodule N of M may conceivably carry more than one structure as a Gmodule. In order to prevent this we shall prefer to make special assumptions on our group and not on the modules. An affine scheme X over k is called flat if k[X] is a flat k-module. A k-group scheme is called flat, if it is so as an affine scheme. This property is obviously preserved under base change. Assume now that G is a flat k-group scheme. If N is a submodule of a Gmodule M, then N O k [ G ] is a G(k[G])-stable submodule of M O k[G] (by our assumption of flatness). Then we obviously get

(1)

AM(N) c N 0 k[G]

and (2)

AN

= (AM)IN.

Together with 2.8 the second equality implies that the G-module structure on N is unique. O n the other hand, if N is a k-submodule of M satisfying (l),then ( 2 )defines a G-module structure on N and N is a G-submodule of M . So the Gsubmodules of M are exactly the k-submodules N satisfying (1). Using 2.8(4) one easily checks:

( 3 ) Let G be a flat k-group scheme. For each homomorphism cp: M + M' of G-modules its kernel ker(cp) and its image im(cp) are G-submodules of M resp. M'. We get from this that the G-modules form an abelian category (for G flat). Under the same assumption intersections and sums of submodules are again submodules. Note that inductive limits exist in the category of G-modules (for G flat): Just take the inductive limit as k-modules. This is a factor module of the direct sum (which is O.K. by 2.7(1)) where we divide by a sum of images of homomorphisms. 2.10 (Fixed Points) Let G be a k-group scheme and M a G-module. Set

(1) M G = { m ~ M I g ( m O 1 ) = m O 1 forall g E G ( A ) andall A } . This is a k-submodule of M and its elements are called the fixedpoints of G on M . We call M a trivial G-module if M = M G .In the notations of 2.6 one has M G = (M,)G(k).If we take g = id,[,] E G(k[G]) in (l), then we get

(2)

M G = { m e MlA,(m) = m @ l } .

34

Representations of Algebraic Groups

This description of M Gas kernel of AM - idMQ 1 yields

(3) Let k' be a k-algebra which is flat as a k-module. Then (M Q k')Gk' = M G Q k'. In case k is a field, this implies, of course, = (MG),. (See [ D G ] , 11, $2, 1.6 for a generalization to k-group functors.) If cp:M --* M' is a homomorphism of G-modules, then obviously In this way M H M Gis a functor from {G-modules} to { k cp(MG)c modules} that we call a fixed point functor (relative to G). It is certainly additive. We get from (2):

If G is f l a t , then the fixed point functor is left exact.

(4)

Furthermore, it commutes with taking direct sums, intersections of submodules, and direct limits (but in general not with arbitrary inductive limits). If we consider k [ G ] as a G-module via the left or right regular representation, then the definition immediately yields:

k [ G I G= k l

(5)

(for p 1 and p,).

Let M ' be another G-module, and suppose that M is finitely generated and projective over k. We can then regard Hom(M, M') as a G-module and easily get Hom(M,

(6)

= HOmG(M,M').

Therefore ( 3 ) implies

( 7 ) Let k' be a flat k-algebra and let M be finitely generated and projective as a k-module. Then the canonical map HOmG(M,M') Q k'

+ HOmGk.(M

k', M' 0 k ' )

is an isomorphism for all G-modules M'. Suppose that k' is finitely generated and projective as a k-module. Then & l (and commute with ? 0 k'. So if M is a direct limit of G-modules Mi, which are finitely generated and projective over k , then

9)

k'), M'Q k ' ) HomGk,(M@k', M ' O k ' ) 1: H o m G , , ( l ~ ( M i @ N

@ HOmG,,(Mi Q k', M' Q k ' )

N

@(HomG(Mi,M ' ) O k ' )

= (I& 1

HomG(Mi,M')) Q k'

HornG(& Mi, M') Q k = HOmG(M,M') 8 k.

Group Schemes and Representations

35

So (7) extends to any such M. It will follow from 2.13(3) that we can take any M , if k is a field, or any torsion free M, if k is a Dedekind ring and G is flat. We can generalize (1)-(4) as follows. For each A E X ( G ) set (1')

MI = { m E MI g(m Q 1) I = m Q 1(g)

for all g E G ( A )and all A } .

Then: MA = {m E M 1 AM(m) = m 0 A}.

(2')

(3') For k' as in (3) we have (M Q k')18 = MI Q k'. (4') I f G is p a t , then the functor M ++ MI is leji exact. Furthermore, we have (8) I f k is a j e l d , then the sum of all MI is direct, (If E l m , = 0, where each m, E M I , then 0 = A M ( C I m , )= E l m , Q 1.Now apply 2.4(5).) 2.11 (Representations of Diagonalizable Group Schemes) Let A be a commutative group and take G = Diag(A) as in 2.5. As k[G] is a free k-module with basis A, we can write the comodule map A M for any G-module M as

for suitable p i E End(M). Using the description of AG, EG in 2.5 and the = idM formulas 2.8(2), (3) one easily checks (cf. [DG], 11, §2,2.5) that and p I p I t = 0 for A # A’ and p : = pi, for all A. This implies that M is the direct sum of all p,(M), that

zIeApI

(2)

p I ( M ) = ( m € M 1 AM(m)= m @ A} = M A

(using 2.10(2')), and (3)

M = @MI. IsA

It follows easily that for all G-modules M, M'

(4)

Hom,(M, M') z

n Hom(M,, M i )

ISA

and that the functor M H MI is exact for all 1. If we consider for example k[G] and p r , then we get (5)

k[GII = k 1

for all 1E A.

36

Representationsof Algebraic Groups

Let (e(A))asA be the canonical basis of the group ring Z[A] over Z. So e(A)e(A’)= e(1 + A’), if we agree to write A additively. If M is a G-module such that each Ma is a projective module of finite rank, then we define its formal character ch M

=

1 rk(M,)e(E,).

1EA

For an exact sequence 0 + M’ + M one has (7)

+M” +0

of G-modules of this type

ch M = ch M ’ + ch MI’.

For two G-modules M,, M, projective of finite rank over k, also M, Q M, has this property, and one has (8)

ch(M, Q M,) = (ch M,)(ch M,).

One uses for (8) that for any M,, M, and all A, A’ E A

(One can generalize (6) to the case where the MAare only assumed to be finitely generated over k and where we replace Z by the Grothendieck group of these k-modules.) If k’ is a k-algebra, then obviously one has for all A: (10)

(Mk,)a= (Ma) 0 k .

If ch(M) is defined, then so is ch(M,.) and it is equal to ch(M).

2.12 (Centralizers and Stabilizers) Let G be a k-group scheme and M a Gmodule. For any subset S c M we define its centralizer ZG(S)as the subgroup functor of G with (1)

ZG(S)(A)= { Q E G ( A )I g(m Q 1) = m Q 1

for all m E S } .

Obviously ZG(S)depends only on the k-module generated by S . It is equal to the intersection of all z G ( m ) with m E S. For any k-submodule N c M we define its stabilizer StabG(N) in G as the subgroup functor of G with

(2)

StabG(N)(A)= { g E G ( A )I g(n 0 1) E N @ A

for all n E N } .

Here N Q A is the canonical image of N Q A in M Q A.

Group Scheme and Representatiolls

37

For two k-submodules N' c N of M, we define another subgroup functor GNp,N of G through (3)

GN,,N(A)= { g E G(A) I g(n €3 1) - n €3 1 E N' 6 A

for all n E N } .

Obviously GO,N= ZG(N) and GN,N= StabG(N). Suppose that MIN' is a projective k-module. Then N' is a direct summand of M N N' 0 MIN'. By adding to M a suitable k-module with trivial G-action, we can assume that MINI is free (without changing GN,,N).Choose a basis ( e j ) j s of some complement to N' in M. For any m E M there are aj(m)E k and A,, E k[G], almost all 0 in both cases, such that m E aj(m)ej+ N' and AM(m)E ej €3 fj,,, + N' 8 4 G I . Then

cjE

g(m 8 1) - m 8 1 E

cje

1 ej €3 (h+rn(g) - aj(m))+ N' €3 A

j eJ

for any A and any g e G(A). So GN,,Nis the closed subgroup scheme defined by the ideal generated by all fj," - aj(n)l with n E N and j E J. One gets similar descriptions for Z&) and Stab,(N), hence: (4) If M is a projective k-module, then each ZG(S)is a closed subgroup scheme of G. ( 5 ) If M I N is a projective k-module, then StabG(N) is a closed subgroup scheme of G. ( 6 ) If MINI is a projective k-module, then GN,,Nis a closed subgroup scheme of G. 2.13 (Local Finiteness) Let G be a j u t k-group scheme and M a G-module. We know that any intersection of G-submodules of M is again a Gsubmodule. So for each subset S of M there is a smallest G-submodule of M containing S. It is called the G-submodule generated by S and usually denoted by kGS. (Note that in general kGS # kG(k)S, the k-G(k)-submodule of M generated by S.)

Now take m E M and write AM(m)= Cf=,mi €3 A with mi E M and k[G]. We claim kGm c

fi E

r

kmi. i=l

c;=,

Let us write M' = Z;= km,. As l m = m we have m = &(l)m,E M'. The same argument proves N c M ' where we set N = { m , E M IAM(ml)E M ' 8 k[G]}. Obviously m E N. So it will be enough to show that N is a G-submodule of M , i.e., that A M ( N )c N 8 k[G]. By definition N = A&'(M'@ k[G]). Using the flatness of k[G] we get N 8 k[G] = (AM8 idkfGl)-'(M'@ k[G] @ k[G]). Therefore it is enough to show

Representations of Algebraic Groups

38

(A, 0 idk&,(N) c M‘ 0 k [ G ] 0 k [ G ] . By 2.8(2)the left hand side is equal to (id.+,0 AG)A.+,(N)c (id, 0 AG)(M’0 k [ G ] ) c M 0 k [ G ] 0 k [ G ] . As kGm is a G-submodule, we have A,(m) E (kGm)0 k [ G ] . We therefore may choose the mi above all in kGm. Then kGm = kmi. This shows:

I:=

(2) Each kGm with m E M is a finitely generated k-module and (3) Each finitely generated k-submodule in M is contained in a G-submodule of M that is finitely generated over k. This property is usually expressed as “any G-module is locally jinite” In the case of a field one can show:

x:=

(4) If k is a jield and if A,(m) pendent, then kGm = kmi.

=

mi 0 fi with

(fill 5 i s r linearly inde-

(We may assume also that the mj are linearly independent. If s j s s is a ajimifor all j (by (1)) and basis of kGm, then there are aji E k with m: = there are f S E k [ G ] with AM(m)=xy=lm;@ f ~ . = x : = l m i O ( x I = l a j i f S ) , hence fi = ajif for all i. Hence r = s and the claim.)

xJ=

xi=

5

2.14 (Simple Modules) In this subsection we assume that k is a field. Let G be a

k-group scheme. As usual, a G-module M is called simple (and the corresponding representation is called irreducible) if M # 0 and if M has no G-submodules other than 0 and M. It is called semi-simple if it is a direct sum of simple G-submodules. For any M the sum of all its simple submodules is called the socle of M and denoted by SOCGM(or simply by SOC M if it is clear which G is considered). It is the largest semi-simple G-submodule of M. For a given simple G-module E, the sum of all simple G-submodules of M isomorphic to E is called the E-isotypic component of SOCGM(or the isotypic component of type E ) and denoted by (socGM)E. By 2.13(3) each element in a G-module is contained in a finite dimensional submodule. This implies:

(1) Each simple G-module is finite dimensional. (2) If M is a G-module with M # 0, then SOCGM# 0.

For any G-module M and any simple G-module E, the map cp 0 e H cp(e) is an isomorphism (3)

HomG(E,M ) O D E2 (socGM),

where D = EndG(E).

39

Group Schemes and Representations

(Of course the algebra D over k is finite dimensional and a skew field by Schur’s lemma. If k is algebraically closed, then D = k.) Each one dimensional representation is irreducible. The isotypic component of SOCGMof type ka is just Ma. We get especially M G = (socGM),. The discussion in 2.1 1 shows: (4) If G is a diagonalizable k-group scheme, then each G-module is semi-simple.

The socle series or (ascending) Loewy series of M

0 c SOCIM= SOCGMc S O C ~ c MS O C ~ c M...

is defined iteratively through soc(M/soci- M ) = sociM/soci- M. Again because of 2.13(3)one has

(5)

u

S O C ~= M M.

i>O

Any finite dimensional G-module M has a composition series (or JordanHolder series). The number of factors isomorphic to a given simple G-module E is independent of the choice of the series. It is called the multiplicity of E as a composition factor of M and usually denoted by [M:E] or [M: E l G . The radical radGM of a G-module is the intersection of all maximal submodules. If dim M < co,then radGMis the smallest submodule of M with M/radGMsemi-simple. If G is an algebraic k-group, then it is called trigonalizable (resp. unipotent)if it is isomorphic to a closed subgroup of T, (resp. V,) for some n E N (cf. 2.2). One can show ([DG], IV, $2, 2.5 and 3.4):

( 6 ) G trigonalizable o Each simple G-module has dimension one. (7) G unipotent o Up to isomorphism k is the only simple G-module. If we assume G be to an arbitrary k-group scheme, then we may take these results as definitions. For unipotent G we deduce SOCGM= M Gfor each Gmodule. We get using (2): (8) G unipotent o For each G-module M # 0 we have M G # 0.

Any decomposition of M into a direct sum of two submodules leads to the corresponding decomposition of soc M. If soc M is simple, then M has to be indecomposable. Therefore (8) and 2.10(5) imply (9)

If G is unipotent, then k[G] is indecomposable ( f o r p l and pr).

40

Represeotationa of Algebraic Groups

2.15 (Twisting Representations) Any homomorphism a: G + G’ of k-group functors leads to a functor a* from {G’-modules}to { G-modules}. If M is a G’module, then set a*(M) equal to M a s a k-module, and let any G ( A )operate on M @ A via the homomorphism a @ ) : G ( A )+ G ’ ( A )and the given action of G’(A).Obviously a* is exact and commutes with the elementary constructions as in 2.7(1)-(6). Suppose that G, G’ are group schemes. The comodule map A , * ( M ) : M -P M @ k [ G ] is the composition of A M : M + M @ k[G ]with i d M €3a* where a* is the comorphism corresponding to a. For any 3, E X(G )one has obviously (1)

a*&,)

= k,*(A)

where on the right hand side a* is the comorphism, i.e., a*@) = 3, a. If k is ) X(G)has finite kernel, and integral, if G, G’ are diagonalizable, if a*:X(G + if M is a G’-module such that ch(M) is defined (cf. 2.11(6)), then 0

ch(a*M) = a*(ch M)

(2)

-+ where on the right hand side a* is the ring homomorphism Z[X(G )] Z[X(G)]with a*(e(3,)) = e(3, a) for all 3, E X(G ). In case G = G’ (so if a is a group endomorphism) then we say that a*(M) arises from M by twisting with a. If a E Aut(G), then we usually write “ M = (a-’)*M. Then “(BM) = @fl)Mfor all a, j? E Aut(G). Note that (for both the left and right regular representation) 0

k[G]N k [ G ]

(3)

via f H f a, i.e., the comorphism of a. If G is a normal subgroup scheme of some k-group scheme H,then each h E H ( k ) induces by conjugation an automorphism Int(h) of G. We usually So h(h’M)N (hh’)Mfor all h, h’ E H(k). We usually say that write hM = ‘nt(h)M. hM arises from M by twisting with h. If M extends to an H-module, then hM N M for all h E H ( k ) ; the isomorphism being given by the action of h on M. 0

2.16 (Twisting with Ring Endomorphisms) A representation over k of an abstract group can also be twisted by a ring endomorphism cp of k. If M is a k-module, then let M(@be the k-module that coincides with M as an abelian group, but where u E k acts as cp(u) does on M. Then End,(M) c End,(M(+”),so any group acting linearly on M automatically acts linearly on M(@.Suppose, for the moment, that k is an algebraically closed field and that cp is bijective. Consider a group of the form G(k) with G a reduced algebraic k-group. Let M be a G(k)-module and let (mi)ip, be a basis of M .

Group Schemes and Representations

41

There are functions cij on G ( k ) such that gmj = xicij(g)mi for all j. Then M is a rational G(k)-module (cf. 2.8, remark) if and only if cij E k[G] for In all i, j . As we assume cp to be bijective (mi)icl is also a basis of M(vD). M @ )one has gmj = Eicp-'(cij(g))mi,so M(') is a rational G(k) = module if and only if cp-l 0 cil E k[G] for all i,j. So, M H M ( @ will map rational G(k)-modules to rational G(k)-modules if f H cp-' 0 f maps k[G] to k[G], e.g., if cp-l is a polynomial. This can be generalized to arbitrary k and k-group schemes G as follows. be a morphism such that each $ ( A ) is a ring endomorLet $ : A ' + A ' phism on A'(A) = A and such that $(k) is bijective. Set cp = $(k)-'. Then $ * : f$~0 f is a ring endomorphism on k[G] = Mor(G,A'), but not, in general, k-linear. If we change the k-structure on k [ G ] to that of k [ G ] ( @ , then $* is a homomorphism of k-algebras k[G](@ + k[G]. If M is a Gmodule, then the comodule map A M : M + M 6 k[G] can also be regarded as a k-linear map M ( @+ ( M 6 k[G])('+')3 M(+"6 k[G](@. If we compose with i d M @ $* we get a k-linear map M('+')+ M('+')6 k [ G ] that can be checked to be a comodule map for M('+')(one uses ($* 6 I)*) 0 A G = AG 0 $* and eG 0 $,,, = cp-l o eG), hence a structure as a G-module on M ( @ ,For any k-algebra A , with $ ( A ) bijective, we can identify M ( @6 A with M @ A via m @ a H $(A)-'(a)m as an abelian group with any a E A operating as $ ( A ) - ' @ ) does on M 6 A . Under this identification any g E G(A) acts on M('+')@ A as on M 6 A . If M has a finite basis (mi)l s i s r ,and if ci, E k[G] are the matrix coefficients for this basis and the given operation (Lea,any g E G ( A )has the matrix (cij(A)(g))i,with respect to the basis (mi 6 l)i of M @ A), then (mi)iis also a basis of M ( ( pand ) here the matrix coefficients are the $ 0 c,,. We shall apply twists of this second kind only in the following situation. Let p be a prime number and suppose that pl = 0 in k. Choose r E N and let $:A' + A ' with $(A)(a)= a*' for all a E A and all A . Then our assumptions are satisfied if a H u p is bijective (e.g., if k is a perfect field of characteristic p). In this case we usually write M(') instead of M('+'). So M(') is M with any a E k operating as ap-' on M .

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3

Induction and Injective Modules

In the representation theory of finite groups or of Lie groups the process of inducing representations from a subgroup to the whole group is an important technique. The same holds for algebraic group schemes. So we start this section with the necessary definitions (3.3), prove elementary properties (3.4-3.6), and describe some easy special cases (3.7/8). All this is a more or less straightforward generalization of what is done in the finite group case or the Lie group case. We have, however, to assume that the group G and its subgroup are flat. We then use the induction functor to show that the category of G-modules contains enough injective objects, i.e., that each G-module can be embedded into an injective one (3.9). In the case where our ground ring k is a field we can be more precise. Then the injective G-modules are determined up to isomorphism by their socle and any semi-simple G-module M occurs as a socle of such an injective G-module; the injective hull of M . The indecomposable injective G-modules are just the injective hulls of the simple G-modules. We get especially a decomposition of k[G] generalizing the decomposition of the regular representation of a finite group into principal indecomposable modules. (These results are proved in 3.10-3.17.) Let me mention as a source [Green 13 for the last part (3.12-3.17). For the first part one may compare [Haboush 21, [Oberst], [Cline, Parshall and

43

44

Representations of Algebraic Groups

Scott 31 or [Donkin 13. (There is not much point in attributing priorities for these generalizations.) We assume from 3.2 on that G is a$at k-group scheme and from 3.10 on that k is a field. 3.1 (Restriction) Let G be a k-group functor and H a subgroup functor of G. Each G-module M is an H-module in a natural way: Restrict the action of G(A) for each k-algebra A to H(A). In this way we get a functor

resg: { G-modules} + { H-modules}, which is obviously exact. It is a special case of an u* as in 2.15 and commutes (as any a * ) with the operations described in 2.7(1)-(6). If G and H are group schemes, then we get the comodule map for resgM from AM as (id, 0 y ) 0 AM where y:k[G] --* k[H] is the restriction of functions. 3.2 Lemma: Let H, H' be subgroup schemes of a k-group functor G such that H' normalizes H and is flat. Let M be a G-module. Then M is an H'-submodule of M.

Proof: It is easy to check that the comodule map AM: M + M 0 k[H] of M, considered as an H-module, is a homomorphism of HI-modules if we regard k[H] as an H'-module under the conjugation action. The same holds for the map m H AM(m)- m 0 1. Therefore its kernel M H is an HIsubmodule. 3.3 (Induction) Let H be a subgroup scheme of G. For each H-module M there is a natural (G x H)-module structure on M 0 k[G]: Let G operate trivially on M and via the left regular representation on k[G], let H operate as given on M and via the right regular representations on k[G], and then take tensor products. Now (M 0 k[G])" is a G-submodule of M 0 k[G] by Lemma 3.2. We denote this G-module by indgM and call it the induced module of M from H to G. Obviously indg: { H-modules} + { G-modules} is a functor. Let me mention that we can interpret the operation of G x H on M 0 k[G] in a different way. We have M 0 k[G] = M,(k[G]) N Mor(G, Ma)by 1.3, and more generally (M 0 k[G]) 0 A z (M 0 A ) @,(k[G] 0 A ) = (M 0 A ) 0, ACG,] 2 Mor(G,,(M 0 A),) for each k-algebra A. Any (9,h) E G ( A ) x H(A)

Induction and Injective Modules

45

acts on some f E Mor(G,,(M @ A),) through for all x E G(A’)and all A-algebras A’. (Let me remind you that there is some abuse of notation going on: We really ought to write ( ( g , h ) f ) ( A ‘ ) ( x=) h,,f(A’)(g;!xh,,) with gas E G(A’) the image of g under the map G ( A )-+ G(A’) defined by the structural map A -+ A’, and similarly for h,..) In this interpretation we have (2)

indgM = {f E Mor(G,M,)If(gh) = h - t f ( g ) for all g E G ( A ) ,h E H ( A ) and all k-algebras A }

and the operation of G is by left translation (in a natural sense). Proposition: Let H be a flat subgroup scheme of G. a) The functor indg is left exact. b) The functor indg commutes with forming direct sums, intersections of submodules, and direct limits. Proof: a) As we assume G to be flat, the functor M H M 0 k[G] is exact. Therefore the claim follows from 2.10(4). b) All these constructions commute with tensoring with a flat k-module and with the fixed point functor (cf. 2.10). Remark: If the fixed point functor ?H is exact, then obviously also indg is exact. So indg is certainly exact whenever H is diagonalizable (by 2.1 1). 3.4 For any k-module M let c M :M @ k[G] + M be the linear map cM = id, @ E ~ If. we take the identification M @ k[G] 1: Mor(G, Ma), then we have c,(f) = f ( 1 ) . We shall also use the notation cM for the restrictions of cM

to various submodules of M @ k[G]. Proposition (Frobenius Reciprocity): Let H be a flat subgroup scheme of G and M an H-module. a) c M :indgM -+ M is a homomorphism of H-modules. b) For each G-module N the map rp H cM 0 rp is an isomorphism HOmG(N, indgM) 3 Hom,(resgN, M).

46

Representatiolls of Algebraic Croup

b) In order to define an inverse, consider for each I/, E Hom,(N, M ) and any x E N the morphism $ ( x ) E Mor(G, Ma) with $(x)(g)= (I/, 0 idA)(g-’(x 0 1)) for all A and all g E G(A). Using the description in 3.3(2), one easily checks that $(x) E indgM c Mor(G, Ma).Another straightforward calculation shows $ E Hom,(N,indzM) and that the maps I/, H $ and cp H cM 0 cp are inverse to each other.

3.5 (Transitivity of Induction) The last result implies (for G, H as above): (1) The functor indg is right adjoint to resg.

This of course determines indg uniquely up to isomorphisms. (One can also say that the pair (indgM,&,) is uniquely determined up to isomorphism by 3.4.b.) Let H‘ be another flat subgroup scheme of G with H c H’. We have obviously res;‘ 0 res$ = resg. Therefore (1) yields: (2) There is an isomorphism ind:.

0

indg‘ 1: indz of functors.

We can express this also in this way: Induction is transitive. For any H module M we can write down isomorphisms indgM 1: ind$ 0 ind;’M explicMor(G,(ind;‘M),) with f ( g ) ( h ’ )= itly. To any f E indgM we associate f (gh’) for all g E G(A), h’ E H ( A ) and all A. To any f E ind&(indg’M) we associate Mor(G, Ma) with f(g) = f (g)(l) for all g E G(A) and all A . The maps f H T and f H turn out to be inverse isomorphisms. Observe that 2.10(3) implies

TE

7

(3) Let k’ be a fiat k-algebra. Then we have for each H-module M a canonical isomorphism (indgM) 0 k’ N indg‘,(M 0 k’). Let a be an automorphism. For any H-module M let “M be the a ( H ) module which coincides as a k-module with M, and where any g E a ( H ) ( A )for any k-algebra A acts on M 0 A as a-’(g) E H ( A ) does. Then Horne(,)(V, “M) = Horn#, M) for any G-module K So (1) easily yields:

(4) “(indgM) z ind&,(“M) for any H-module M. 3.6 Proposition (The Tensor Identity): Let H be a fiat subgroup scheme of G. For any G-module N and any H-module M there is a canonical isomorphism of G-modules

indg(M 0 resgN) r (indgM) 0 N.

41

Induction and Injective Modules

Proofi Both sides may be embedded into Mor(G,(M 0 N ) , ) N M 0 N 0 k [ G ] using 3.3(2), the left hand side as L = {f:G -+ (M 0 N ) , I f(gh) = (h-’ 0 h-’)f(g)

for all g, h }

and the right hand side as

R = {f:G + (M 0 N ) , I f(gh) = (h-’ 0 l)f(g)

for all g, h}.

Here “for all g, h” means “for all g E G ( A ) ,h E H ( A ) and all A”. We define two endomorphisms CI, b of Mor(G, ( M 0 N ) , ) through ( a f ) ( g )= (1 0 g)f(g) and (bf)(g) = (1 0 g-’)f(g) for all g. Obviously, they are isomorphisms and inverse to each other. A straightforward calculation shows that a ( L ) c R and b ( R ) c L, and that CI, B are G-equivariant for the two actions of G we consider. (On L we have gf = f(g-’?) and on R we have gf = (1 0 g)f(g-’?).) This implies the proposition.

Remark: We ought to express the proposition (the tensor identity) as saying: The functors (M, N ) H indg(M 0 resgN) and ( M ,N ) H (indgM) 0 N from { H-modules} x { G-modules} to { G-modules} are isomorphic. 3.7 (Trivial Examples) We can apply all this especially to the subgroup schemes H = 1 and H = G. The first case yields

(1)

indFM = M 0 k [ G ]

for any k-module M

(where M is considered as a trivial G-module on the right hand side),especially

(Here and below k[G] is considered as a G-module via p i . ) Combining (2) with 3.4.b (Frobenius reciprocity) we get for any G-module M (3)

Hom,(M,k[G]) z M*.

(This can also be shown directly using matrix coefficients, cf. [DG], $2, 2.3.) Taking M = k in 3.6 we get for each G-module N an isomorphism

(4)

N 0 k [ G l s N , , 0 kCGl

where N,, denotes the k-module N considered as a trivial G-module. Going back into the proof and the definitions one checks that this isomorphism is given by

48

Representatious of Algebraic Groups

If we restrict this map to the G-submodule N 6 k l see:

= N of N @I k[G],

then we

( 5 ) AN:N --* N,, @I &[GI is an injective homomorphism of G-modules.

(This can be checked directly, of course). As resgM = M for each G-module M we also have by 3.5(1) (canonically) ( 6 ) M 3 indgM

for each G-module M .

This isomorphism M 3 (M @ k[G])G c M @ k [ G ] is given by (id, 6 oG)0 A,. (In other words, any m E M is mapped to the morphism G + Ma with g H g-'(m @ 1) for all g E G ( A )and all A,) 3.8 (Induction and Semi-direct Products) Let G' be a flat k-group scheme operating on G through automorphisms and let H be a flat subgroup scheme of G stable under G'. We can then form the semi-direct products H >Q G' and G >Q G', and we can regard H XI G' as a subgroup scheme of G >Q G'. Let M be an (H x G')-module, i.e., a k-module that is simultaneously an H-module and a G'-module so that these two operations are compatible: g'(hm) = (g'hg'-')(g'm). Then G' acts naturally on Mor(G, Ma) N, k[G] B, M via ( g ' f ) g )= g'(f(g'-'gg')), i.e., through the tensor product of the conjugation action with the given action on M. This defines a structure of an (H >Q G')-module and also of a (G XI G')-module where H, G operate as usual in the construction of indZM. As G' normalizes H, it operates also on indgM = Mor(G, Ma)H,cf. 3.2. Therefore we get on indgM a structure as a (G XI G')-module. We claim that we have an isomorphism of (G XI G')modules (1)

indg M 3 indg:

z:M.

We simply associate to f E indgM c Mor(G, Ma)the map F E ind; :Z:M c Mor(G XI G', Ma) with F(g,g') = 9 ' - ' F ( g ) , and to any F the map f with f ( g ) = F(g, 1). The claim follows now from elementary calculations. Taking H = 1 we get especially for any G'-module M:

(2)

indg." "M

N

k[G] 6 M

N

Mor(G, Ma)

with G acting via pt on k[G] and trivially on M, and with G' acting via the conjugation action on k[G] and as given on M. We can also describe ind:""N for any G-module N. There is an isomorphism (3)

indg" "N 3 Mor(G', Na) 'v k [ G ' ] @ N

Induction and Injective Modules

49

mapping any F E indz w C ‘ N c Mor(G 24 G ,Nu) to f: G + Nu with f ( 9 ’ ) = F(1,g’) and any f to F with F(g,g’) = g’-lgg’f(g’). This isomorphism is compatible with the G’-actionif we let G’act on k[G’] via pl and trivially on N . The action of G on some f: G + N, is given by ( g f ) ( g ’ ) = (9’99’- ’) f (9’).This implies:

(4) If N is a trivial G-module, then G acts trivially on indg “ N . 3.9 We define an injective G-module to be an injective object in the category of all G-modules. Proposition: a) For each flat subgroup scheme H of G the functor indz maps injective H-modules to injective G-modules. b) Any G-module can be embedded into an injective G-module. c) A G-module M is injective if and only if there is an injective k-module I such that M is isomorphic to a direct summand of I Q k[G] with I regarded as a trivial G-module. Proof: a) This is obvious as ind: is right adjoint to the exact functor resz. b) Let M be a G-module. We can embed M as a k-submodule into an injective k-module I. Then I@ k[G] 1: indFl is injective by (a) and indyM N M,,@ k[G] is a submodule of I @ k[G]. Now combine this with the embedding of M into M,, @ k[G] from 3.7(5). c) If M is injective, then the embedding M + I @ k[G] constructed in the proof of (b) has to split. This gives one direction in (c).The other is obvious, as I @ k[G] is injective by (a), hence also each direct summand. 3.10 Let us assume from now on in Chapter 3 that k is a Jield. Then we can

simplify the last result: Proposition: a) A G-module M is injective if and only if there is a vector space V over k such that M is isomorphic to a direct summand of V Q k[G] with V regarded as a trivial G-module. b) Any direct sum of injective G-modules is injective. c) If M , Q are G-modules with Q injective, then M Q Q is injective.

Proof: (a) is just 3.9.c and (b) is an immediate consequence of (a). If Q is a direct summand of V @ k[G] as in (a), then M @J Q is a direct summand of M Q V Q k[G], which is isomorphic to M,, Q V @ k[G] by 3.7(4). This yields (c).

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Representations of Algebraic Groups

3.11 Before looking at indecomposable injective G-modules in general, let us treat one important example. Suppose G = H pc G’ with H a diagonalizable and G’ a unipotent group scheme. We set for each 1. E X ( H ) :

(1)

Y, = indgk,.

We have k[G] E indyk E indgindyk E indgk[H] by the transitivity of induction, and k[H] N @,,,(,,k, by 2.11(5) (also with respect to p l , of course), hence

We know by 3.8 that Y, is isomorphic to k[G’] when considered as a G‘module. Therefore 2.14(9) implies:

(3) Each Y, is an indecomposable and injective G-module. Each 2 E X ( H )can be extended to an element of X ( G )with G’ in the kernel. We also denote this extension by 3, and the corresponding G-module, by k,. For each G-module M the subspace MG‘is a G-submodule by 3.2. Because of 2.1 1 it is a direct sum of one dimensional G-submodules of the form k, with A E X ( H ) . This shows especially that MG‘is a semi-simple G-module. On the other hand, we have MG’# 0 for any simple G-module because of 2.14(8). Therefore the k , with A E X ( H ) are all simple G-modules (up to isomorphism) and we have

(4)

SOCGM= MG’

for any G-module M. The discussion in 3.8 shows that Y, N k, @I k[G’] where H operates on k[G’] via the conjugation action. Then (YJG‘ N k, 0 (k“’]‘’) = k, 0 k l 3: k,, hence by (4): (5)

SOCG

YA = k,.

This shows that in this case there is, for each simple G-module E, an indecomposable and injective G-module with socle isomorphic to E . We want to generalize this result. At first we shall prove the uniqueness of such a module (up to isomorphism). 3.12 Proposition:

Let M, M’ be injective G-modules and cp E HOmG(M, MI). Then cp is an isomorphism i f and only i f cp induces an isomorphism soc,M --t S O CM’. ~

51

Induction and Injective Modules

Proof: The “only if” part is obvious, so let us look at the “if”. We know by 2.14(2) that ker cp # 0 =+-0 # socG(kercp) = ker(cplsocGM). Assuming cp to induce an isomorphism of the socles we get ker cp = 0 and the injectivity of cp. Therefore cp(M) N M is an injective G-module, hence a direct summand of M‘. If M , is a G-stable complement, then M‘ = q ( M )0 M , implies SOC,(M’)’= s0cGcp(M)@ soc&f,. The assumption SOC,M’ = cp(socGM) yields S O C ~ M=, 0, hence M , = 0 by 2.14(2). Therefore cp is bijective.

Two injective G-modules are isomorphic if and only if their socles are isomorphic.

3.13 Corollary:

Proof: Because of the injectivity, any isomorphism of the socles can be extended to a homomorphism of the whole modules. Then apply 3.12.

Let M be an injective G-module and let cp, E EndG(socGM) be idempotent. Then there is cp E EndG(M)idempotent with cplsoccM = cpl. 3.14 Proposition:

Proof: Consider the socle series of M as in 2.14(5). Let us abbreviate Mi = sociM. Each endomorphism of M has to preserve all M i . Therefore the injectivity of M yields, for each i, an exact sequence (1)

0 -+ mi + EndG(M)

EndG(Mi) + 0

where m i is the two-sided ideal

(2)

I

mi = {cp E EndG(M) cp(kfi) = o}.

Any cp E m i maps Mjinto Mj-i for all j 2 i. This implies

(3)

mimj c mi+j

We deduce from M =

(4)

u

for all i, j 2 1.

,Mi that EndGM = lim EndG(Mi). t

Therefore the proposition follows from a version of Hensel’s lemma proved below. 3.15 Proposition: Let R be a ring and let m, I> m2 I> ... be a chain of twosided ideals of R with mimj t m i + jfor all i, j 2 1 and R = @ R/mi naturally.

52

Representatiom of Algebraic Groups

Then there is for each idempotent element el E R/ml an idempotent element e E R withe, = e m,.

+

Proof: Because of R N @ R/mi it is enough to construct e 2 , e 3 ,... E R such that each ei + mi E R/mi is idempotent and such that e, + mi- = ei-l mi-.l for each i > 1. We define iteratively ei+l = 2ei(ei - e:) ez. As e, + mi is assumed to be idempotent we have ei+l + mi = e: + mi = ei + mi. Furthermore, we get e;+ E 4ei3 (ei - e;) + e f + m; = 3ei(e, - e?) + e:(ei - e f ) m; c 2ei(ei - e;) ez m i + , , hence ei+l m i + l is idempotent. Therefore we can go on.

+

+

+

+ +

+

3.16 Proposition: a) For each simple G-module E there is an injective Gmodule QE (unique up to isomorphism) with E 1: soc QE. b) An injective G-module is indecomposable if and only if it is isomorphic to QE

for some simple G-module E. c) Any injective G-module Q is a direct sum of indecomposable submodules. For each simple G-module E the number of summands isomorphic to QE is equal to the multiplicity of E in SOCGQ.

Proof: Let Q be an injective G-module. Any decomposition S O C G Q = M, 0 M, leads by 3.14 to a decomposition Q = Ql 0 Q2. As we can embed any G-module into an injective G-module by 3.9.b we get the existence of the QE in (a) immediately. The uniqueness follows from 3.13. The other parts of the proposition are now obvious. 3.17 The module QE from 3.16.a is called the injective hull of E. More generally, we can find for each G-module M an injective G-module QM (unique up to isomorphism) with SOCGM= SOCGQM.The embedding of SOCGMinto Q can be extended to an embedding of M into QM.We call QMthe injective hull of M. It is clear that this is compatible with the general definition, e.g., in [Bl], ch. X, 01, no 9. In the situation of 3.16.c, the number of summands isomorphic to QE is equal to

dim HomG(E,Q)/dim EndG@), cf. 2.14(3). If we take especially Q = k [ G ] , then we get from 3.7(3) k[G]

1:

@Q”,’”’ E

where (2)

d(E) = dim(E)/dim(End,(E)),

Induction and Injective Modules

53

and where the direct sum is taken over a system of representatives of all simple G-modules. (If k is algebraically closed, then d ( E ) = dim@) of course.) In the situation of 3.11 we have obviously YA= QkA,and 3.1 l(2) illustrates (1) very well. In the case of an unipotent group one has k [ G ] = Qk,cf. 2.14(9). Let us mention one standard property of injective hulls: Let E be a simple Gmodule and M a finite dimensional G-module. Then (3)

[ M : E ] , = dim(HOm,(M, Q,))/dim(End,(E)).

(For the notation cf. 2.14.)

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4

Cohomology

Throughout this chapter let G be a flat k-group scheme. We have shown in the last chapter that each G-module has a resolution by injective G-modules. Therefore we can define (right) derived functors of left exact functors on the category of G-modules. We can, for example, describe the Ext-functors as derived from the Hom-functor, and we can introduce the cohomology functors H"(G, ?) as derived from the fixed point functor. Furthermore, there are for each flat subgroup scheme H of G the derived functors R"indg of the induction functor. After recalling some general facts about derived functors (4.l), and making the definitions (4.2) we prove many elementary properties of the derived functors mentioned above (4.3-4.13,4.17). We prove equalities between two derived functors and mention several spectral sequences. We show that the cohomology can be computed using an explicit complex; the Hochschild complex (4.14-4.16). Besides proving a universal coefficient theorem (4.18), this complex is used for the computation of the cohomology of the additive group over a field (4.20-4.27). Because of later applications we formulate the results at once not for G, but for direct products G, x G, x x G,. As in the last chapter there is not much point in attributing priorities for generalities. In addition to the papers listed there, one ought to mention [Andersen 123 where some results were extended to the case of an arbitrary

55

56

Reprwntatiom of Algebraic Group

ground ring (instead of a field.) When discussing the Hochschild complex I follow [DG] more or less. The computation of H'(G,,k) is due to [Cline, Parshall, Scott, and van der Kallen]. 4.1 (Derived Functors) Let % be an abelian category containing enough

injectives, i.e., such that each object can be embedded into an injective object. Then certainly each object admits an injective resolution. We can then define the (right) derived functors R n 9 of any additive (covariant) functor 9from % into some other category W'. We have R o 9 = 9if and only if 9is left exact. An object M in 9 is called acyclic for 9 if R n 9 ( M )= 0 for all n > 0. Any short exact sequence in %? gives rise to a long exact sequence in 'Y. Suppose now that 9: %' --t V’ and 9’: %' V" are additive (covariant) functors where W, %",W' are abelian categories with %, W' having enough injectives. --f

Proposition (Grothendieck'sSpectral Sequence): If 9' is left exact and if 9 maps injective objects in %' to objects acyclic for 9’, then there is a spectral sequence for each object M in % ' ? with diflerentials d, of bidegree (r, 1 - r), and Enm = ( R n 9 " ) ( R m F ) * M R n + m ( 90' 9)M. (1) One can find a proof (and more background material) in the second edition of S. Lang's Algebra or in [Ro]. Let me mention two trivial special cases:

(2) If 9' is exact, then 9’R " 9 0

= Rm(9'

0

9 )for all n E N.

(This is obvious). ( 3 ) If 9 is exact and maps injective objects to objects acyclic for 9’, then ( R " 9 ' )0 F N B"(9' 0 9 )for all n E N. (This can be proved by degree shifting, i.e., induction on n using the long exact sequence.) Consider an arbitrary spectral sequence (E:."') with differentials d:,": E;.m+~;+r,m+l-r, with El." = 0 for n < 0 or rn < 0 converging to some abutment (Er).As d:,' = 0 for all n and r (resp. as d;',"' = 0 for all m and r), we get epimorphisms El.' + E:' (resp. monomorphisms E:" E:'"). Combining this with the monomorphisms E:' + E n resp. the epimorphisms E m + E:" we get natural maps --f

E;O + E n

and E m --t E:.",

called the base maps of the spectral sequence. In the situation as in (1) they have

57

Cohomology

the form ( R n 9 ' ) ( 9 M )-+ R " ( 9 ' 0 9 ) M and R"(B' 0 9 ) M + F ' ( ( R " 9 ) M ) . If we take the base maps for n = m = 1 and for n = 2 together with d$*', then we get an exact sequence (cf. [Bl], ch. X, $2,exerc. 15c)

(4)

0 -+ E ; . O - + E' + E:*' + E2-O -+ E 2 ,

called the five term exact sequence. 4.2 Throughout this chapter let G be a flat group scheme over k and H a flat

subgroup scheme of G . We know by 2.9 and 3.9.b that the G-modules form an abelian category containing enough injective objects. So we can apply the general principles from 4.1. For example, the fixed point functor from {G-modules} to {kmodules} is left exact. We denote its derived functors by M H H"(G,M), and call H"(G,M) the nth(rational) cohomology group of M. For any G-module M the functor Hom,(M,?) is left exact. Its derived functors are denoted (as usual) by Ext",M,?). They can (as always) also be defined using equivalence classes of exact sequences of G-modules. For the trivial module k the functor HOmG(k, ?) is isomorphic to the fixed point functor: For each G-module M we have an isomorphism HomG(k,M) r M Gwith cp H cp(1). We therefore get isomorphisms of derived functors (1)

ExtL(k, ?) N H"(G, ?).

The induction functor from H to G is left exact. We can therefore define also its derived functors R'ind;. 4.3 Lemma: Suppose that G is diagonalizable. Let A be an abelian group with G 1: Diag(A). Then one has for all G-modules M, N : a) Ext",M, N ) N HA,,Ext;(M,, NJ for all n E N . b) H"(G,M) = 0 for all n E N , n > 0. c) I f k is a Jield, then Ext",M, N ) = 0 for all n E N , n > 0.

Proof: The first claim follows easily from 2.1l(4).The other statements are immediate consequences. 4.4 Lemma: Let M, N , V be G-modules. If V is finitely generated and projective as a k-module, then we have for all n E N a canonical isomorphism

Ext",M, V Q N ) 3 Ext;(M Q V*,N ) .

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Representations of Algebraic Groups

Proofi We have a canonical isomorphism Hom(M, V @ N ) 7 Hom(M @ V * ,N ) sending any cp to the map m @ a H (a @ id,)(cp(rn)). It is easy to check that this induces an isomorphism (1)

HOmG(M, I/ @ N ) 3 HOmG(M @ V * , N ) .

This is functorial in N and can be interpreted as an isomorphism of functors Hom,(M, ?) (V @ ?) 3 HOmG(M@ V * , ?). 0

The functor V 0 ? is exact and maps injective G-modules to injective Gmodules (cf. 3.9.12). We can therefore apply 4.1(3). 4.5 Proposition: Let M be an H-module. a) For each G-module N we have a spectral sequence with

En." = Ext",N, R"indg M) =-Ext;l+"(N, M )

b) There is a spectral sequence with E?" = H"(G, R"indgM)

H"+"(H, M).

c) Let H' be a $at subgroup scheme of G with H c H'. Then there is a spectral sequence with E;" = (R"ind&)(R"ind$')M

* (R"+"ind;)M.

Proofi a) The Frobenius reciprocity in 3.4 can be interpreted as an isomorphism of functors Hom,(N, ?) 0 indg 1: Hom,(N,?).

As indg maps injective H-modules to injective G-modules by 3.9.a, we can apply 4.1( 1). b) This is the special case N = k of a). c) Take the isomorphism in 3.5(2) and argue as in the proof of a). 4.6 We call H exact in G if indg is an exact functor. For example, any diagonalizable subgroup scheme of G is exact in G. (See the remark to 3.3.) The

last proposition implies obviously: Corollary: Suppose that H is exact in G. Let M be an H-module. a) For each G-module N and each n E N there is an isomorphism Ext'&(N,indg M) N Ext;l(N, M ) .

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Cohomology

b) For each n E N there is an isomorphism H"(G,indgM) N H"(H, M).

Remark: These results are also known as "generalized Frobenius reciprocity" and "Shapiro's lemma". 4.7 When we regard k[G] as a G-module and do not mention the representation explicitly, we will deal with pl or p r . As both structures are equivalent it is most of the time not necessary to specify which of these two we consider. The same applies to H instead of G.

Lemma: Let n E N. a) We have for each G-module N: H"(G, N 0 k[G])

N

N

if n = 0 ,

0

if

n>0.

b) We have for each H-module M : R"indE(A4 0 k[H])

N

{r@k[G]

if

if

n=0, n>0.

Proof: a) The trivial subgroup 1 of G is exact in G as it is diagonalizable (or even more trivially, as indy = k[G] 0 ? is obviously exact). Therefore a) is an immediate consequence of 4.6.b (applied to H = 1) and of the tensor identity. b) Apply the spectral sequence 4.5.c to (H, 1) instead of (H', H). As 1 is exact in H,the spectral sequence together with the tensor identity yields isomorphisms R"indg(M 0 k[H])

2:

R"indy(M).

As 1 is exact in G , the right hand side is 0 for n > 0 and equal to M 0 k[G] for n = 0 by the tensor identity. This implies b). Remark: If k is a field, then N 0 k[G] is an injective G-module by 3.10.c. Similarly, M @ k[H] is an injective H-module. So the lemma is obvious in this case. 4.8 Proposition (The Generalized Tensor Identity): Let N be a G-module that is flat as a k-module. Then we have for each H-module M and each n E N an

isomorphism

R"indg(M Q N ) N (R'indgM) @ N .

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Representatiom of Algebraic Groups

Proof: The tensor identity may be interpreted as an isomorphism of functors

indg o (resgN @ ?) N ( N @ ?) o indg. Tensoring with N is exact and maps, because of 3.9 and 4.7.b, injective Hmodules to modules acyclic for indg. So we can apply 4.1(2), (3).

4.9 (Semidirect Products) Let G' be a flat k-group scheme that operates on G. We can therefore form the semi-direct product G ZQ G'. By 3.2, we may also regard the fixed point functor ?G as a functor from {(G 24 G')-modules) to {G'-modules}. There is an obvious isomorphism resy' ?G N ?G 0 resg*G' of functors. The isomorphism of k-algebras k [ G XI G'] N k [ G ] @ k [ G ' ] is compatible with the action of G via pI on k[G ZQ G'] and k [ G ] , and with the trivial action on k [ G ' ] . Therefore, 3.9 and 4.7.a imply that re$ * ' maps injective modules to modules acyclic for the fixed point functor. We therefore get isomorphisms of derived functors by 4.1(2),(3). So we have for all n E N and any (G ZQ G')-module M a natural structure as a G'-module on H"(G, M). Suppose now that G' stabilizes the subgroup scheme H of G. We can interpret 3.8(1) as an isomorphism resg G' 0 indgz $: = indg 0 res; G' of functors. As above, 3.9 and 4.7.b imply that resi- G' maps injective modules to modules acyclic for indg . Therefore, 4.1(2), (3) yield isomorphisms of functors (for all n E N): 0

2o

(1)

re$;

G'

0

R"indgz $:

1:

R"indg o res;

G’.

For H = 1 this shows that G' is exact in G 24 G', which is already clear by 3.8(2). Similarly G is exact in G ZQ G' by 3.8(3). We have for each H-module M and each n e N an isomorphism of k-modules

4.10 Proposition:

H"(H, M 0 k [ G ] ) r (R"indg)M. Proof: The definition of indg yields an isomorphism of functors

9 0 indg N

?H

0

(k[G] @ ?),

where 9 is the forgetful functor from {G-modules) to {k-modules}. As k[G] @ ? is exact and maps injective H-modules to modules acyclic for the fixed point functor (by 4.7.a), we can apply 4.1(2), (3).

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Cohomology

4.11 Corollary: If k[G] is an injective H-module, then H is exact in G.

Proof: Under our assumption k[G] is a direct summand of some M, Q k[H], hence M 0 k[G] of some M, 0 k[H] (for suitable H-modules M , , M,). Then 4.10 and 4.7.a imply the claim. Remarks: 1) Suppose that k is a field. Then the corollary can be proved directly as follows. If 0 + MI + M , + M, + 0 is an exact sequence of Hmodules, then 0 + M, Q k[G] + M 2 Q k[G] + M, 0 k[G] + 0 is an exact sequence of injective H-modules (by 3.10), hence split as a sequence of Hmodules. Therefore, the sequence of all (Mi Q k[G])H = indg(Mi) also has to be exact. 2) The example H = 1 shows that the converse will not hold in general. However: 4.12 Proposition: Suppose that k is a jield. Then H is exact in G if and only if k[G] is an injective H-module.

Proof: Because of 4.1 1 we have to prove one direction only. Suppose that H is exact in G. We have for each finite dimensional module V, by 4.4, 4.2(1) and 4.10 ExtL(V,k[G])

N

ExtL(k, V* Q k[G])

N

H"(H, V* Q k[G]) = 0

for all n > 0. Therefore the functor Horn,(?, k[G]) is exact when restricted to finite dimensional H-modules. This easily implies the exactness on all Hmodules (ie., the injectivity of k[G]) because each H-module is the direct limit of finite dimensional H-modules. Let k' be a p a t k-algebra. Let n E N . a) For each G-module N there is an isomorphism

4.13 Proposition:

H"(G, N ) Q k'

2:

H"(Gk,,N 0 k').

b) For each H-module M there is an isomorphism R"(indgM) 0 k'

= (R"ind%',)(M 0 k').

Proof: We get from 2.10(3) and 3.5(3) isomorphisms of functors to which we want to apply 4.1(2),(3).This is possible as ? 0 k' is exact and maps injective Gmodules to modules acyclic for the Gk.-fiXed point functor (by 3.9 and 4.7.a),

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Representations of Algebraic Groups

and also maps injective H-modules to modules acyclic for the induction from Hkfto Gkr(by 3.9 and 4.7.b). 4.14 Let M be a G-module. The cohomology H'(G, M ) can be computed using

the Hochschild complex C'(G, M ) which we are going to describe now. k[G] for all n E N,and define boundary maps We set C"(G, M ) = M 0 a": C"(G, M ) -+ C"+'(G, M ) of the form a" = (- 1)'a; where

0"

~~~~

a;(m O fl O

* *

0 fl 0 * * * 0 fn ,

O f,) =

a ~ ( m O f i O . . . O f n =mOflO...Of;:-iOAG(fi)Of;:+lO...Qfn ) 1 I i I n,

for O f1O

* * a

O fn O 1.

0 fn) = m O fl O

We can also interpret Cn(G,M ) as Mor(G", Ma)where G" is the direct product of n copies of G, cf. 3.3. Then the 8; look like a",f(gl, 92

3. *

*

alf(gl,g2

9

*

9

gn+ 1) = glf(g2

gn+ 1) = f(g1,

9 .

3

* * 3

gi - 1, gigi+ 1, gi+2

*

9n+1) = fbl, *

9

* * 9

gn + 1)

1 I i I n,

for

a:+ 1f(s1,92,.

gn+ 1 1 9

7

9n).

It is easy to check that a"a"-l = 0 for all n. Therefore (C'(G,M),a') is a complex. We want to prove that its cohomology is just H'(G, M ) . 4.15 If our last claim is true, then C'(G, k[G]) ought to be exact except in degree 0 by 4.7.a. Let us consider k[G] as a G-module via pr so that Ak[G]= AG.

We define for each n a linear map sn:

cn + 1(G,k[G])

= @"'2k[G]

--*

@"" k[G]

= C"(G,k[G])

through s" = E~ 8 @"+ idk[G].An elementary calculation using 2.3(2) shows s"a" = id - 8"'1~n-1 foralln > 0.Thisimpliestheexactnessof C'(G,k[G])at each point n > 0 whereas

8 ’ : C'(G, k[G]) = k[G] maps f to A ( f ) - f sequence

-+

C'(G, k[G]) = k[G] O k[G]

1, hence has kernel k l . Therefore we have an exact

g3

0 + k -P k[G] + @k[G] -+ k[G] This sequence can be regarded as a sequence of homomorphisms of Gmodules when we let G operate on @"[GI via p1 on the first factor and (1)

-+

a * * .

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Cohomology

trivially on all the other factors. It is for this operation that k [ G ] + k [ G ] 0 k [ G ] , f H AG(f ) - f 0 1 is G-equivariant. If we now tensor (1) with M , then we get a resolution

(2)

0 +M

+M

@ k [ G ] -+ M @ @ ' k [ G ] +

of M by acyclic modules. Furthermore, by 3.7(4)we can make the operation of G on the factor M in any M 0 k [ G ] trivial, hence get a resolution

Oi

(3)

0 +M

+ M,,

0 k [ G ] -+ M,, 0 8' k [ G ] -+

* - *

using the same notation as in 3.7(4).Therefore H'(G, M ) is the cohomology of the complex (4)

O+(M,,@ k [ G ] ) G + ( M , , @ @ ' k [ C ] ) ' + * * * .

As G operates trivially on all but one factor, and as k [ G I G = k, the nrhterm in (4) is equal to (M,, 0 @ " + I k[C])' = M,, 0 @"k [ G ] N C"(G, M ) .

Furthermore, tracing back the maps one finds that 8"is just the map from C"(G, M ) to C"' '(G, M ) occurring in (4).(The shortest way of doing it is via the interpretation as functions G" + M.) This proves our claim.

4.16 Let M be a G-module.

Proposition:

The cohomology of the complex C'(C,M ) is equal to H'(G, M ) .

Remark: In [DG], 11, 53, the case of arbitrary group functors (instead of our flat group scheme) is treated and more general coefficients are considered. 4.17 Lemma:

Let (Mi)is,be a directed system of G-modules. Then there are natural isomorphisms for all n E N :

l&H"(G,M i ) 7 H"(G,9Mi).

Proof: As tensoring with k [ G ] is exact, we have natural isomorphisms lim C"(G, M i ) 3 C"(G,l& Mi)for all n E N . So the claim follows from the * exactness of 9,cf. [ B l ] , ch. 11, §6, prop. 3. Remark: For any direct system ( N i ) i E Iof H-modules one similarly gets isomorphisms of G-modules (1)

* lim R"indg(Ni) 7 R"indg(l& %),

cf. also [Donkin 91, 1.1.1.

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Representations of Algebraic Groups

4.18 We can identify C'(Gk.,M Q k') for any k-algebra k' with C'(G, M ) Q k'. Suppose that M is a flat k-module. Then all C"(G,M )are also flat. If k has the

property that any submodule of a flat module is flat, then we get a universal coefficient theorem, e.g., by [Bl], ch. X., 54, cor. 1 du th. 3 (after re-indexing). Any Dedekind ring has this property, as for such a ring the notions "flat" and "torsion free" coincide (e.g., by [B2], ch. VII, $4, prop. 22). We therefore get the first part of Suppose that k is a Dedekind ring. Let k' be a k-algebra and let n E N. a) There is for each G-module N which is flat over k an exact sequence Proposition:

0

-+

H"(G, N) Q k'

+ H"(Gkt,N Q k')

--+

Tor:(H"+'(G, N), k')

-+

0.

b) There is for each H-module M which is flat over k an exact sequence of Gktmodules 0

-+

(R'indZM) Q k'

-,R"ind%:(M Q k') + Tort(R"+'indZM, k')

-+

0

Note that b) follows on the level of k'-modules from a) and 4.10. It is left to the reader to find the Gk.-module structure on the Tor-group and to prove the equivariance of the maps.

Remark: If k' is flat over k, then we get from part a) that H o ( G , N ) Q k' 2: Ho(Gk,,N Q k'), which we know already from 2.10(3) to hold for all N. If k' is not flat, however, such a statement will not be true, even for flat N (in spite of the lemma 1.17 in [Andersen 121). Take, e.g., G = G, and its representation a H

k' = F2.

(L

:")

on k 2 and get a contradiction for k = Z,

Such a formula will, however, hold for acyclic modules as the last term in part a) is then zero. We can, for example (by 4.7.a), take for N a direct summand of some E Q k[G], where E is a flat k-module regarded as a trivial G-module. If N' is another G-module which is finitely generated and projective over k, then Horn(", N) N (N')* Q N is again of this type because of the tensor identity. This shows (for any Dedekind ring k): (1) Let N, N' be G-modules such that N' is jinitely generated and projective over k, and such that N is isomorphic to a direct summand of some G-module E Q k[G] with E flat over k. Then we have for all n E N and for each k-algebra k' a natural isomorphism Ext:(N', N) Q k'

N

Ext'&,(N' Q k', N Q k').

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Cohomology

eiz0

4.19 For any k there is on H'(G,k) = H'(G,k) a structure as an (associative) algebra over k. The multiplication is called the cup product, and it satisfies the usual anti-commutativity formula: If a E H'(G,k) and b E H j ( G , k) then ab = ( - 1)'jba. Furthermore, there is (for each G-module N ) a natural structure of a H'(G, k)-right module on H ' ( G , N ) = @izoHi(G,N). Let us describe these structures using the Hochschild complexes for k and N . We can obviously identify C"(G, k) = k [ G ] and then have to write 8; in the form a;(x) = 1 0 x. Furthermore, we identify C"(G, N ) 0 Cm(G,k) and C"+'"(G,N ) for all n, m E N. For all a E C"(G, N ) and b E Cm(G,k) one easily checks C?"+"'(a0 b) = ( P a ) 0 b (- 1)"a0 (P'b).Hence a 0 b is a cocycle if a and b are. Another simple computation shows, then, that the cohomology class [a 0 b] of a @ b depends only on the classes [a] of a and [b] of b. Then the action of [b] E H"(G, k) on [a] E H " ( G , N ) is defined through [a][b]= [a 0 b]. In the case N = k we get thus the cup product on H'(G, k). Let G’ be a flat group scheme operating on G through group automorphisms. If N is a (G w G')-module (e.g., N = k), then G’acts on each H"(G, N ) , cf. 4.9. This operation can be described using the Hochschild complex. The discussion above shows that G’ acts on H'(G, k) through algebra automorphisms, and that the action of H'(G, k) on an arbitrary H'(G, N ) is compatible with the G'-action, i.e., that H'(G, N ) 0 H'(G, k) + H'(G, N ) is a homomorphism of G'-modules.

0"

+

4.20 We want to discuss H'(G,, k) or (more generally) H'(V,, k) for a free kmodule V of finite rank, say rk(V) = n. Of course, there is a Kunneth formula,

reducing the second problem to the first one. But we shall prefer to formulate our results at once for V in order to keep track of the GL(V)-operation on the cohomology groups (as in 4.19). Choosing a basis we identify k[K] with the polynomial ring k[Tl, T2,., .,T,,]. We get then an N"-grading and an N-grading on the complex C'(V,, k). For each a = (a1,u z , , . . ,a,) E N" let Ci(V,,k), be spanned by all tensor products of monomials such that the degrees of in the factors add up to ai for each i. Set C'(V,, k), equal to the sum of all Ci(V,,k), with m = la( (where ((al,.. .,an)[= ai). Obviously, the Ci(y,k), are G L ( V ) stable whereas the C'(K, k), are not (for n > 1). As the comultiplication is given by A(?) = 1 0 Ti + T j 0 1 for all j , the formulas for the 8’ in 4.14 show aiCi(K,k), c C'''(V,,k), for all a and C?'C'(K,k), c C'+'(K,k),,,. Therefore, we also get gradings for the cohomology groups

(Note that these gradings simply describe the representations of the diagonal

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Representations of Algebraic Croups

subgroup of G L ( V ) on the cohomology resp. of the subgroup of scalar diagonal matrices.) 4.21 We can now easily compute

H’(V,, k).

Lemma: Suppose that k is an integral domain. k q N V as a GL( V)-module. a) If char(k) = 0, then H '( V,,k ) = b) If char$) = p # 0, then H'( V,,k ) = kTP'.

crm=o

Proof: We have obviously H o ( V , , k ) = k and d o = 0, hence H ' ( V , , k ) = ker(8'). This map is given by a'(f) = 1 0 f - A(f) + f 0 1. Because of 4.20(1), the monomials TI"' with a'(fl;=, Ti"))= 0 form a basis of T;" occurs ker(8'). If at least two r(i) are positive, then each T;" 0 with coefficient - 1 in a'(fl;= Ti'") so that this element is different from 0. As a'(1) = 1 0 1 we have to look only at

fl;=,

flj+i

This is certainly 0 if r = 1. We then have to determine all r > 1 with all those binomial coefficients equal to 0. The result is well known and implies the lemma. 4.22 Keep the assumption of Lemma 4.21. The cup product induces a

homomorphism of GL( V)-modules

H ' ( L k ) 0 H'(V,, k ) -,HZ(V,,k). Because of the anti-commutativity of the cup product (i.e., because of = -a'($') for f,f’ E ker(8')) this map has to factor through A2H'(V,, k ) if char@) # 2, and through S2H'(V,, k ) if char@) = 2. Let us denote the image of this map by M . We want to show

f 0 f’ + f’ 0 f

(1)

M z {

A2H'(V,,k) S2H'(V,,k)

if if

char(k) # 2, char(k) = 2.

The image of 8’ in C z ( V , , k )= k [ V , ] 0 k [ V , ] consists of symmetric elements, i.e., of elements stable under f 6f’ H f’ 0 f.If we take two different basis elements f,f’in 4.21, then f @ f’ is not symmetric, hence the class [f][f'] = [f 0 f’] E H 2 ( V , , k) is nonzero. In order to get their linear independence we just have to observe that these tensor products are homogeneous of pairwise different degrees (except for the trivial equality Cf 0 f'l = - Cf’ 0 fl).

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Cohomology

This proves ( 1 ) for char(k) # 2. For char(k) = 2 we have still to show f 0 f # im(8') for any basis element f in 4.21. We can do something more general. Suppose char(k) = p # 0, set

{fi i(f)s =

for 1

i Ip - 1 and

for all f E k [ V , ] , (So P(f) = f @I f if char(k) = 2.) This map is of course induced from the map f H ((1 Q f + f 0 l ) P - 1 0 f p - f p 0 l ) / p on ZITl, ..., T,]. Using this fact (or a direct calculation), we get that p maps ker(d') = H1(V,,k)into ker(d2),hence we get a map P:H'(V,,k) + H2(V,,k). A simple computation shows

B

for all f1,f2 E H1(V,,k). Therefore is additive. Obviously, f? is GL(V)equivariant and satisfies P(af) = ap$(f) for all a E k. Take now for f a basis element from 4.21. Then P(f) is homogeneous with degree p-times the degree off. The only element (up to scalar multiple) in k [ K ] having this degree is f p . As dl(fp) = 0, we get B(f) 4 im(d'). This concludes the proof of ( 1 ) and shows for p # 2 that the P(TP') with 1 < i In and r E N span as a basis a GL(V)submodule in H 2 ( V , ,k ) intersecting M in 0. We claim that we have found all of H2(V,, k ) in case k is a field. We refer to [DG], II,43,4.6 for the proof and just state the result: Lemma: Suppose that k is a jield. a) If char&) = 0, then H 2 ( V , , k ) N A2H1(V,,k). b) Ifchar(k) = 2, then H 2 ( V , , k ) N S 2 H 1 ( V , , k ) . c) Ifchar(k) # 2 , 0 , t h e n H 2 ( V , , k ) - A 2 H ' ( V , , k ) O k B H ' ( I / , , k ) . 4.23 In order to get all of H'( V,,k), we shall reduce its computation to that of the cohomology of finite cyclic groups. This is done using a filtration of the Hochschild complex. Set k [ K , m] for all m E N equal to the span of all monomials

..

~rfl)~r(2)T . r(nf 1 2 n

with r(i) < m for all i. Then the formula A(7J = 10 A(kCV,,ml) = k [ V , , m l 0 k[V,,ml. Set

+ ?;.@I

c j ( E , k , m ) = @ k [ I / , , m ] c @ k [ K ] = cj(V,,k).

1 implies

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Representations of Algebraic Groups

Then we see that ajCj( V,,k , m) c C j + l (V,,k , m).

Hence C'( V,, k, m) = Oj2Cj( V,,k, m) is a subcomplex of C'( V,, k, m). Let us denote its cohomology by H ' ( V , , k , m ) = @ i 2 0 H i ( V , , k , m ) . For m, m' E N with m' Im we have an inclusion C'( V,, k, m') c C'( V,,k, m), hence a homomorphism a,,,,: H’ (V,,k, m') + H'(V,, k, m).We have obviously a,,,. 0 a,.,,.. = a,,,,, for any m" Im'. Similarly, the inclusion C*(V',k, m) -, C'( V,, k ) induces a homomorphism a,: H ' ( 5 , k, m)+H'( V,, k ) with a, 0 a,,,, = a,,, We thus get a homomorphism a: l&H' (V,, k , m)+ H'(V,, k). Obviously, H'(V,, k ) is the union of all a,(H'(V,, k,m)) and for each f E ker(a,) there is m' 2 m with f E ker(a,,,,). This implies limH'(V,, k,m) 5 H'(V,, k). (1) 4 Note that C'(V,,k , m) 0 Cj( V,, k, m) = C i + j (V,,k, m). Therefore we can define a cup product on each H'(V,, k, m), and the a, are homomorphisms of algebras. Hence so is the isomorphism (1). Let me point out that this construction can be generalized to any V,-module M that is finitely generated over k. For such an M there is some r ( M ) E N with A M ( M )c M 0 k [ V,,r ( M ) ] . Then all C'( V,,M , m) with m 2 r ( M ) are subcomplexes of C'(V,,M ) , and we get as above (2)

limH'(V,,M,m) 2 H'(V,,M).

4

4.24 Obviously, we can define a complement Cj(V,, k,m)c to Cj(V,,k , m ) in Cj( V,, k): Take the span of all tensor products of all monomials not belonging to Cj( V,, k, m),i.e., where in at least one factor some occurs with an exponent 2 m .In general the Cj(V,,k, m)’ do not form a subcomplex. Suppose, however, that p is a prime number and that p l = 0 in k. Then A(Tf') = 1 @ TP' + Tf' @ 1 for all i and r. This implies that all Cj(V,, k,p')' are subcomplexes, and that I f J ( & , k, p') is a direct summand of Hj(V,,k). We

may write 4.23(1) in the form (1)

H'(V,, k ) =

u H'(V,, k , p r )

(if p k = 0).

r>O

(We can generalize 4.23(2) in a similar way.) Of course our computations in 4.21/22 are compatible with this formula. In the situation of 4.21.b we have

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Cohomology

4.25 The groups H'(V,,k,p') in 4.24 have a different interpretation. Let p be still a prime and suppose p l = 0 in k. Identify V with k" via the T , and consider the (Frobenius) endomorphism F of V, with F(a,, . ..,a,) = (uy,. ..,a;) for all (al,.. .,a,) E A" = k" 6 A N K ( A ) and all A. This is an endomorphism of algebraic k-groups with F*(ZJ = Tf for all i. The kernel V,,' of F' is therefore also an algebraic k-group with k[V,,r] N k[Tl,. . . ,7J

(T!?,. . ., Ti').(Obviously, E,’ is independent of the choice of the identification I/ N k". Notice that V,,' is isomorphic to the direct product of n copies of the algebraic k-group Go,’introduced in 2.2.) Obviously, the restriction of functions k[V,] + k [ V,,'] induces an isomorphism k [ V,,r] + k [ V,,'] compatible with the comultiplication, hence an isomorphism C'( V,, k, p') + C'(V,,,, k ) of complexes and an isomorphism of algebras

(1)

H * ( V , , ~ P ' ) H'(V,,r,k)-

Any Cj(V,,k,p')' is just the kernel of the restriction map C j ( V , , k )+ Cj(V,,',k). This gives a better reason for 0 C j ( V , , k , p ' ) to form a subcomplex and hence for the injectivity of the map H’(V,,k, p') + H’(V,, k). Again we can generalize (1) to any V,-module M , finitely generated over k, and get (2)

H'(T/,,M,p')%H'(V,,,,M)

if

p' > r ( M ) .

Notice that the gradings on Ha(%, k ) considered in 4.20 induce similar gradings on H’(V,,,, k). 4.26 Let us assume that k is a field of characteristic p # 0. It will be convenient

to suppose for the moment that k contains an algebraic closure of F,. Consider the endomorphism F of V, as in 4.25, and define for each r E N, r > 0 a closed subgroup V,(p') of V, via V,(p')(A)= { u E V,(A)I F'(u) = u}.

It is defined by the ideal generated by all Tf' - with 1 Ii In. Therefore the restriction of functions induces also an isomorphism k [ V,,p'] + k [ V,(p')]

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compatible with the comultiplication, hence an isomorphism (1)

H'(V,,k,p') r H'(V,(p'),k).

If A is an extension field of k, then V,(p')(A) is simply the group of all points in A" having all coordinates in the finite field Fpr.Let us denote this group by V(p'). It is an elementary abelian p-group of order p'". We may regard k[ V,(p')] as the algebra of all functions from V(p') to k. The comultiplication on k [ K(pr)] is given by the group law in the finite group V(p'). Therefore the Hochschild complex for V,(p') computes the cohomology of the finite group V(p'). (Equivalently one can say that the category of I/,(p')-modules is "the same" as the category of k- V (p')-modules.) Now the cohomology of a cyclic group is well known (cf., e.g., [HS]) and the cohomology of an elementary abelian group follows using the Kunneth formula. The results can be formulated as follows: (2) if p = 2, then H'(K,k,p')

N

SH1(V,,k,pr).

We denote here by S ( M ) resp. A ( M ) the symmetric resp. exterior algebra of a k-module M given its natural grading. If we put each element of S'M in degree 2i, then we write S ' ( M ) . (3) if p # 2, then H * ( K ,k,p') H 2 ( V , ,k, P ' ) / A 2 H (V,,k, P').

N

AH1(Va,k,p') @ S'(V') with

V' N

These results are certainly also true if k is finite, e.g., by 4.13.a. 4.27 Combining 4.26(2), (3) with 4.24(1) we get a complete description of H'(G,, k). Before formulating the result, we want to introduce some notation to describe the operation of GL(V ) on the spaces kTP' and kg(T7). Let us assume for the sake of simplicity that k is perfect. We shall use the notation M(') as in 2.16. The map y:f H f P r is an isomorphism y: V*(')-+ kT:' of vector spaces over k. The map y @ i d , on V * ( ' ) Q k = ( V * @ &)(').for some algebraic closure & of k commutes with GL(V ) ( k ) .As G L ( V ) is reduced, this implies that y is an isomorphism of GL(V)-modules. Similarly, one checks that f H p ( f P r ) is an isomorphism V*('+')+ kp( T F ) of GL( V)-modules. This shows:

cI=

c3=

Proposition: Suppose that k is a perfect j e l d of characteristic p # 0. a) if p = 2, then

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Cohomology

and (for all r > 0)

H’(V,,,, k) 2: s(

@;

V*(j').

b) If p # 2, then

and (for all r > 0)

Remarks: 1) The explicit description of H' and H Zalso gives the gradings of the generators of H*(V,,k) and H'(J$k). All elements in V * ( j )are homogeneous of degree p j with respect to the N-grading. 2) If k is a field of characteristic 0, then H ' ( K , k ) 'Y A ( V * ) . This follows, e.g., from the proposition applying the universal coefficient theorem to Z".

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Quotients and Associated Sheaves

Some properties of the derived functors of induction can be proved only by interpreting the R"indzM as cohomology groups H " ( G / H ,9 ( M ) )of certain quasi-coherent sheaves on G / H . Before we can define these "associated sheaves" (5.8/9), and prove the equality R"ind$M = H " ( G / H ,d p ( M ) )in 5.12, we have to introduce the quotients G / H . This is a nontrivial problem. Assuming G to be a (flat) group scheme and H a (flat) subgroup scheme we want G / H to be a scheme. The choice at first sight, the functor A H G ( A ) / H ( A ) ,will in general not be a scheme. On the other hand, there is an obvious definition of a quotient scheme via a universal property (cf. 5.1) that, however, gives no information about existence and what the quotient looks like, if it happens to exist. It has turned out to be useful to construct quotients not at once in the category of schemes over k but in the larger category of all k-faisceaux. These are the k-functors having a sheaf property with respect to the faithfully flat finitely presented (Grothendieck) topology, cf. 5.2/3. The quotient faisceau G / H has a not too complicated description (5.4/5). In the most important cases (e.g., over a field) the quotient faisceau is a scheme (hence the quotient scheme) and has nice properties (5.6/7).It is only in this case that we can prove the relation between sheaf cohomology and the derived functors of induction mentioned above.

73

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One consequence of this relation is that indg is an exact functor if G / H is an affine scheme. One can find direct proofs in [Oberst] and [Cline, Parshall, and Scott 31. The converse is also proved in these papers. The contents of the sections 5.14-5.21 are applied in only a few places later on. They deal mainly with a description of 9(M) as the sheaf of sections of a certain fibration G x H M , + G / H and with computing inverse and (also higher) direct images of 9(M) in special situations. I more or less follow [DG] in the sections 5.1-5.7. Proposition 5.12 was first proved in [Haboush 21. Let me add that closely related material is treated in [Cline, Parshall, and Scott 91. 5.1 (Quotients) For a linear algebraic group G over an algebraically closed

field and a closed subgroup H of G, it is well known how to make the coset space G / H into a variety. We should like to have a generalization to the case where G is a k-group scheme and H a closed subgroup scheme. Unfortunately, the “obvious” choice, i.e., the functor A H G ( A ) / H ( A )turns out to be the wrong one (in general) as it will in general not be a scheme. Let us define instead a quotient via a universal property. This can be done in the more general situation of a k-group scheme G operating on a scheme X over k. A quotient scheme of X by G is a pair ( Y ,n) where Y is a scheme and n: X -, Y is a morphism such that n is constant on G-orbits, and such that for each morphism f:X -+ Y’ of schemes constant on G-orbits there is exactly one morphism f’: Y + Y’ with f ’ 0 n = f. (“Constant on G-orbits” means that each n(A):X ( A ) + Y ( A ) is constant on the G(A)-orbits.) Of course, such a quotient scheme is unique up to unique isomorphism-if it exists (and that is the problem). Let me give another formulation of this definition. We want to assume that G operates from the right. (The necessary changes for left actions will be obvious.) Consider the two morphisms a, a’: X x G + X with a ( x , g ) = x g and a ’ ( x , g ) = x . Then a morphism f:X + Y’ will be constant on G-orbits if and only iff 0 a = f 0 a‘. So ( Y ,n) is a quotient scheme if and only if n a = n a’, and if for all morphisms f : X + Y’ with f 0 a = f 0 a‘ there is a unique morphism f’:Y + Y’ with f ’ 0 n = f. (We assume Y, Y’ are schemes.) So a quotient scheme of X by G is (in categorical language) the cokernel of the pair (a,a ’ ) in the category of schemes over k. This way of formulating the universal property allows for generalizations. Take, for example, a “schematic” equivalence relation on X , i.e., a subscheme R c X x X such that each R ( A ) is an equivalence relation on X ( A ) . Then a quotient scheme of X by R is the cokernel in the category of schemes of the pair of the projections from R to X . There is a generalization of these two 0

0

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Quotients and Associated Sheaves

situations (i.e., of X x G 3 X for group actions and of R 3 X for equivalence relations) called groupoid. This is discussed, e.g., in [DG], III,42, no 1.

5.2 (The fppf-topology) Of course, we can also define quotients by group actions in larger categories than {schemes over k} using the same type of universal property as before, but allowing any Y, Y‘ in that larger category. If we take, e.g., the category of all k-functors, then certainly A H X(A)/G(A)is the quotient. If we now had a functor from {k-functors) to {schemes over k) left adjoint to the inclusion, then it would map A H X(A)/G(A)to the quotient scheme. But we do not have such a functor. It has proved to be useful in this situation to replace the category {schemes over k} by a larger one for which there is such a functor with nice properties. Any scheme X is by definition local (cf. 1.Q i.e., Y H Mor( Y,X )is a sheaf in some sense: If (q)jis an open covering of Y , then any a E Mor(Y,X) is uniquely determined by its restriction to the 5, and one can glue morphisms aj E M o r ( 5 , X ) together if they coincide on intersections. The open coverings were defined using the Zariski topology. One can now consider more general topologies, called Grothendieck topologies, where the property “open” is no longer attached to subsets (or rather subfunctors) but to certain morphisms. We shall consider only the faithfully flat, finitely presented topology (for short “fppf” as the French is much more symmetric in this case), and the k-functors with the sheaf property for this topology will be called faisceaux (reserving the term “sheaf” to objects related to the Zariski topology). As in 1.8, it is enough to consider open coverings of affine schemes by affine schemes. Let R be a k-algebra. An fppf-open covering of R is a finite family R,, R , , , . .,R, of R-algebras such that each Ri is a finitely presented R-algebra, and such that R, x R, x ... x R, is a faithfully flat R-module. (An R-algebra is finitely presented if it is the quotient of a polynomial ring in a finite number of variables by a finitely generated ideal.) Note that R, x R, x ... x R, is faithfully flat over R if and only if each Ri is a flat R-module and Spec(R) is the union of the images of all Spec(Ri), cf. [B2] ch. II,§2, cor. 4 de la prop. 4. For any f E R the R-algebra Rf is finitely presented (as Rf N R[T]/(Tf- 1)) . . ,f,E R satisfy Rfi = R, then Rf, and a flat R-module. If fl,fi,. is faithfully flat, cf. [B2], 11, $5, prop. 3, hence the Rf, form an fppf-open covering of R.

,

nl=,

5.3 (Faisceaux) A k-functor X is called a faisceau if for each k-algebra R and each fppf-open covering R,, R,, . . .,R, of R the sequence

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Representations of Algebraic Groups

is exact. (The maps are the obvious ones, induced by the structural maps R + Ri and by R i - +RiQRRj resp. R i t RjQRRi with a w a Q 1 resp. a H 1 0 a.) (A k-faisceau is defined as a k-functor that is a faisceau.) Ri)For any k-algebras R,, R,, . ..,R, we can regard each Rias a algebra via the projection. The Ri obviously form an fppf-open covering of Ri = R. As Ri QR Rj = 0 for i # j , the exactness of (1) amounts in this case to:

(nl=,

nl=,

(2) The projections induce for all k-algebras R,, ...,R, a bijection X(R, x . * . x R,)G X(R,) x . * . x X(R,). A single R-algebra R’ is an fppf-open covering of R if and only if it is a faithfully flat R-module and a finitely presented R-algebra. Let us call this an “fppf-Ralgebra”. So, the exactness of (1) implies:

(3) If R is a k-algebra and if R’ is an fppf-R-algebra, then X(R) -+ X(R’) 3 X(R’ QR R’) is exact.

So, the arguments above prove one direction of (4)

A k-functor X is a faisceau

if and only if it satisfies (2) and (3).

nLetiRi

nl=,

ni

For the converse one applies (3) to R‘ = Ri and (2) to Ri and QR n j R j * f,,. . . ,f , E R with RA = R. Then the R,,(1 S i n) form an fppf-open covering of R, cf. 5.2. We can identify each R,, ORRf, with Rf,f,. Therefore a comparison of the definitions above and in 1.8 yields: ( 5 ) Each faisceau is a local functor.

Suppose that R’ is a faithfully flat R-algebra. We have then an exact sequence 0 t R + R’ + R’ QRR’ where R -+ R’ is the structural map, and where any a E R’ is mapped to a 0 1 - 1 Q a. (This is only the beginning of a long exact sequence, see [DG], I, $1, 2.7. It is enough to show the exactness of 0 - R Q R R ‘ + R‘ QR R‘ + R‘ QR R’ QR R‘. The last map sends a Q a’ to a Q 1 Q a’ 1 0 a Q a’. If this is 0, then 0 = a Q a’ - 1 Q aa’, hence a Q a’ is in the image of the previous map.) We can also express the exactness above as: (6)

R t R’ 3 R’ QR R’ is exact.

(Here the two maps are a H a 0 1 and a H 1 0 a.) Now the left exactness of Homk-,,,(A,?) shows that each affine scheme SpkA over k is a faisceau.

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Quotients and Associated Sheaves

More generally, one can show ([DG], 111, $1, 1.3): (7) Any scheme over k is a faisceau.

Let M be a k-module and k‘ a faithfully flat k-algebra. Then the same argument as above gives as an exact sequence 0 +M

+M

Q k ’ + M Q k ’ Q k’

with maps rn H m Q 1 and m Q b H m Q b Q 1 - m Q 1 Q b. Applying this to all M Q A we get: (8) For each k-module M the functor M, is a faisceau. The following property is obvious: (9) Let X be a k-functor and k’ a k-algebra. If X is a faisceau, then X,. is a faisceau.

5.4 (Associated Faisceaux) There is a natural construction that associates to each k-functor X a k-faisceau 8 (called the associated faisceau) together with a morphism i : X + 3 such that for all k-faisceau Y the map f H f 0 i is a bijection Mor(8, Y )+ Mor(X, Y ) . We get thus a functor X H 8 from { k-functors} to {k-faisceaux}left adjoint to the inclusion of { k-faisceaux} into {k-functors). This construction should be regarded as an analogue of the construction of a sheaf associated to a presheaf. The details may be found in [DG], 111, $1, 1.8-1.12. I shall describe 2 only in a particularly simple case where X is already close to being a faisceau. To be more precise, I want to assume the following:

(1) X satisjes 5.3(2) and X ( R ) + X ( R ’ )is injective for each k-algebra R and each fppf-R-algebra R’. Under this assumption 3 has the following form. Take a k-algebra A and consider for each fppf-A-algebra B the kernel X ( E , A ) of X ( E ) 3 X ( E 0, B). If E’ is an fppf-E-algebra, then B’ is also fppf over A , and the natural inclusion from X ( E ) into X ( E ’ )maps X ( B , A ) into X ( E ’ ,A ) . More precisely E’ Q, B’ is fppf over EQ,E, hence the standard map X ( E Q , E ) + X ( E ’ Q , E ’ ) is injective, and we can identify X ( E , A ) with the intersection of X ( B ’ , A ) and X ( E ) .The X ( E , A ) with E fppf over A form a direct system. (If El, E , are fppf over A, then El 0, B , is fppf over El and 8 , .) So, we can form the direct limit of these X ( E ,A ) and this is our 8 ( A ) : 8 ( A )=

9X ( E ,A ) .

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Representations of Algebraic Croup

z(A),

As all maps X(B, A ) -+ X(B’, A ) are injective, so are all maps X(B, A ) -, we can identify X ( B , A ) with its image in z ( A )and regard as the union of

z(A)

all X(B, A ) . We see especially:

(3) For X as in (1) each X(A) -+

z(A)is injective.

(For arbitrary X this will not be true.) If A -+ A’ is a homomorphism of k-algebras, then B OAA’ is fppf over A‘ for any fppf-A-algebra B, and the natural map X ( B ) -P X(B OAA ’ ) maps X(B, A ) to X(B OAA’, A’). Taking direct limits we get a map % ( A ) + r?(A’), which is easily checked to be functorial. In this way r? is a k-functor. It is rather obvious that 2 inherits the property (1) from X. Consider any element in the kernel of ?(B) 3 Z ( B OAB ) for some fppf-A-algebra B. Then it belongs to X(B’, B ) for some fppf-B-algebra B‘. The restrictions of the two maps from r?(B) to X(B’, B ) are induced by the maps X(B’, B ) -+ X(B‘ O,(B OAB), B OAB ) 1 X(B’ OAB, B OAB ) c X(B’ OAB’, B OAB ) c OAB ) and X(B‘, B ) X(B’ OA( B OAB), B OAB) 3 X(B OAB’, B OAB) c X ( B ’ OAB‘, B OAB ) c r?(BO,B), where the isomorphism in the second step is induced by b ’ O ( b , 0 b,) H (b’b,)0 b2 in the first case, and by b‘ 0 ( b , 0 6 , ) H 6 , 0 (b’b,) in the second case. (We use here that B’ OAB’ N B’ O,(B OAB‘) N (B’ OAB ) 0, B’ is fppf over B OAB‘ and B’ OAB.) Therefore the intersection of ker(z(B) 3 r?(B OAB)) with X(B’, B ) is equal to ker(X(B’, B ) 3 X(B’@,B’, B Q B ) )= ker(X(B’)=tX(B’OAB‘))= X(B‘, A ) , hence contained in This shows that r? is a faisceau. For any morphism f: X -+ Y into a k-faisceau Y, any f ( B ) x with x E X(B, A ) as above has to belong to Y ( A ) c Y(B),so we can define fg -+ Y through f”(A)x = f ( B ) x E Y ( A ) .This is easily checked to be a morphism and to be unique with J X = f.So, 2 has indeed the universal property we wanted. Notice: If each f( A )is injective, then so is each ?(A). So we can regard X as a subfunctor of Y. One easily gets the following:

z(B

-+

z(A).

(4) Let X be a subfunctor of a k-faisceau Y such that X satisfies 5.3(2). Then

2 is a subfunctor of

Y. One has r?(A) = {x E Y ( A )1 there is a fppf-A-algebra B with x E X(B)}. It is clear in a situation as in (l), but can be proved also in the general

situation, that taking the associated faisceau commutes with base change:

( 5 ) Let X be a k-functor and k‘ a k-algebra. Then associated to Xkr.

(2)k, is the

faisceau

5.5 (Images and Quotients) Let f:X -+ Y be a morphism of k-faisceaux. The subfunctor A I-+ im( f ( A ) ) = f(A)X(A) of Y obviously satisfies 5.3(2). So,

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Quotients and Associated Sheaves

5.4(4) yields a rather precise description of the associated faisceau which is called the image faisceau of f.We shall usually denote this associated faisceau by f ( X ) or im(f). So, in general, f ( A ) X ( A )is properly contained in f ( X ) ( A ) . Let G be a k-group faisceau acting on a k-faisceau X (say from the right). We define the quotient faisceau X / G as the associated faisceau of the functor A H X(A)/G(A). The universal property of the associated faisceau implies that X / G is a quotient of X by G in the category of k-faisceaux. If X is a scheme and if X / G also happens to be a scheme, then X / G is a quotient in the category of k-schemes. In general the functor A H X ( A ) / G ( A )will not satisfy 5.4(1), so, in general, the description of X / G is more complicated than what is done in 5.4. However, let us assume that G acts freely on X , i.e., that each G ( A ) acts freely on X ( A ) . We claim: (1) If G acts freely on X , then the functor A H X ( A ) / G ( A )satisfies 5.4(1). The part about direct products is obvious (for any operation). Consider now some k-algebra A and an fppf-A-algebra B. In order to prove the injectivity of X ( A ) / G ( A )+ X ( B ) / G ( B ) we , have to take x , x' E X ( A ) with xG(B) = x'G(B) in X ( B )and show x G ( A ) = x'G(A).(We use here and below the same notation for x and its canonical image in X ( B ) . ) So, there is g E G ( B ) with x = x'g in X ( B ) . Let g l , g 2 be the images of g in G ( B 6 , B ) derived from the two homomorphisms B + B 0, B with b H b 0 1 resp. b H 1 0 b. We have then x = x ' g , = x'g2 in X ( B @A B), hence g 1 = g 2 as G(B 0, B ) acts freely. The faisceau property of G yields that we may assume g E G ( A ) . The faisceau property of X then implies that x = x'g already in X ( A ) , hence that x G ( A ) = x ' G ( A ) as claimed. We get now from 5.4(3) that the natural map X ( A ) / G ( A )+ ( X / G ) ( A )is injective in the case of a free action. This inclusion will in general be strict. We can express this injectivity obviously as follows: The maps X ( A ) x G ( A )+ X ( A ) x X ( A )with ( x ,g ) H ( x ,x g ) induce an isomorphism: (2)

X x G 7 X xXiGX

(for a free action).

(The fibre product is taken with respect to the canonical projection X + X / G taken twice.) Let us describe an example. Denote by A",+' the open subscheme of A"" that associates to each A the set of all (ao,a,, . . . ,a,,) E A"" with x l = o A a i= A . For each such a = (ao,al,.. . , a n ) there is a linear map rp: A"' + A with q ( a ) = 1. Then An+' = Aa 0 ker(rp) and so a defines a point Aa E P"(A), cf. 1.9. Two vectors a, a' E A"" define the same submodule

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Aa = Aa' if and only if they differ by a factor in A x = G,(A). So a H Aa yields an embedding A;+'(A)/G,(A) 4 P"(A). We claim that this induces an isomorphism (with the obvious, free action of G , on A;+ l )

(3)

A ; + + / G , 2 Pn.

Because of 5.4(4)we only have to show the following: If M E Pn(A)for some A , then there is some fppf-A-algebra B and some P E A;+ ' ( B )with BP = M OAB, where we identify M OAB c A"+' OAB with its image in B"' Equivalently, we have to find B, fppf over A , such that M OAB is free. But by [B2], 11, $5, thm. 1 there are fl, . . . ,f , E A such that A = AJ and such that each M f i is free over A f i .(One can even assume that ( A n + ' / M ) fisi also free. This is used A f i will do, cf. 5.2. when proving 5.6(3).) Now B = If X ' c X i s a G-stable subfaisceau, then A H X ' ( A ) / G ( A )is a subfunctor of A H X ( A ) / G ( A ) This . leads to an embedding of X ' / G into X / G . If we denote the canonical map x H x G ( A ) for all x E X ( A ) by n:X -+ X / G , then X ' / G is identified with the image faisceau n(X'). Notice that 5.4(5) implies for any k-algebra k':

'.

nr=

(4)

xk,/Gk, = ( x / G ) k , .

If a direct product G I x G2 of k-group faisceaux acts on a k-faisceau X , then G1 acts in a natural way on X / G 2 and we get a natural isomorphism (5)

( X / G Z ) / G l 3 X / ( G l x G2).

This follows from the universal property of a quotient faisceau. In case G2 x G2 acts freely one also gets ( 5 ) from the explicit construction in 5.4. In that case the action of G 1 on X / G 2 is also free. Suppose for the moment that k is an algebraically closed field. One can show ( X / G ) ( k )= X ( k ) / G ( k ) ,cf. [DG], 111, $1, 1.15. Suppose that X and G are reduced (hence correspond to varieties).There will be, in general, orbits of G(k) on X ( k ) that are not closed. If so, X ( k ) / G ( k )= ( X / G ) ( k ) cannot have a topology for which the natural map X ( k ) -+ ( X / G ) ( k )is continuous and where all points are closed. Therefore X / G is not a scheme. It is only for very nice operations (as that in (3)) that the quotient faisceau leads to the quotient scheme. Let us mention without proof one example where this happens. We call an affine group scheme G finite if k [ G ] is a finitely generated and projective k-module. ( 6 ) Let X be an affine scheme and G an affine k-group scheme acting freely on X . If G is finite, then X / G is an affine scheme isomorphic to S p k ( k [ X I G ) .

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Though not stated in this way, this follows easily from combining [DG], 111, $2, no 4 and $1, 2.10. The results at the first place imply that k[X] is finitely generated and projective as a k[XIG-module and that X/G is a subfunctor of Spk(k[XIG). The second result quoted implies that the inclusion k[XIG c k[X] induces an epimorphism X+Spk(k[XIG) in the category of kfaisceaux, while on the other hand the image faisceau is equal to X/G. (In [DG], 111, $2, the case where X is not affine or where G does not act freely is also discussed.) 5.6 (Quotientsby Subgroups and Orbit Faisceaux) Let G be a k-group faisceau. The action of any subgroup faisceau H of G on G by right multiplication is obviously free. So we can apply the explicit construction in 5.4 in order to get G/H (cf. 5.5(1)).We see especially that each map G(A)/H(A) --t (G/H)(A) is injective, and that we get an isomorphism (cf. 5.5(2)): (1)

G

X

H

3 G XGiH

G.

Let X be a k-faisceau on which G acts (from the left). Take x E X(k) and consider its stabilizer Stab,(x) in G, i.e., the subgroup functor of G with Stab,(X)(A) = {g E G(A) I gX

= X)

for all A . (In the notation of 2.6(3) this is Stab,(Y), where Y c X is the subfunctor with Y ( A ) = {x} for all A.) It is easy to check that Stab,(x) is a faisceau. We can identify A k+ G(A)/Stab,(x)(A) with a subfunctor of X, hence also its associated faisceau G/Stab,(X). More precisely, the morphism n,: G -+ X, g H gx factors through G/Stab,(x) and induces an isomorphism (2)

G/Stab,(X) 3 im(n,)

onto the image faisceau of n,, which is also called the orbit fuisceuu of x. Take for example G = GL, +,, for some r, n E N, # 0. The natural action of G on A"" = (kr+"),induces an action on the Grassmann scheme 4,,, cf. 1.9. Let x E $,,,,(k) be the direct summand {(xi,x2,.. .,xr,O,O,. ..,0) E kr+"I x i , . . ., x, E k } . Set P, = Stab,(x). Then G(A)x consists of all direct summands M of A"" such that M is free of rank r and A'+"/M is free of rank n. Arguing as in 5.5(3) one shows (3)

GLr+nIPr3 4 , n .

Let us consider another example. Let A be a finitely generated abelian group and A' be a subgroup of A. We can regard (cf. 2.5) Diag(A/A') as a subgroup

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functor of Diag(A): Diag(A/A')(A) = { cp E Horn,,(& A " ) I cp(A') = 0} c

Hom,,(A, A " ) = Diag(A)(A)

for each k-algebra A . Then cp H rp,,I defines an embedding Diag(A)(A)/Diag(A/A')(A)4 Diag(A')(A),

which we claim to yield an isomorphism (4)

Diag(A)/Diag(A/A') r Diag(A').

We have to show that there is for any k-algebra A and group homomorphism cp: A' -+ A x an fppf-A-algebra B and a group homomorphism (p: A -+ B" extending cp. By induction we can assume that A = Z l + A' for some , IE A. There is n E N such that Z l n A' = Znl. Then take the polynomial algebra A[t] and its homomorphic image B = A[t]/(t" - cp(nA)) and map 1 to the residue class of t. More generally, consider a homomorphism a: A2 -+ A1 of finitely generated abelian groups, and the corresponding homomorphism Diag(a): Diag(A,) + Diag(A2) of diagonalizable group schemes. One can obviously identify (5)

ker(Diag(a)) N Diag(Al/a(A2))

and the discussion above yields (6)

im(Diag(a)) N Diag(A,/ker(a)).

In this situation, one can also define a cokernel of Diag(a) as the quotient of Diag(A,) by im(Diag(a)) and get (7)

coker(Diag(a)) N Diag(ker(a)).

It has the usual properties of a cokernel. Recall that an affine scheme X over k is called algebraic if k[X] is a finitely presented k-algebra. (8) Suppose that G is an algebraic k-group. If k is a j e l d , then G / H is a scheme for each closed subgroup scheme H of G.

One can find a proof in [DG], III,§3, 5.4. In the case where G and H are reduced, then so is G / H and one gets the same object as in [Bo], 6.8 in a different language. The general case is reduced to this special case using a generalization of 5.5(6).

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For arbitrary k, G, H the quotient faisceau G / H will not be a scheme, cf. the counter examples in [DG], 111, $3, 3.3 and in [Ra], p. 157. Let us however mention: (9) Let G be an algebraic k-group. If k is a Dedekind ring, then GIH is a scheme for any closed and flat subgroup scheme H of G.

We shall use this result only in cases where it is clear for other reasons. A proof is contained in [A], Thm. 4.C. 5.7 (Flatness of Quotients) Let X be a k-scheme and let G be a k-group scheme acting freely (on the right) on X. Let n:X -+ X / G be the canonical map. Then:

(1) Suppose that X/G is a scheme. For any open and affine subscheme Y of X / G the inverse image n- ( Y) is open and affine in X. If G is flat (resp.flat and algebraic), then k[n-’(Y)] is faithfully flat over k[Y] (resp. an fppf-k[Y]algebra).

One usually expresses this in the form: n is affine, faithfully flat (for G flat) and finitely presented (for G flat and algebraic). In [DG], 111, §3,2.5/6 one can find a proof of (1) in the special case where a subgroup operates by right multiplication on the whole group. Let us describe the main steps of the proof in order to show that it generalizes to our situation. To start with, n-’(Y) is a G-stable and open subscheme of X with n-’(Y)/G N Y. So we can replace X by n-’(Y) and can assume Y = X/G. Set A = k[Y]. Let us regard id, as an element of Y(A). By the construction of X/G there is an fppf-A-algebra B such that id, is in the image of X(B). Set X = SpkB for such a B. Using the identification Mor(X‘, X) = X(B) we get a morphism a: X’ + X such that n: o a: X = SpkB + S p k A = Y is given by the structural homomorphism A --t B. The isomorphism X x G + X x, X from 5.5(2) induces an isomorphism X’ x G -+ X’ x,X given by ( x ’ , g )H (x’,a(x‘)g).So, the faithfully flat base change X’ -+ Y applied to n : X -P Y yields the first projection X’ x G * X’. As G is affine, so is this projection, hence also X, cf. [DG], I, 82, 3.9. Furthermore, we get B OAk[X]

N

k[X’ x y X ]

N

k[X’ x G]

N

k[X’] O k[G].

So, if k[G] is flat (hence faithfully flat as k is a direct summand of k[G]) over k (resp. fppf over k), then so is B OAk[X] over B, hence k[X] over A, cf. [DG], I, §3, 1.4. Let me mention one consequence of (1): (2) If X and G are frat and if X/G is a scheme, then X/G is flat.

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Indeed, for any Y as above k [ n - ' ( Y ) ] is flat over k and faithfully flat over k [ Y ] . So k [ Y ] is flat over k. Another corollary of the flatness is the openness of n: (3) Suppose that X / G is a scheme and that G is flat and algebraic. if Y is an open subfunctor of X , then n( Y ) is open in X I G . Let me quote [ D G ] , I, §3,3.11 together with the proof of 111, §3,2.5 for this result. (In the noetherian case one can also use [Ha], 111, exerc. 9.1.) 5.8 (Associated Sheaves) We mentioned in 1.11 that to each k-scheme X there corresponds a topological space 1x1 together with a sheaf of k-algebras. The open subsets of correspond bijectively to the open subfunctors of X . So we can describe a sheaf on as a contravariant functor from {open subfunctors of X } (with inclusions as morphisms) to some other category having the usual sheaf property with respect to open coverings of open subfunctors (defined in 1.7). For example, the structural sheaf Ox associates to each open subfunctor Y the k-algebra Ox(Y)= Mor(Y,A') = k [ Y ] . Let G be a flat k-group scheme acting freely (from the right) on a flat kscheme X such that X / G is a scheme. Let us denote the canonical morphism by n:X + X / G . We want to associate to each G-module M a sheaf 9 ( M ) = Y x i G ( M on ) X / G . We set for each open subfunctor U c X / G

1x1

(1)

1x1

9Ww)= {fE Mor(n-' u,Ma)I f ( x g ) = s-'f (4 for all x E (n-’ U ) ( A ) ,g

E

G ( A ) and all A } .

If n-’ U is affine, then Mor(n-' U, M,) N M,(k[n-' U ] ) = M @ k[n-' U ] . This is a G-module via the given action on M and the operation on k[n-’ U ] derived from the action on n-’ U c X . Then obviously (for n-l U affine) (2)

Y ( M ) ( U )= ( M @ k[n-'U])G.

Any fl E Oxlc(U )can be regarded as a G-invariant element in OX(n-' U ) . So we can multiply any f E 9 ( M ) ( U )with f l and still get an element of Y ( M ) ( U ) . Thus Y ( M ) ( U )is an OxiG(U)-module.For two open subfunctors U ' c U of X / G we have an obvious restriction map 9 ( M ) ( U )-+ 9 ( M ) ( U ' ) .Thus Y ( M ) is a presheaf of OxiG-modules. Consider the morphisms y u : ( n - ' U ) x G + C ' U , ( x , g ) H xg and fl: M, x G + M,, ( m , g )H g-'m. Then some f E Mor(n-'U,M,) belongs to Y ( M ) ( U )if and only if f 0 yv = fl 0 (f x idG).So we have an exact sequence (3)

Y ( M ) ( U )+ Mor(n-' U, M,) 3 Mor(n-' U x G, Ma).

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Because of 5.3(8) the functors U H Mor(n:-' U, Ma) and U H Mor(7t-I U x G, Ma) are sheaves. An elementary argument yields now that Y ( M ) is also a sheaf (of OX/,-modules). It is called the associated sheaf (to M on X / G ) . 5.9 Let X , G, n: be as in 5.8. Any homomorphism cp: M induces a homomorphism of Lo,/,-modules:

(1)

%cp):

B(M)

+

w w ,f

H%

-P

M' of G-modules

f.

So 9is a functor from {G-modules} to {Oxl,-modules}. Proposition: a) The functor Y is exact. b) For each G-module M the OxlG-moduleY ( M ) is quasi-coherent. c) If a G-module M is Jinitely generated over k, then Y ( M ) is a coherent Ox,cmodule. Proof: a) It is enough to show that M H Y(M)(U) is exact for each open and affine U c X / G . For such U, U' = n:-' U is also affine and k[U'] is faithfully flat over k[U], cf. 5.7(1). It is therefore enough to show that

(2)

M H k[u'] @k[u]y(M)(U)= k[u'l @k[u](k[U'] 8 MIG

is exact, cf. 5.8(2). The isomorphism in 5.5(2) induces an isomorphism U' x G + U' x u U' compatible with the actions of G (on the second factors). So the corresponding isomorphism k[U'] @,,,,k[U'] 'v k[U'] @ k[G] is also G-equivariant. As (k[G] 0 M)' N M, cf. 3.7(6), the functor in (2) can be identified with M H k [ U ' ] @ M. It is exact as X is assumed to be flat. b) For each scheme Y and each k-module M, the sheaf U H Mor(U, Ma) is quasi-coherent. (If Y is affine, then it is the quasi-coherent sheaf associated to the k[Y]-module k[Y] 0 M, cf. Yoneda's lemma 1.9.) The sheaves U H Mor(n:-' U, Ma) and U H Mor((7c-I U ) x G, Ma) on X / G are direct images of such sheaves, hence quasi-coherent, cf. [Ha], 11, 5.2.d. (We can reduce to the affine case using 5.7(1).)By 5.8(3) we can now regard 9 ( M ) as a kernel of a homomorphism between quasi-coherent OX/,-modules. So it is quasi-coherent itself, cf. [Ha], 11, 5.7. c) We have to show for all U c X/G (open and affine), that Y(M)(U) is finitely generated over k [ U ] .As k[n:-' U] is faithfully flat over k[U],cf. 5.7(1), it is enough (cf. [B2], ch. I, $3, prop. 11) to show that k[n:-' U] @k[u] Y(M)(U) is finitely generated over k[n:-'U]. This module is isomorphic to k[n:-' U] 0 M as we saw in the proof of part a), so the claim is obvious. 5.10 (Examples) Let us mention a few trivial cases. For the trivial G-module k one has 9 ( k ) ( U )= Mor(n-'U,A')' = Mor(n:-'U/G,A') = Mor(U,A'),

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hence (1) Consider, on the other hand, the G-module k[G] under pl or, more generally, M 0 k[G] for any k-module M regarded as a trivial G-module. For any U c X/G open and affine we can identify Mor(n-'U,(M 0 k [ G ] ) , ) z Mor(n-' U x G , Ma).Then the G-invariance condition translates into f(x, 9') = f(xg,g-'g') for all x, g, 9'. The map (x, g ) H (xg,g ) is an automorphism of n-’ U x G and transfers the condition into f(x, gg') = f(x, 9'). In this way 9 ( M ) ( U ) is identified with Mor(n-'U,M,) = LF'x,l(M)(~-'U)= (n*9x,l (M))(U).This implies

5.11 Let X, G, K be as in 5.8.

Proposition: The functor M H Ho(X/G,9(M)) from {G-modules} to {kmodules) is left exact. If X is afine, then its derived functors are M H H"(X/G, =WM)). Proofi The first claim is clear from 5.8.a and the left exactness of Ho(X/G,?). In order to get the second claim, it is enough to show that LF' maps injective G-modules to acyclic sheaves, cf. 4.1(3). By 3.9.c it is enough to consider G-modules of the form M 0 k[G] with G operating trivially on M. As n is affine (cf. 5.7(1)) we get from 5.10(2) and [Ha], 111, exerc. 4.1

H"(X/G, Y ( M 0 k[G]))

N

H"(X, 9,,1(M)).

If X is affine, these cohomology groups vanish, cf. [Ha], 111, thm. 3.5. Let G be a Jlat k-group scheme and H a subgroup scheme of G such that G/H is a scheme. a) There is for each H-module M and each n E N a canonical isomorphism of k-modules 5.12 Proposition:

(1)

R"indgM z H"(G/H, 2&(M)).

b) If G/H is noetherian, then R'indg = 0 for all n > dim(G/H). c) Suppose that k is noetherian and that GfH is a projective scheme. For any H-module M that is Jinitely generated over k, each R"indgM is also finitely generated over k.

Quotients and Associated Sheaves

87

Proofi As the forgetful functor from {G-modules} to {k-modules} is exact, the R"indg are also the derived functors of indg regarded as a functor from {H-modules} to {k-modules}, cf. 4.1(2). Comparing 5.8(1) and 3.3(2) we get (2)

ind$M = YG,H(M)(G/H) = Ho(G/H, Y G / H ( M ) ) .

So (1) follows from 5.1 1. Now b) and c) are translations of well known results on sheaf cohomology, cf. [Ha], 111, 2.7 and 5.2.a. Remark: We get from (1) a G-module structure on each H"(G/H, Y G I H ( M ) ) . The existence of such a structure (satisfying (1)) is, however, already clear for other reasons: Each Y G , H ( Mis) a G-linearized sheaf in the sense, say, of [MF], p. 30. For such sheaves one gets the G-module structure on the cohomology groups using the functoriality of the cohomology.

5.13 Corollary: Let G be a flat k-group scheme and H a JEat subgroup scheme of G. a) r f G/H is an afine scheme, then indg is exact. b) If H is a j n i t e group scheme, then indg is exact. Proofi a) Combine 5.12 with [Ha], 111, 3.7. b) Because of 5.5(6) this is a special case of a). Remarks: 1) One can prove a) directly using 5.7(1) without going through the construction of the associated sheaves, cf. [Oberst] or [Cline, Parshall, and Scott 31. 2) If k is a field, then the converse of a) also holds, i.e., if indg is exact, then G/H is affine. This is proved in [Oberst]. The special case where k is algebraically closed and where the groups are reduced was proved independently in [Cline, Parshall, and Scott 31. There is a more recent proof in [Donkin 91, 12.4.

5.14 (Associated Fibrations) Let G be a k-group faisceau acting freely (from the right) on a k-faisceau X . Let Y be a k-faisceau with a left G-action (not necessarily free). Then G acts via (x,y)g = (xg,g-'y) freely on X x Y. We denote the quotient (X x Y ) / G for this action by X x G Y. Let x : X + X / G be the canonical map x H xG(A). Then the map X x Y -,X / G , (x,y)H n(x) is constant on G-orbits, hence induces a morphism n,:X xGY --$ X / G with (x,y)G(A)--$ xG(A). For any k-algebra A and any x E X ( A ) the map y H (x,y)G(A)is a bijection from Y ( A ) onto the inverse

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image n;'(n(x))in ( X x G Y ) ( A ) .That is why we call xy a fibration with fibre Y, or more precisely the fibration over X / G associated to the faisceau with G-action Y. Let me mention two trivial cases. If Y = G (resp. X = G) with the G-operation via left (resp. right) multiplication, then ( x , g )H xg (resp. ( 9 ,y ) H gy) induces an isomorphism (1)

X xGGrX

resp. G x G Y r Y.

(2)

The inverse maps are x H ( x , l ) G resp. y H (1, y)G. If we let G operate on the left hand side in (1) resp. (2) via right (resp. left) multiplication on G, then the maps in (1) and (2) are G-equivariant. The map ( x ,y ) H ( x ,( x ,y)G) induces (for all X , Y ) an isomorphism (3)

x

x Y r X XX/G(X X G Y ) .

The inverse maps any (x',( x ,y)G) to (x',gy), where g is the unique element in G with x' = xg. (In the special case (1) we get just 5.5(2).) So, if we apply the base change n to ny,then we get the first projection X x Y + X . If Y is an affine scheme, then zy is an affine morphism, e.g., by [ D G ] , 111, $1, 2.12. This means by definition ( [ D G ] , I, §2, 3.1) that for any affine scheme 2 and any morphism 2 + X / G , 2 x ~ , ~ x( GXY ) is also an affine scheme. So if (UJi is an open covering of X / G by affine subschemes, then ( z ; ' U ~ )is~ an open covering of X x G Y by affine schemes. As X x G Y is a faisceau, hence a local functor, we get the first parts of

(4) Suppose that X / G is a scheme and that Y is an ajine scheme. Then X x G Y is a scheme. For any U c X / G open and afine, n;'(U) c X x G Y is open and afine. If Y is flat (resp.flat and algebraic),then k [ z ; ' ( U ) ] is flat over k [ U ] (resp. an fppf-k[U]-algebra). For the proof of the last part we have to argue as in the proof of 5.7( 1). This is left to the reader. As in 5.7, we get from (4)that X xG Y is flat if X / G and Y are so (and if the other assumptions in (4)are satisfied). We recover the results in 5.7 by taking Y = A'. 5.15 (Sheaves and Fibrations) Let G be a k-group scheme acting freely (from the right) on a scheme X such that X / G is a scheme. We can apply the construction

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of 5.14 for any G-module M to Y = Mawith the obvious G-operation. We shall write (by abuse of notation) X x G M = X x G Ma and n, = n,,. A section of ny on an open subscheme U of X / C is a morphism s: U + X x G M with nM s = id,. Let us denote the set of all such s by T(U, X x G M ) . We claim that there is a canonical bijection (for each U ) 0

Indeed, each f E Y ( M ) ( U )c Mor(n-'U,M,) defines a morphism f l : n-lU + X x G M via x H ( x , f ( x ) ) G .Then n, 0 f l = f. As f l is obviously constant of G-orbits, it induces a morphism sf: U + X x G M with n, 3 sf = id,, i.e., with sf E T(U, X x G M ) . On the other hand, take any s E T(U,X x G M ) . We compose the morphism x H (x,s(n(x)))from n-lU to n-'U x X I G ( Xx G M ) with the isomorphism y : X x X I G ( Xx G M ) % X x Ma inverse to the map in 5.14(3), and then take the second projection X x Ma + Ma. This yields a morphism f,:n-lU -+ Ma that can be checked to be in LF(M)(U). In more down to earth terms: If x E n-'U(A), then f , ( x ) is the unique element in M 0 A such that s(xG(A))= ( x , f , ( x ) ) G ( A ) .Now it is elementary to show that s H f, and f~ sf are inverse maps. For more details one can consult [Cline, Parshall, and Scott 91, 1.3. We get via (1) a structure as an OxIG(U)-moduleon r(u,x x G M ) .It is left to the reader to construct this structure directly.

5.16 (Local Triviality) Let G be a k-group faisceau acting freely (from the right) on a k-faisceau X and let n:X + X / G be the canonical map. We call n locally trivial if there is an open covering (Ui)iof X / G such that there exist morphisms a,:Ui -,X with n 0 a, = id,, for all i. (So the a, are local sections of n.)Then (u,g) H a,(u)g is an isomorphism Ui x G % n-'(Q) for all i. It is compatible with the action of G, which is given on the left hand side by right multiplication on G. The inverse map is given by x H (n(x),g), where g is the unique element in G with oi(n(x))g= x . Suppose from now on that n is locally trivial and let Ui, ci be as above. Let Y be a k-faisceau with a G-action from the left. We want to consider X x G Y and n y : X x G Y + X / G as in 5.14. We now get for each i , isomorphisms

n;'(ui)= n-'(ui) X G Y

N

(Q x G)

XG

Y

N

ui x

(G

XG

Y )N

u; x Y

such that Ui x Y -,n; '( Ui)is given by ( x ,y ) H (ai(x),y)G. Under this identification the restriction of ny to x i 1 ( & ) is just the first projection Ui x Y -+ Ui.

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That is why we call I I in ~ this case a locally trivial Jibration. As the zi1(Ui)form an open covering of X x G Y we can conclude: (1) Suppose that X/G and Y are schemes and that IT is locally trivial. Then X x G Y is a scheme. If both X/G and Y are frat (resp. smooth, resp. algebraic) then so is X x G Y.

Consider now the special case where Y = Ma for some G-module M . For all i and J the two identifications (Uinq)x M a S n ; ; I 1 ( Q nL$) derived from oi and ojdiffer by the map (x,m) H (x,gm), where g is the unique element in G with oi(x)g = aj(x). This shows: If M is a projective k-module of rank n < 00 (and if IT is locally free), then X x GM is a vector bundle of rank n over X / G (in the sense, say, of [Ha] 11, exerc. 5.18). It is called the bundle associated to the G-module M. Furthermore, 5.15(1) implies that Y ( M ) is the locally free sheaf associated to this vector bundle in the usual way. Let us state explicitly: (2) If IT is locally trivial and if M is a projective k-module of rank n < 00, then Y ( M )is a locally free sheaf of @,,,-modules of rank n. 5.17 (Inverse Images) Let H be a flat subgroup scheme of a flat k-group scheme G. Let X (resp. X ’ ) be flat k-schemes with free actions (from the right) of G (resp. H) such that X/G and X‘/H are schemes. Denote the canonical maps by 11: X -,X / G and d:X’ + X’/H. Suppose that we have an H-equivariant morphism cp: X’ X. Then cp induces a morphism @: X‘/H + X / G with x’H + cp(x’)G. For any G-module M we can look at the inverse image @*9x/G(M) of the Lo,,,-module Y‘/G(M). We want to compare it with the @xgl,-moduleY x , , H ( M )where we regard M as an H-module (via resg). Consider open subschemes U c X / G and U’ c X’IH with @(U’)c U. Then c p ( d - ’ U ’ )c z-’U and the restriction to 71 - ‘ U of f 0 cp for any f E Y x / G ( M )U ( ) belongs to Y x f / H ( M ) ( U ’ )This . yields a homomorphism ) Y X c I H ( MWe ) . claim: from @ * Y x / G ( M to (1) If II is locally trivial, then @*Yx/G(M)3 Y x , / H ( M ) .

As this is a local problem, we can assume that X’/H and X/G are affine and that X N X/G x G (compatible with the G-action). By 5.7(1) X’ is also affine.

Now Y x / G ( M )is the unique quasi-coherent sheaf on X/G with global sections (k[X] 0 M)‘

2:

Mor(X/G x G, Ma), = Mor(X/G, Ma) E k[X/G] 0 M .

Therefore, @*9x,G(M)is the unique quasi-coherent sheaf on X’/H with global

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sections k [ X ’ / H I @ r x / ~ l ( k [ X / GO ] M ) N k [ X ’ / H ]O M N

N

Mor(X’/H,M,)

Mor(X’, Ma)H= Y x , / H ( M ) ( X ’ / H ) ,

It is left to the reader to check that these identifications yield the same homomorphism as constructed above. Remark: One can generalize (1) to the case where H is no longer a subgroup scheme of G,but where one has a homomorphism a: H -+ G and where cp is compatible with a (ie., cp(x’h) = cp(x’)a(h)).Then one has to replace the right hand side in (1) by Yx,,H(a*(M)),with a * ( M ) as in 2.15. 5.18 (Direct Images) Let G , H be flat k-group schemes and X , Y flat k-

schemes, such that G (resp. H) operates freely from the right on X (resp. Y ) , and such that G operates from the left on Y. Furthermore, assume that both operations on Y commute (so that (gy)h = g(yh) always). Then (x,y)(g,h)= (xg,g-’yh) yields a free operation of G x H on X x Y. Set Z = ( X x Y ) / ( G x H ) . As we can take the quotient in steps (cf. 5.5(5)), we have isomorphisms (1)

z N (X

XG

Y)/HN

x

xG(Y/H).

We want to assume that Z , X / G , and Y / H are schemes and that Y is Z -+ X / G be affine. Let n:X --t X / G be the canonical map, and let n’ = nYiH: the fibration with fibre Y / H arising from the identification Z N X x G ( Y / H ) as in (1). For any open subscheme U c X / G the inverse image of n’-’U c Z in X x Y is equal to n-’ U x Y. So we have for any (G x H)-module M : (2)

nTI;Y’(M)(U) = Mor(n-’U x Y , M , ) G x H .

We have a natural action of G on H o ( Y / H ,Y y / H ( M ) ) = M o r ( Y,

= (k[ Y ] @ M ) H

derived from the representation of G x H on k [ Y ] 0 M via 3.2. (Recall that for any affine scheme Y’ Yoneda’s lemma implies Mor( Y’,Ma)N k [ Y ’ ]O M.) The associated sheaf on X / G , for any U as above, satisfies

We can associate to any f E Mor(n-’ U,( M o r (Y,M J H ) , ) a morphism F : n-’U x Y -+ M, via F ( x , y ) = f ( x ) ( y ) . One easily checks that F is

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Representations of Algebraic Groups

H-equivariant and that f

f

E

H

F is G-equivariant. So, for

Mor(n-lU,(Mor(Y,M,)H),,)G

we get F E Mor(71-l U x Y, M J G x H . Hence f (4)

HF

defines a homomorphism

L?X/G(Ho(y/H, d % Y / H ( M ) ) ) 3 x;L?Z(M)

that we claim to be an isomorphism. In order to check this, we can restrict ourselves to affine U. Then n-’ U c X i s also affine by 5.7(1). So, the right hand side in (3) is isomorphic to ( k [ n - ' U ] 0 Mor(Y,

N

( k [ n - ' U ] 0 ( k [ Y ]0 M ) H ) G

N

(k[n-'U] 0 k [ Y ] 0 M ) G X H

N

(k[n-'U x Y ] 0 M ) G " H

N

Mor(n-' U x Y,

H,

using the flatness of k[n-' U ]for the second step. This yields the isomorphism in (4). It is clear that the construction is functorial in M , i.e., that we have an isomorphism of functors

5.19 (Higher Direct Images) Keep all the assumptions and notations from 5.18. We want to discuss the derived functors of those in 5.18(5). As Y is affine, the functor Ho(Y/H,?)0 L?y/H has, by 5.11, as derived functors the Hi(Y/H,?)0 ZylH. This is still true if we regard Ho(Y/H,?)0 LYYlHas a functor on { G x H-modules) with values in {G-modules}, arguing as in the proof of 5.12.a. So the left hand side in 5.18(5) has derived functors L?x/G H i ( Y / H ,?) 0 L?yiH as L?x,Gis exact, cf. 4.1(2). We want to prove for all i E N: 0

By the remarks above, by the exactness of SfZ, and by 4.1(3) we just have to show that YZmaps injective G x H-modules to sheaves acyclic for x;. Let nz:X x Y -,2 = (X x Y ) / ( G x H ) be the canonical map. Then n’ 0 nz = n p r , where prl:X x Y + X is the first projection. By 5.7(1) both n, and n are affine. So is prl as Y is affine. Hence, 0

0 = R i ( x 0 p1)* = R * ( d 0 nz)* = Rin; 0 nz

for all i > 0 because of [GI, Theoreme 11, 3.1.1.

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Quotients and Associated Sheaves

For any injective (G x H)-module M , the sheaf Y z ( M )is a direct summand of (nz)*2Ffor some sheaf 9on X x Y, cf. 5.10(2). So Rin’,(Yz(M))= 0 for all i > 0. This concludes the proof of (1). If we apply the Leray spectral sequence H j ( X / G ,R’n’,?) =. H i f j(2, then we get for any (G x H)-module M a spectral sequence (2)

H j ( X / G ,Yx,G(H’(Y/H,2 y / ~ ( M ) ) )* ) Hi+’ (2,= K Z W ) ) *

Remark: Let G’ be a flat group scheme acting on X from the left such that the actions of G’ and G commute (i.e., g’(xg) = (g’x)g always holds). Then G’ also acts on X x Y (compatibly with G x H), on X / G , and on 2. The maps in 5.18(1) are G’-equivariant. Assume also that X is affine. Then G’ acts on any H o ( X / G ,Y x , G ( M ’ )and ) H o ( Z ,Y z ( M ) )for any G-module M ‘ and any (G x H)-module M . We can regard the functors in 5.18(5) as functors with values in { G’-linearized sheaves} and then still get an isomorphism, similarly for 5.19(1). Especially the isomorphism

H o ( X / G ,Y X / G ( H o ( Y / H ,

YY/H(M))))

H o ( Z ,yZ(M))

is G’-equivariant. In the special case where H c G are subgroups of G’ and where Y = G, X = G’ (with the operation by left or right multiplication), then we get back the transitivity of induction (3.5(2)) and the corresponding spectral sequence (4.5.c).

5.20 Keep all assumptions and notations from 5.18. Suppose additionally that we have a homomorphism a: G 4H and y o E Y(k)with gy, = yoa(g) for all g E G ( A ) and all A . Let us embed G into G x H via d : g H (g,a(g)). Then the map cp: X + X x Y, x H (x, y o ) is G-equivariant. We can therefore apply 5.17(1)to any (G x H)-module M and get:

(1) If X x Y 4 2 is locally trivial, then @ * Y z ( M )z gx/G(a *kf). If we are in the situation of the remark to 5.19, then this is an isomorphism of G’-linearized sheaves. 5.21 Let k be noetherian. Consider an algebraic k-group G acting freely (from

the right) on a k-scheme X such that X / G is an algebraic scheme, and such that the canonical map n:X + X / G is locally trivial. (Some of these assumptions are made only to simplify some proofs.) Let X c X be a G-stable subscheme. We claim: (1) I f X ’ is closed in X , then so is X‘/G in X / G .

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Representations of Algebraic Groups

Using 1.13 we can assume that X z X / G x G, hence X ' z X ' / G x G implies XG /' = X'/G. = X'/G x G, cf. 1.14. So X ' = and In general we get: (2)

r is G-stable with X ' / G = X'/G.

Indeed, the operation X x G --+ X is continuous and maps X ' x G to X ' , is G-stable. Furthermore, F I G =I X'/G hence X m = x G to F.So is obvious. In order to prove equality it is enough to do so on an open covering (cf. 1.12(5)).So we may assume X z X / G x G, hence X ' = X ' / G x G and = X'/G x G = F / G x G and, finally, F / G = X'/G. Furthermore 5.7(3) implies: (3) If X ' is open in F,then X ' / G is open in F I G . If one wants to prove similar results without assuming n: to be locally trivial, then one has to apply results like [DG], I, 52, 5.3, 111, 51, 2.12, and I, 52, 3.3.

6

Factor Groups

If G is a k-group faisceau and N a normal subgroup faisceau of G, then GIN is again a k-group faisceau and has the universal property of a factor group. This and related things are described in 6.1/2 following [DG]. In this chapter we discuss the relation between the representation theories of G , N , and GIN under the assumption that they all are flat group schemes. The results are usually generalizations of known theorems in the case of abstract group theory like, e.g., the Lyndon-Hochschild-Serre spectral sequence in 6.6 or the Clifford theory in 6.15/ 16. More or less all necessary references have been given before. Let me add that 6.12 generalizes 3.1 in [Andersen and Jantzen]. 6.1 (Factor Groups) Let G be a k-group faisceau and N a normal subgroup faisceau of G. Obviously A H G ( A ) / N ( A )is a k-group functor. Then so is the associated faisceau G I N . This follows (on one hand) from the universal

property (cf. [DG], 111, 83, 1.2) and is (on the other hand) clear from the construction in 5.4/5: For any g, g’ E ( G / N ) ( A )there is an fppf-A-algebra B with g, g’ both in the kernel of G ( B ) / N ( B )3 G ( B B ) / N ( B@A B ) . As these maps are group homomorphisms gg’ and g-l also belong to the kernel. This easily yields the group structure on each ( G / N ) ( A ) Furthermore, . it is simple to see that all maps ( G / N ) ( A )+ ( G / N ) ( A ’ )and G ( A )-,( G / N ) ( A ) are group

95

96

Representations of Algebraic Groups

homomorphisms. Hence, G I N is a k-group faisceau and the canonical map n:G -, G / N is a group homomorphism. We call GIN the factor group of G by N . Note that G I N has the universal property of a factor group: If cp: G -, G' is a homomorphism of k-group faisceaux with N c ker(cp), then there is a unique group homomorphism (p: G / N + G' with (p o n = cp. (As cp is constant on the N-cosets, the universal property of GIN as a quotient faisceau gives the existence of @ as a morphism. It is immediately seen from the construction that (p is a group homomorphism. This also follows from the uniqueness of (p.) For any homomorphism c p : G + G ' of k-group faisceaux the kernel ker(cp) is a normal subgroup faisceau of G. We can identify G/ker(cp) with the image faisceau im(cp), which is a subgroup faisceau of G'. This is really a special case of an orbit faisceau as we can make any g E G ( A ) operate on G ' ( A )as multiplication with cp(g). 6.2 (Product Subgroups) Let G be a k-group faisceau and let H , N be subgroup faisceaux of G such that H normalizes N. We can then form the semi-direct product H M N , and have a natural homomorphism H DC N + G, (h,n) H hn with kernel isomorphic to the intersection H n N (cf. 2.6). Both H M N and H n N are k-group faisceaux. We denote the image faisceau of the homomorphism H M N --* G by H N and call it the product of H and N. It is a subgroup faisceau of G with (1)

(H

DC

N ) / ( H n N ) 1: H N .

The definitions imply for any k-algebra A ( H N ) ( A )= { g E G ( A )I

(2)

there are an fppf-A-algebra B and h E H(B),n E N ( B ) with g = hn in G(B)).

Obviously N is a normal subgroup faisceau of H N . The canonical homomorphism H N + ( H N ) / N has kernel N , hence its restriction to H has kernel H n N . We thus get an embedding H / ( H n N ) + ( H N ) / N that has to be an isomorphism: For all g, h, n as in (2) the element h(H(B)n N ( B ) ) defines an element in ( H / ( H n N ) ) ( A )that is mapped to g N ( A ) . Therefore, all ( H N ) ( A ) / N ( A )are in the image, hence all ( ( H N ) / N ) ( A )are in the image faisceau. So, we get the isomorphism theorem

(3)

H / ( H nN ) 2 ( H N ) / N .

Suppose now that N is normal in G and let n:G + G / N be the canonical map. Let us denote by n ( H )the image faisceau of q H .Then

(4)

H N = n-'(n(H)).

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Factor Groups

Indeed, if g E n-'(n(H))(A),then there is a B (fppf over A ) with n(g) E n(H(B)), hence h E H ( B ) with gh-' E ker(n)(B) = N(B) and g E (HN)(A) by (2). The other inclusion is even more obvious. If H 2 N, then obviously H N = H and H = n-'(n(H)). So we have for normal N the usual bijection between {subgroup faisceaux of G containing N} and {subgroup faisceaux of GIN). Furthermore, one can then show (for H 3 N) that H is normal in G if and only if HIN is normal in GIN, and that one has a canonical isomorphism (G/N)/(H/N) s GIH of faisceaux, which is a group isomorphism if H is normal, cf. [DG], III,§3, 3.7.

6.3 (G/N-modules) Let us assume from now on until the end of this chapter that G is a flat group scheme over k , and that N is a normal and flat subgroup scheme of G. The general construction from 2.15 applied to the canonical homomorphism n: G + GIN yields for any GIN-module M an operation of G on M. As in 2.15 we use the notation n*M for this G-module. (If there is no confusion possible we shall simply write M instead of n*M.) Obviously n* is a functor from {GIN-modules} to {G-modules} that is exact and faithful, i.e., we have for all GIN-modules M, M': (1)

H0mGiN(M,M') = HOmG(n*M,n*M').

(Any E (G/N)(A) has a representative g E G(B) with B fppf over A . If cp E HomG(K*M,n*M'), then cp Q idB commutes with g, hence cp 0 id, with S as M Q A is mapped injectively into M 0 B.) The image of n* consists of all G-modules V on which N operates trivially. For such V,the k-group functor A H G(A)/N(A)operates naturally on V,,and this operation can be extended uniquely to the associated faisceau GIN as V ,is itself a faisceau. This follows from the universal property of GIN and also from its explicit description in 5.4/5. The full subcategory of all G-modules on which N operates trivially is obviously an abelian category. So we see that the category of all GIN-modules is an abelian category even without knowing whether GIN is a flat group scheme (what we needed in 2.9) or not. 6.4 For any G-module V the subspace V N is a G-submodule of V, by 3.2, on which N operates trivially. We therefore can regard V Nas a GIN-module and V H V N as a left exact functor from {G-modules) to {GIN-modules).

Lemma: The functor V H V N from {G-modules) to {GIN-modules) is right adjoint to n*. I t maps injective G-modules to injective GIN-modules. The category of GIN-modules contains enough injective objects.

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Representations of Algebraic Groups

Pro08

We have for any GIN-module M and any G-module V HOmG(z*M, V ) N H0mG(Z*M,V") = H0mGiN(M,VN),

by 6.3(l), where the first isomorphism is induced by the inclusion V Nc V. This shows that V H V N is right adjoint to the exact functor z*, hence that injective objects are also mapped to injective objects. Any embedding of z*M into an injective G-module Q induces an embedding of M into the injective GIN-module QN. Therefore { GIN-modules} contains enough injective objects. Remark: We can generalize the above as follows. Let E be a G-module that is finitely generated and projective over k. Then M H n*(M) 0 E is an exact functor from {GIN-modules} to {G-modules}. The functor Vt+ Hom,(E, V ) N (E* 0 V ) N ,cf. 2.10(6), is right adjoint to it. It is therefore left exact and maps injective G-modules to injective G / N-modules. Indeed, one has for any G-module V and any GIN-module M

H0mGiN(M,(E* @ V ) N )= H0mG(M,(E* @ V ) N )N HOmG(M,E* 0 V ) , 3:

HOmG(M 0 E, V ) ,

using 4.4 for the last step. Notice that we can also regard this as an isomorphism of functors (1)

HomGiN(M,?) 0 Hom,(E, ?) 3: HOmG(M 0 E, ?).

6.5 (Factor Groups as Affine Schemes) Let me quote the following result from [DG], 111,§3, 5.6: (1) If k is a j e l d and if G, N are algebraic k-groups, then GIN is an algebraic k-group. Notice that in our convention an algebraic k-group is assumed to be affine. Another case where we know GIN to be affine is when N is a finite group scheme (by 5.5(6)). Let us recall from 5.7(2) and 5.8: (2) If GIN is an afJine scheme, then it is $at and N is exact in G.

Of course, in this case we do not need 6.314 to see that {GIN-modules} is an abelian category and has enough injective objects. The functor V H V Nmaps M 0 k[G] for any k-module M to M 0 k[GIN N M 0 k[G/N] if k is a field. Therefore we can also use 3.9.c to show that it maps injective G-modules to injective GIN-modules (in that case).

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Factor Groups

6.6 Proposition: Suppose N i s exact in G. Let E be a G-module that is finitely generated and projective over k. Then the derived functors of V H Hom,(E, V ) from {G-modules} to {GIN-modules} can be identijied with V H Ext",E, V ) . There are for each GIN-module M and each G-module V spectral sequences (1) (2)

(3)

E2'" = EXtEIN(M,ExtG(E, V ) )* Ext~+'"(MQ E, V ) , E;'"

=

EXt",,(M, H'"(N, V ) )* ExtL+'"(M, V ) ,

ET'" = H"(G/N,H"(N, V ) )* H"+'"(G, V ) .

Pro08 As N is exact in G, the functor res: maps injective G-modules to modules acyclic for the fixed point functor ?N. (Use 3.9s and 4.10.) The composition of ?N from {N-modules} to { k-modules} with res: is isomorphic to the composition of res?" with ?N from {G-modules} to {GIN-modules}. Therefore 4.1(2), (3) imply that all VwH"((N,V) can be regarded as the derived functors of V H V Nfrom { G-modules} to { GIN-modules}. The same is true for VHH"(N, E* 0 V ) 2: Ext;l(E, V ) , cf. 4.4, and VH(E* 0 V ) , N HOmN(E,V ) . As HomN(E,?) maps injective G-modules to injective GIN-modules, we can apply 4.1( 1) to 6.4(1) and get the spectral sequence in (1). Taking E = k we get (2), and setting M = k yields (3). Remark: The spectral sequence in (3) is known as the Lyndon-HochschildSerre spectral sequence. Let us be a bit more precise about its construction. There is a standard injective resolution

0 -P V + V 0 k[G] + V Q

o2k[G]

-P

VQ

o3k[G]

+*

* *

of V as a G-module, cf. 4.15(2). This is also an injective resolution as an Nmodule, see the beginning of the proof. So we get H'(N, V ) as the cohomology of the complex (V Q 8'k[G]),. Constructing the standard injective resolution of the GIN-modules ( V Q @ik[G])N we get a double comk[G])N Q @k[G/N]. The spectral sequence associated plex @ i , j ( V Q to this double complex is just the one from 6.6(3). Using this, the edge maps H'(G, V ) = E' + Es*' = H'(N, V)'" can be checked to be the restriction maps H'(G, V )+ H'(N, V ) that take values in the GIN-fixed points, cf. [Mac], XI. 9.1.

oi

6.7 In the special case E = k, the proposition 6.6 implies that each H"(N, V ) for any G-module V has a natural structure as a GIN-module. This can be constructed using the Hochschild complex. We make G act on each

100

Representations of Algebraic Group

C"(N, V ) 1: V @ @"k[N] via the given representation on V and via the conjugation action on each factor k [ N ] . Then each 8"is a homomorphism of G-modules as A" and A, are so. This makes each H"(N, V ) into a G-module. One can now check that all connecting maps H"(N, V " )-+ H"+'(N, V ' ) for any exact sequence 0 -+ V’ -+ V -+ V" -+ 0 of G-modules are homomorphisms of G-modules. (See [Sullivan 31, 4.1 for the case of a field.) The universal property of derived functors (via &functors) shows, then, that the G-modules H"(N, V ) constructed in this way yield the derived functors of V H V N from { G-modules} to { G-modules}. This functor can be written as the composition of V H V N from {G-modules} to {GIN-modules} with the natural inclusion of { G/N-modules} into { G-modules}. The last functor being exact implies that the GIN-structure on H"(N, V ) given by the proposition must lead to the same G-structure as the construction using the Hochschild complex. Notice that this implies, in the case G = N, that the action of G on the H"(G, V ) constructed with the conjugation action on the Hochschild complex is trivial. 6.8 Corollary: Suppose that N is diagonalizable. Then we have for all Gmodules V and E with E finitely generated and projective over k, f o r all GINmodules M and all n E N isomorphisms (1)

Exf&,(M, Hom,(E, V ) )N Ext2;(M 0 E, V ) ,

(2)

Ext;,N(M, V N )N Ext",M, V ) ,

(3)

H " ( G / N ,VN) N H"(G, V ) .

Proof: All this follows immediately from 6.6 and 4.3 as each E, is a projective k-module and as N is exact in G (cf. 4.6). Remark: If we apply (3) to the G-module n*M, then we get (4)

H"(G/N, M ) = H"(G, M).

6.9 Corollary: Suppose that G f N is a diagonalizable group scheme. Then we have for all G-modules V and E and for all Gf N-modules M with E, M projective over k and rk,(E) < co isomorphisms f o r all n E N: (1)

HomGi,(M, Ext",E, V ) ) N Extz(M 0 E, V ) ,

(2)

HomG,,(M, H"(N, V ) )2! Ext$(M, V ) ,

(3)

H"(N, V)'"

N

H"(G, V ) .

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Factor Groups

Proof: As GIN is affine, hence N exact in G, we can apply 6.6.The formulas follow now immediately from 4.3.

Remark: Suppose GIN 2.1 l(3)decompositions

N

Diag(A) for some abelian group A. We have by

Ext;(E, V ) E' @ Ext;(E, V ) ,

(4)

,€A

(for all n E N).The map cp H cp( 1) is for any G / N-module M' an isomorphism HornGlN(k,,M') 1 M i , cf. 2,11(4). We can therefore identify the direct summands in (4) using (1) and 4.4 as follows: Ext",E, V ) , 'V ExtE(E 0 1,V ) 'V Ext",E, V 0 (-A)),

(5)

(We use the convention E 0 A = E 0 k, etc.) In the special case E = k we get (for all A E A and n E N) H"(N, V ) , N H"(G, I/ 0 (-A)).

(6)

6.10 Let M be a GIN-module. The spectral sequence 6.6(3)yields base maps of

the form

H"(G/N, M ) + H"(G, M). They are induced by the map C'(G/N,M ) -+ C'(G, M ) of the Hochschild complexes arising from the inclusions k[G/N] N k[GIN c, k[G]. This is just the usual "inflation" map in group cohomology, cf. the proof of [Mac], XI, 10.2. We shall show that these (and more general) base maps are injective provided certain conditions are satisfied. At first the conditions may look artificial, but we shall see later on (11. 10.11/12/18) that they are satisfied in some important cases. Proposition: Let N be exact in G. Let Q be a G-module and E a G-submodule of Q such that E is finitely generated and projective over k. I f Q is injective as an N-module and if Horn,@, Q ) = k, then the base maps

(1)

EXt:/N(M1, M2)

-+

Ext",E 0 Mi, E 0 Mz)

are injective for all n E N and all GIN-modules M,, M2. Furthermore, we get an exact sequence (2)

0

+

Ext&;v(M1,Mz) + Ext$(E 0 Mi, E 0 M 2 ) -+

HomGiN(M1, Exti(E, E) 6 M2)

-+

0.

102

Proofi

(3) and (4)

Representations of Algebraic Groups

From 6.6( 1) we get two spectral sequences: E?m = EXtZlN(M1,Ext!(E, E ) @ M2)* Ext:+"'(E @ M , , E @ M2), E'flsm 2

= EXt:/N(M1, Ext{(E,

Q) 0 M 2 ) * ExtE+"'(E 0 Mi Q 0 M2).

N EXtEIN(M1, M2) cz E;"". So we have to prove that the base maps for (3) are injective. Then (2) will follow from 4.1(4), as the map E;*' + E 2 there is one of the base maps. As Q is injective for N , the spectral sequence (4) degenerates and yields isomorphisms E;",' N E:' N E"'. The embedding of E into Q induces homomorphisms E:," + E:"*'" for all n, m, r. These maps are compatible with the differentials of the spectral sequences. Thus we get a commutative diagram &yo ,E;",O

AS HOmN(E,E ) C HornN(,!?,Q) = k, we have E?'

I

1

,I770

,EzO

We know E?' + E'"vo and E;",' + E'",' to be isomorphisms. This implies that E?' -+ E;' is injective, hence so is the base map E?' -+ E".

Remarks: 1) The special case E = M , = k yields: If there is a G-module Q injective for N such that QG = QN= k, then the base maps H " ( G / N ,M ) + H"(G,M )

(5)

are injective f o r each GIN-module M . Furthermore, we have an exact sequence (61

O+H'(G/N,M)

+H'(G,M)+(H'(N,k)@

M)'IN+O.

2) One can generalize the proposition as follows: Let V be a G-module that is finitely generated and projective over k. Drop the similar assumption on E and that HornN(&Q) = k, but assume that Horn,( V,E ) = Horn,( V,Q) N k. Then the spectral sequence Ext",,,(M,,Ext",V,E)O

M2)*Ext:+m(V0 M,,E 0 M 2 )

will yield injective base maps EXftjN(M1,M 2 ) + Ext",V O Mi, E O M z ) ,

(7)

and an exact sequence

(8)

0

-+

-+

Exth/,v(M,,M,)

+

Exth(J'O

M,,E 0 M2)

HomG/N(M,,EXtA(V,E ) @ M 2 ) + 0.

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Factor Groups

Let H be a .flat subgroup scheme of G with N c H. Suppose that both GIN and HIN are afine. Then one has for each HIN-module M and each n E N an isomorphism of G-modules 6.1 1 Proposition:

(1)

(R'indg) M

1:

(R"indg',$)M.

Proofi Let n: G -+ GIN and n':H + H/N be the canonical maps. Our claim ought to be formulated as an isomorphism of functors: (1')

(R'indg)

o

rc'*

n* o R"ind:$;.

N

First let us consider the case n = 1, i.e., get an isomorphism (2)

indg(M) N indg$(M).

The right hand side is a subset of Mor(G/N, M,), which we may identify with Mor(G, because of the universal property of GIN. (Remember that M, is a faisceau.) Any f E Mor(G, Ma) will belong to indg$(M) if and only if f ( g n ) = f ( g ) and f ( g h ) = h-'f(g) for all g E G ( A ) , h E H(A), n E N(A) and all A . As N(A) c H(A) operates trivially on M 0 A, we can drop the first part of the condition. The second one alone describes just indg(M) so that we get (2). As above, we ought to have formulated this as an isomorphism of functors (2')

indg n' 0

'Y

n indg$N,. 0

This formula implies (1) using 4.1(2), (3) as soon as we can show that n'* maps injective HIN-modules to H-modules acyclic for indg. By 3.9.c it is enough to look at HIN-modules of the form Q 0 k[H/N] z indY"(Q) for injective k-modules Q. Applying (2) to (H, N) instead of (G,H ) we can identify n'*(indY"Q) with ind{(Q), where we regard Q as a trivial N-module. By our assumption N is exact in H and G. The spectral sequence 4.5.c yields therefore

(3)

(R"indg) (ind!) 0

=0

for all n > 0.

This certainly implies the required acyclicity of ind!(Q) (1).

above, hence

Remark: We use often only the following part of the proposition: Let M be an H-module. If N operates trivially on M, then it operates trivially also on indgM and even on all (R"indg)M. 6.12 The isomorphism in 6.1 l(2) can be regarded as a special case of a more

general statement that we are going to prove now.

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Proposition: Let H be a flat subgroup scheme of G. Suppose both GIN and H / ( H n N ) are afjfne schemes. a) The functors Fl,F2from {H-modules) to {GIN-modules}with %(M) = (indgM)N and

9 2 ( b f ) = hdz$yHnN)(MHnN) are isomorphic. b) For each H-module M there are spectral sequences (1)

E?" = H"(N,R"ind$M) +(R"+"Fl)M

and (2)

= (R"ind~$yH,,,)H"(H n N , M )

* (R"+"F2)M.

Proof: a) Let rc: G --+ GIN and rc': H + H / ( H n N ) be the canonical maps. Obviously resz 0 x * = n’* reszrHnN).This yields an isomorphism of the adjoint functors, i.e., of Fiand F2. b) Both Fland F2are compositions of two left-exact functors. It is enough to show that the first one maps injective objects to acyclic objects with respect to the second one. Then we can apply 4.4( 1). The functor ind: maps injective H-modules to injective G-modules (3.9). This gives the claim for pi. Notice that we have to apply 6.6 in order to regard the H"(N,?)as derived functors on the category of G-modules. In the second case we have to apply 6.4 to ( H , H n N ) instead of (G, N ) . 0

Remark: Notice that a) implies R"Fl N R"F2 for all n, so the two spectral sequences (1) and (2) have the same abutment. 6.13 Proposition: Let H be a flat subgroup scheme of G such that H N is an afjfne scheme. Then there are isomorphisms of functors

and

Proof: Let H' be the kernel of the obvious homomorphism H LX N -+ G that can be identified with H n N via h H (h,h - l ) , cf. 2.6. We have an isomorphism (H LX N ) / H ' = H N , so H N is flat by our assumption and by 6.5(2).

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Factor Groups

Let M be an H-module and M' an N-module. Because of H' n N = 1 = H' n H (in H N ) we get from 6.12.a isomorphisms indiN(M)N (ind: " N M ) H ' and ind$,(M')

'v

(indf" N M ' ) H ' .

Now 3.8(2), (3) yield i n d i N MN ( k [ N ] 0M ) H ' and indfN(M')N ( k [ H ] 0 Here any h E H ( A ) and n E N ( A ) operate via pc(h)0h resp. p,(n) 0 1 on k [ N ] 0M , where pc is the conjugation action. If h E H ( A ) n N ( A ) , then (h, h - ' ) E H ' ( A )acts therefore as p,(h) 0 h. So N and H n N act on k [ N ] 0M as in the definition of indin,. This yields (1). Similarly, any h E H ( A ) will operate on k [ H ] 0M' as in the definition of indE,,(M'). Some (n,n-') E H ' ( A ) with n E N ( A ) n H ( A ) will not operate in that way, but the set of fixed points will be the same. (Regard f E k [ H ] 0M' as morphism H + M,. Then ( n , n - ' ) f ( h ) = (h-’n-’h)f(h h-'n-'h).) From this we get (2).

-

Remark: Suppose also that H / ( H n N ) is affine. Then resin, maps injective H-modules to modules acyclic for indi,, as (R"indi,,)(Q

0k [ H ] ) N H"(H n N , Q 0 k C H l 0 k"1) N

0k [ N ] ) = 0

(R"ind:,,)(Q

for all n > 0 and all k-modules Q. We therefore get from (1) and 4.1(2),(3) isomorphisms of derived functors (for each n E N) (3)

resf,

o

R"indiN N R"ind$,,,

0

resHnN. H

In (2) the higher derived functors are 0 (for H / ( H n N ) N ( H N ) / N affine). 6.14 Keep the notations of 6.12. The inclusion of H into H N induces by 6.2(3)

an isomorphism H / ( H n N ) N ( H N ) / N . Similarly, one can show that the inclusion of N into H N induces an isomorphism of faisceaux N / ( H n N ) N ( H N ) / H . (One can regard ( H N ) / H as an orbit faisceau of N , cf. 5.6(2).) Suppose now that these quotient faisceaux are schemes. Then any N module M' resp. any H-module M defines a sheaf Y,,,,,,(M') resp. Y(HN),H(M) as in chapter 5. The isomorphisms above identify it with YH,(HnN)(resi,,M') resp. YN,(HnN)(res~nNM). This is a consequence of 5.17(1). Using 5.12 one gets another approach to 6.13(3) and the symmetric statement with H and N interchanged.

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This can be generalized as follows: Let H , H ’ be flat subgroup schemes such that the multiplication map m: H x H’ + G has image faisceau equal to G. Then one gets an isomorphism of faisceaux H / ( H n H’) s G / H ’ . If these quotient faisceaux are schemes, then one gets as above reSs 0 R”ind$ N R“indE,,, 0 reSHnH’. H (1)

A (slightly) more general result is proved in [Cline, Parshall, and Scott 91, 4.1. 6.15 As explained in 2.15, we can twist any N-module V with any g E G(k). As there, we denote the twisted module by g V, Recall that “ V N V for all n E N(k). More generally, if V is an N-submodule of a G-module M, then gV is another N-submodule of M that is isomorphic to V. Suppose from now on that k is a field. Any N-module V is simple (resp. semisimple) if and only if gV is so. This implies:

(1) If M is a G-module, then SOCNMis G(k)-stable. Let L , M be G-modules with dim(L) < co.Then Hom(L, M) 1: L* 0 M is also a G-module and HornN& M ) N (L* 0 M ) Nis a G-submodule, cf. 6.3/4. The map cp 0 x H cp(x) from Hom(L, M ) 0 L to M is easily seen to be a homomorphism of G-modules. Therefore 2.14(3) implies (2) If L is simple as an N-module with End&) N k, then we have an isomorphism HomN(L,M) 0 L N (socNM), of G-modules. 6.16 We call G(k) dense in G if there is no closed subfunctor X c G with X(k) 3 G(k) and X # G, cf. the definition of closures in 1.4. If k is an algebraically closed field and G is a reduced algebraic k-group, then G(k) is dense in G (by Hilbert’s Nullstellensatz). The same is true for G reduced connected and algebraic over any infinite perfect field ([Bo], 18.3). For reductive G one may even drop the assumption “perfect”.

Proposition: Suppose that k is a field and that G(k) is dense in G. Then for any G-module M its N-socle SOCN M is a G-submodule of M with SOCG M c S O C M. ~ Proofi As G(k) is dense in G, any subspace of M is a G-submodule if and only if it is G(k)-stable, cf. 2.12(5). So 6.15(1)implies that SOCNMis a G-submodule. If M is a simple G-module, then SOCNMis a G-submodule that is f O by 2.14(2), hence equal to M. This implies that every semi-simple G-module is also semi-simple for N, especially that SOCGMc SOCNMfor any M .

Factor Groups

107

Remark: This proposition generalizes to the following more general situation. Suppose there is a subgroup scheme H of G such that G = H N and such that H ( k ) is dense in H.Then an N-submodule of M is a G-submodule if and only if it is an H-submodule, hence if and only if it is H(k)-stable. Therefore we can argue as in the proof above. 6.17 Any homomorphism a: G + G' of flat k-group schemes gives rise (by 2.15) to an exact functor a*: {G'-modules} -+ {G-modules}. In case a is an inclusion (resp. a induces an isomorphism G/ker(a) 3 G') we have constructed a right adjoint functor a* = indg' (resp. a* = ?ker(a)). In general, ct is a composition of maps of this type, cf. [DG], 111,53, 3.2, so we get always such a right adjoint. See [Donkin 11, section 3 or [Cline, Parshall, and Scott 61, 1.2 for a unified treatment.

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7

Algebras of Distributions

Over a field of characteristic 0 the representation theory of a connected algebraic group G is very well reflected by the representation theory of its Lie algebra g. Any representation of G gives rise to a representation of g. Then the notions of “submodule”, “fixed point”, or “module homomorphism” give the same result whether applied to G-modules or to g-modules. This is no longer true in characteristic p # 0. Any G-module still yields a g-module in a natural way, but now there may be g-submodules that are not G-submodules, or g-homomorphisms that are not G-homomorphisms, etc. It is, however, still possible to save some of the advantages of the linearization process (of going from G to g) by looking not only at g, but at the algebra Dist(G) of all distributions on G with support at the origin. (See 7.1 and 7.7 for the definition.) In characteristic 0 it will not contain more information, as then Dist(G) is isomorphic to the universal enveloping algebra of g. This is no longer true in characteristic p # 0 and there Dist(G) will do everything that g does not do (7.14-7.17). In this chapter we give at first the definitions of distributions with support in a rational point on an affine scheme, prove elementary properties, and then go over to distributions on group schemes with support in the origin.

I09

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Representations of Algebraic Groups

The definitions and results are more or less contained in [DG], [Ta], and [Y]. In [Ta] and [Y] there are many more results on distributions on schemes over a field than I could include here. In some cases it was necessary to extend their results from fields to rings. There [Haboush 31 was very useful. 7.1 (Distributions with Support in a Point) Let X be an affine scheme over k and x E X(k). Set I, = {f E k[X] I f ( x ) = O}. Then k[X] = k l 0 I, N k 0 I,. A distribution on X with support in x of order I n is a linear map p: k[X] + k with p ( I : + ' ) = 0. These distributions form a k-module that we denote by Dist,(X, x). We have (1)

(k[X]/I:+ +')*N Dist,(X, x) c k[X]*.

Obviously Dist,(X, x)

(2)

N

k*

1:

k, and for any n

Dist,(X, x)

1:

k 0 Dist,f (X, x),

where (3)

Dist;(X, x) = { p E Dist,(X, x) 1 p(1) = 0} _N (l,/l:+')*.

For each p E Dist,(X,x) we call p(1) its constant term and elements in Dist: (X, x) are called distributions without constant term. The k-module Dist:(X,x) % (I,/I;)* is called the tangent space to X at x and is denoted by T,X. (Cf. [DG], II,94, 3.3 for another description.) The union of all Dist,(X,x) in k[X]* is denoted by Dist(X,x) and its elements are called distributions on X with support in x:

(4)

Dist(X, x) = { p E k[X] * I3n E N:p(I:+ ') = 0} =

u Dist,(X,

x).

n2O

This is obviously a k-module. Similarly, Dist+(X,x)= Un2,Dist;(X,x) is a k-module. For each f E k[X] and p E k[X]* we define fp E k[X]* through (fp)(f+')= p(ff+') for all fi E k[X]. In this way k[X]* is a k[X]-module. As each I",+' is an ideal in k[X], obviously each Dist,(X,x) and hence also Dist(X, x) is a k[X]-submodule of k[X] *. We have restricted ourselves above to the case of affine schemes. There is, however, a definition available for all schemes. In general one defines 5.2), I, shows that all these distributions as special deviations ([DG], 11, & deviations form a k-module ([DG], 11, $4, 5.4), and uses [DG], 11, $4,5.7 in order to prove that in the affine case one gets the same definition as above. In the case of a ground field, however, we can easily give another description that works for all schemes. Suppose that k is a field. Then we can associate to

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Algebras of Distributions

x E X(k) the local ring Ox,,and its maximal ideal m,. In the affine case these are localizations Ox,,=k[X], and m, =(IJX. Furthermore, the natural map 'v Ox,x/(nt,)n+l for all n. k[X] + Ox,,induces then isomorphisms k[X]/I:'' So we can in general define Dist,(X,x) as (OX,,/(m,)"+')*. Similarly, we get Dist(X, x), Dist+(X,x), and Dist: (X, x). 7.2 (Elementary Properties) Let q : X + Y be a morphism of affine schemes over k and let cp*: k[Y] + k[X] be its comorphism. Then ( q * ) - ' I , = IP(,) for all x E X(k), hence q*(Z;&:) c I:" and q* induces a linear map k[Y]/I;&; + k[X]/I:+'. The transposed maps for all n yield a linear map (1)

(dq),: Dist(X,x) -+ Dist(Y,q(x)),

with (dq),(Dist,(X, x)) c Dist,( Y, q(x)) and (dq),(Dist: (X, XI) c Dist:(Y, q(x)) for all n. We get on T,X = Dist: (X, x) the usual tangent map, and in general, call (dq), the tangent map of q in x. One easily checks d($ 0 cp), = ( d ~ ) ) ~ 0( ,dq, ) for any morphism $: Y + 2 into another affine scheme. Let X be an affine scheme over k and x E X(k).Suppose I is an ideal in k[X] , with I c I,, cf. 1.4 for the notation. We can then apply the with x E V ( l ) ( k )i.e., construction above to the inclusion of V(Z) into X. We have k[V(I)] = k[X]/I. The ideal of x is & / I , and its nth power is ( I : + Z)/Z. This implies that the inclusion yields isomorphisms (2)

Dist,( V ( I ) x) , 2: { p E DistJX, x) I p ( I ) = 0)

and

(3)

Dist(V(Z), x) N { p E Dist(X, x) I p ( I ) = 01,

similarly for Dist: and Dist'. We shall usually identify both sides in ( 2 )and (3). If I' is another ideal with x E V(I')(k),then 1.4(5)implies

(4)

Dist(V(l) n V(I'),x) = Dist(V(I),x)n Dist(V(I'),x),

(5)

Dist(D(f),x) N Dist(X,x),

similarly for Dist,, Dist', and Dist:. If x E D ( f ) ( k )for some f E k[X], then the canonical map k[X] -+ k[X], induces an isomorphism of each k[X]/I:+' onto the corresponding object for D(f). Therefore the inclusion of D( f )into X induces an isomorphism

similarly for Dist,, etc.

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Representations of Algebraic Groups

Let cp:X + Y again be a morphism of affine schemes, let Y' c Y be a closed subscheme defined by an ideal J c k[Y], and let x E X(k) with cp(x) E Y ( k ) . Then cp-'(Y') is the closed subscheme V(k[X]cp*(J)) of X. As X E cp-'(Y')(k) we have I, =I k[X]q*(J). We get therefore from k[X] = k l @ Z, that k[X]q*(J)

+ rf = cp*(J) + Z,q*(J) + If = cp*(J) + zf.

Any p~ T,(X) = DistT(X,x) belongs to T.(cp-'(Y')) if and only if p(k[X]q*(J)) = 0, hence if and only if p(cp*(J)) = 0, i.e., if ( d q ) , p ( J ) = 0. This yields:

Here and below we regard ( d q ) , as a map defined on T,(X). The reader who is more familiar with varieties than with schemes should be aware that q - ' ( Y ' ) has to be taken as a scheme, even if X, Y, Y' happen to be varieties, so that T'.(q-'(Y')) can be larger than expected. We can always apply ( 6 ) to Y = V(Z+,(J. This closed subscheme is isomorphic to A' such that the unique element in any Y ( A ) is just the canoniY') = 0 for this Y', hence cal image of cp(x). Obviously TqCx,( (7)

ker(dcp), = Tx(~-'(~(X))).

The constructions and results above have generalizations to the case where the schemes are not affine. This is particularly obvious when k is a field and when we can work with Ox,,. One can also generalize ( 5 ) to Dist(Y,x) N Dist(X, x) for any open subscheme Y of X with x E Y(k).

7.3 (Distributionson A") First let us consider as an example X = A ' = S p , k [ T ] and x = 0, hence I, = ( T ) . The k-module k[X]/Z:+' is free and has the . . , T nas a basis. Define ym E k[T]* = residue classes of 1 = T o ,T = T ' , T2,. k[A']* through ym(T")= 0 for n # rn and y,(T") = 1. Then obviously Dist(A',O) is a free k-module with basis (y,JrnoN and each Dist,(A',O) is a free k-module with basis ( y J 0 <, n , If k is a field of characteristic 0, then obviously

This can be easily generalized to A" = Sp,k[ Tl, . . . , T,] for all rn. For each rn and multi-index a = (a(l),a(2),. , , ,a(rn))E N", set T o = T;(1)T$2). To(") denote by yo the linear map with yo( T b )= 0 for all b E N", b f a and y,( T " )= 1. One easily checks that Dist(A", 0) is free over k with all yo as a basis, and that Distn(Am,O) is free over k with all yo with (a1 In as a basis. (For a as above

Algebras of Distributions

set la1 =

113

a(i).)If k is a field of characteristic 0, then

If k is a field, then any Dist(X, x) will only depend on the m,-adic completion of Ox,,. So, for a simple point x all Dist,(X,x) and Dist(X,x) will look like Dist,(A”, 0) and Dist(A”,0) where m = dim, X, cf. [DG], I, §4,4.2. 7.4 (Infinitesimal Flatness) Let X be an affine scheme over k and x E X(k). We call X infinitesimally flat in x if each k[X]/I:+’ with n E N is a finitely presented and flat (or, equivalently, projective) k-module. (In [Haboush 31 this property is called “infinitesimally smooth”. As obviously over a field any algebraic scheme (cf. 1.6) has this property, I think that name is not appropriate.) If X is infinitesimally flat in x, then each l:/l: with n I m is also finitely generated and projective, and each 1: is a direct k-summand of k[X]. Let k‘ be a k-algebra. Any x E X(k) defines a point in X(k’) = Xkr(k’) k’ I N k‘[xk?]. Then k‘[Xkr]/(l, @ k ) + = with ideal I, @ k‘ c k[X] @ (k[X]/I:+ ’) 0 k’. Now ring extension commutes with taking the dual module as long as the module is finitely generated and projective. So we get: (1) If X is infinitesimally flat in x, then Xk,is infinitesimally flat in x for each k-algebra k . There are natural isomorphisms Dist,(X, x) 0 k N Distn(Xk,, x) and Dist(X, X) 0 k Y DiSt(Xk,,X).

Of course, we use here the letter x also for the image of x in Xk.(k’)= X(k’). Consider two affine schemes X, X‘ and points x E X(k) and x’ E X’(k).Then the ideal of (x,x‘) in k[X x X’] N k[X] 0 k[X’] is I(,,,,) = I, 0 k[X’] + k[X] @ l x fIf. X and X’ are infinitesimally flat in x resp. x’, then I;::,!, can be n+l j identified with Cj=o I , 0 I,,n + l - j , and then with n;=o(k[X] 0 I:?’-’ 0 k[X’]). Now some elementary considerations yield:

IF

+

(2) If X and X’ are infinitesimally flat in x resp. x’, then X x X’ is infinitesimally flat in (x, x’). There is an isomorphism Dist(X, x) @ Dist(X’,x’) N Dist(X x X’,(x, x’)) mapping Dist,(X,x) @ Dist,,-,(X’,x’) onto Dist,(X x X’, (x, x’)) for each n E N. We can apply (2) to X’ = X. Consider the diagonal morphism 6,: X + X x X, x H (x,x). Let us regard the tangent map (d6,), as a map A;,,: Dist(X, x) -,Dist(X, x) @ Dist(X, x). It makes Dist(X, x) into a coalgebra, i.e., satisfies 2.3(1) as (id x 6,) 0 6, = (6, x id) 0 6,. This coalgebra is cocommup p(1) tative, i.e., s 0 A;,, = A;,, where s(fi @ f2) = f2 @ fi. The map ~ : : H

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Representations of Algebraic Groups

is a counit, i.e., it satisfies 2.3(2). If q : X + Y is a morphism, then (dq), is a homomorphism of coalgebras, as (q x cp) 0 dx = By 0 q. So we have seen: (3) If X is infinitesimally flat in x, then Dist(X, x) has a natural structure as a cocommutative coalgebra with a counit. Tangent maps are homomorphisms for these structures. Remark: There is another case besides (1) where Dist,(X,x) 0 k' = Dist,(X,.,x) for all n and Dist(X,x) 0 k' 31 Dist(X,,,x). Assume that k is a noetherian integral domain and that k' is its field of fractions. In this case tensoring with k' commutes with taking the dual module for all finitely generated k-modules, cf. [Bl], ch. II,§7, exerc. 29b. If X is an algebraic scheme is finitely generated over k. So we get in this case over k, then each k[X]/I:" the desired isomorphisms. 7.5 An affine scheme X is called noetherian if k[X] is a noetherian ring, and it is called integral if k[X] is an integral domain.

Proposition: Let X be an afJine scheme over k and x E X(k). Let I , I' be ideals in k[X] with x E V(I)(k) n V(I')(k). If V(1) is integral, noetherian, and infinitesimally pat in x, then: V(1) c V(I')o Dist(V(I),x) c Dist(V(I'), x). Pro08 If V ( I ) c V(I'), then I' c I by 1.4(3), hence Dist(V(I),x)c Dist(V(I'), x) by 7.2(3). Suppose now Dist( V ( I ) ,x) c Dist(V(I'), x). We want to show

I' c I

(1)

+ I:"

for all n E N.If not, then (I' + + I)/(Z:+l + I ) # 0 for some n. Now I J I is the ideal of x in k[V(I)] = k[X]/I and its (n + 1)" power is (1:" + I ) / I . So k[X]/(I:" + I ) is a finitely generated and projective module. For any a E (I' + I:+' + I)/(I:+l + I ) , a # 0, there is some p E (k[X]/(I:+' + I))* = Dist,,(V(I),x)with p(a) # 0, hence p ( I ' ) # 0 and p q! Dist(V(I'),x). So we get a contradiction and have established (1). We can now apply Krull's intersection theorem to k[V(I)] 'v k[X]/I and + I;+1)3 1', hence V ( I ) c V(I'). get I =

nnro(Z

Remark: This obviously generalizes to the case where I is no longer integral, but where I, contains all associated prime ideals of I .

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Algebras of Distributions

7.6 Proposition: Suppose that k is a field. Let q : X + Y be a morphism of algebraic schemes over k and let X E X ( k ) . If q is p a t in x, then (dq),: Dist(X, x) + Dist( Y, q(x)) is surjectiue. Proofi Set A = Oy,qp(x) and B = Ox,,. The flatness of q in x amounts to the following:Using the comorphism (we may assume X, Y to be affine) we may regard A as a subalgebra of B such that B is a faithfully flat A-module. This faithful flatness implies ma&: = A n Bm;&,' for all n E N,cf. [BZ], ch. I. $3, prop. 9. As we assume our schemes to be algebraic, the rings A , B are noetherian and each A/m;&,' is finite dimensional. So Krull's intersection theorem yields

Bm;&: = (7 (mi+'

+ ~m;&;),

r2O

hence

m;&;

=

n ((m:+

r>O

+ Bm;&,') n A ) ,

and dim(A/m;&;) c 00 implies that there is some r with A n (mi+' n Bmn+ 1) = m n +). + 1 We can therefore embed A/m;&: - ( A + Bm;&,’+ m:+ ')/ OW (Bm;&. +mi+ ') into B/(Bm;&,' + mi+ '). As k is a field, any p~ Dist,( Y, q(x))= (A/m;&:)* has an extension to B/(Bm;&,' + mi+') that gives some p' E @/mi+ ')* N Distr(X,x). Then obviously (dq),p' = p. Therefore ( d q ) , is surjective. Remark: Note that we do not claim that each Dist,(X,x) is mapped onto Dist,( Y, q(x)). Indeed, it is well known that, e.g., the "classical" tangent map T,X = Dist:(X,x) -,Dist:(Y, cp(x)) = T"+,,,,Ywill not be surjective in general.

7.7 (Distributions on a Group Scheme) Let G be a group scheme over k. In this case we set Dist(G) = Dist(G, 1). We can make Dist(G) into an associative algebra over k. For any p, v E k [ G ] * we can define a product pv as (1)

pv: k [ G ]

k [ G ] 0 k [ G ] @% k 0 k 7 k.

We have obviously pv E k [ G ] * and the bilinearity of ( p , v ) H pv. Furthermore, 2.3(l) implies that this multiplication is associative and 2.3(2) that

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Representatiolls of Algebraic Groups

* has a structure as an associative algebra over k with one. It will in general be not commutative. Now Dist(G) is a subalgebra of k[G]* with eG is a neutral element. So k[G]

Dist,(G) Dist,(G) c Dist,+,(G).

(2)

cf. 2.4(1). (We This follows easily from the formula A(II) c x : , o I : 0 have written here 1; 0 ITwrinstead of its image in k[G] 0 k[G].) More precisely, 2.4(1) implies A(flf2...f,)~ n ( l @ J + f i @ l ) + i= 1

for all f i , f 2 , .

2 I?@ly+l-r

r= 1

. . ,f,E I, and n E N. We get therefore

(3) If p E Dist,(G) and v E Dist,(G), then

[p,v] = pv - vp E Dist,+,-,(G).

So Dist(G) has a structure as a filtered associative algebra over k such that the associated graded algebra is commutative. We call Dist(G) the algebra of distributions on G , dropping the addendum "with support in the origin". (Some people call Dist(G) the hyperalgebra of G.) Because of A ( l ) = 1 0 1 the subspace Dist+(G) is a two-sided ideal in Dist(G). Therefore (3) implies [Dist;(G), DistL(G)] c Dist;+,- l(G). This shows especially that Dist:(G) is a Lie algebra, which we denote by Lie(G), and call the Lie algebra of G . Note that Lie(G) = Tl G as a k-module, cf. 7.1. It can be shown that we have constructed the usual structure as a Lie algebra on Tl G . 7.8 (Examples) Let us look at first at the additive group G = G,. As a scheme we may identify G, = Sp,k[T] with A'. Therefore we have described Dist(G,) as a k-module already in 7.3. As before let yn be the element with y,(T") = 1 and y,(T") = 0 for m # n. We have A(T) = 1 0 T + T O 1, hence A(T")=

I:=, (1) T i0 T"-'. This implies easily hence

(2)

y ; = n!y,.

So Dist(G,,,) can be identified with the polynomial ring C[yl], and Dist(G,,,) Y ; In general Dist(G,) = Dist(G,,,) OZk. with the Z-lattice spanned by all I. n.

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Let us consider now the multiplicative group G, = S p k k [ T , T-'I. Then I, is generated by T - 1. The residue classes of 1, (T - l), ( T - 1)2,. . . ,(T - 1)" form a basis of k[G,,,]/Zy+'. There is a unique 6, E Dist(G,) with 6,,(Z;+') = 0 = 6,((T - 1)') for 0 I i In and 6,,((T - 1)") = 1. From this and the bino-

mial development of T" = ((T - 1) + 1)" one gets 6,(T") = and r E N. If k is a Q-algebra, then obviously

All 6, with r E N form a basis of Dist(G,), all 6, with Y I n one of Dist,(G,). We get A(T - 1) = ( T - 1) 0 ( T - 1) + ( T - 1) 0 1 + 1 Q ( T - 1) from A( T ) = T Q T, hence (3) We get as a special case 6,6, = ( I S,, and inductively

,

(4)

+ l)&+ + rS,, hence (6, - r)6, = (r + 1) x

r!6, = 61(61 - 1).*.(6,- I

If k is a Q-algebra, then 6, =

(3 .

+ 1).

Therefore Dist(G,,,)

31

C[S,], and

Dist(G,,,) is the Z-lattice in Dist(G,,,) generated by all Dist(G,) = Dist(G,,,) Qz k.

7.9 (ElementaryProperties) If a: G -+ G' is a homomorphism of group schemes over k, then (1)

dcz = (da),: Dist(G) -+ Dist(G')

is a homomorphism of algebras. This follows easily from the definition of the multiplication. On Lie(G) = Dist:(G) we get the usual tangent map Lie(G) -+ Lie(G'), which is a homomorphism of Lie algebras. If H,H' are closed subgroup schemes of a group scheme G, then the inclusions of Dist(H) and Dist(H') into Dist(G), cf. 7.2(3), are homomorphisms of algebras, and 7.2(4) implies (2)

Dist(H n H') = Dist(H) n Dist(H'),

similarly Lie(H n H') = Lie(H) n Lie(H'). (The same statement for linear

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algebraic groups is known to be false in general. There the intersection as varieties is considered, not as schemes as we do here.) We call G infinitesimally flat if it is so at 1. Now 7.4(2) easily implies

(3) If G l , G 2 are infinitesimally flat group schemes, then GI x G, is infinitesimally flat and there is an isomorphism of algebras over k Dist(Gl) Q Dist(G,) r Dist(G, x G2). In the case of a semi-direct product there is still an isomorphism of kmod u 1es. If we take G, = G, = G and consider the multiplication map m,: G x G -+ G,then we see easily:

(4) If G is an infinitesimally $at group scheme over k, then d(m,): Dist(G) Q Dist(G) -+ Dist(G) is given by d(m,)(p Q v ) = pv for all p, v E Dist(G). For G as in (4) and any k-algebra k', the isomorphisms Lie(G) Q k' N Dist(Gk.), cf. 7.4(1), are isomorphisms of Lie algebras resp. of associative algebras. Furthermore, the comultiplication A; = A&: Dist(G) -+ Dist(G) Q Dist(G) can be checked to be a homomorphism of algebras over k. The map i,: G -+ G with g H 9 - l has as a tangent map (cf. 2.3) 3 Lie(Gkf)resp. Dist(G) Q k'

One easily checks that ob is an anti-automorphism of Dist(G), i.e., satisfies ob(pv) = ob(v)ob(p)for all p, v. If G is infinitesimally flat, then obis a coinverse

for the coalgebra structure, i.e., 2.3(3) is satisfied by (A;, a&,E , ) instead of (A, C, 4.

7.10 (Distributions and the Enveloping Algebra) To each Lie algebra g over k one can associate its universal enveloping algebra U(g). One may consult [B3], ch. I, §2, or [Dix], ch. 2 for the definition and the elementary properties of this object. It has a natural filtration U,,(g) = k l c U,(g)= kl 0 g c U2(g)c ... where U,,(g) is spanned over k by all products x 1 x 2 ~ ~ *with x, r I n and all x i E g. Let G be a group scheme over k. As Lie(G) = Dist:(G) is a Lie subalgebra of Dist(G), the universal property of U(Lie(G)) yields a homomorphism y: U(Lie(G)) + Dist(G) of algebras that induces the identity on Lie(G). It maps U,,(Lie(G))to Dist,,(G) because of 7.7(2).

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It is not very difficult to prove (cf. [DG], 11, $6, no 1): (1) I f k is a Jield of characteristic 0 and G an algebraic k-group, then y is an isomorphism U(LieG)) 3 Dist(G) and maps each U,,(Lie G ) bijectioely to Dist,,(G). Using this one can then show that algebraic k-groups are smooth and reduced over fields of characteristic 0, cf. [DG], 11, §6, 1.1. If k is a field of characteristic p # 0, then the situation is completely different. In this case for each x E Lie(G) = Dist:(G) its pthpower in Dist(G) also belongs to Lie(G). This is more easily seen by identifying Dist(G) with the algebra of left or right invariant derivations of k[G] as in 7.18 below. Let us denote this pthpower in Lie(G) c Dist(G) by x[P1in order to distinguish it from the pIh power xP in U(Lie(G)). The pair (Lie(G), x ++x [ P I ) is an example of what is called a p-Lie algebra. (One can find the general definition in [DG], 11, $7, no 3.) For any p-Lie algebra (9,x H XIPI) set UIP1(g)equal to the quotient of U ( g )by the two-sided ideal generated by all xP - xIp1with x E Q. This algebra is called the restricted enveloping algebra of g. We can still regard g as a subspace of UIP1(g).If xl,. . . ,x, is a basis of g, then all x;(’)x$’)* * with 0 Ia ( i ) p for all i form a basis of @“(g), cf. [DG], 11, $7, 3.6. So dim u[PI(g) = f m ( g ) . By the definition of x[Pl for x E Lie(G) it is clear that y has to factor through ULP1(Lie(G)). One can show:

-=

(2) I f k is a Jield with char(k) = p # 0 and G an algebraic k-group, then y induces an injective homomorphism UcP1(Lie(G)) + Dist(G). For this and for more details one may consult [DG], 11, $7, no 2-4. 7.1 1 (G-modules and Dist(G)-modules) Let G be a group scheme over k. Then any G-module M carries a natural structure as a Dist(G)-module:One sets for each p E Dist(G) and m E M:

(1)

pm

= (id,

0 p)AM(m),

i.e., the operation of p on M is given by (2)

M

M 0 k[G]

@’*

M Q k 7 M.

It is obvious that ( p , m )H p m is bilinear, and it is easy to see that p(vm) = ( p v ) mand EGm = m for all rn E M and p, v E Dist(G), using 2.8(2),(3)and 7.7(1). Obviously, 2.8(4) implies for all G-modules M , M‘: (3)

H O ~ G ( MM’) , C HOmDi,t(G)(M,M’).

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120

Applying this to inclusions we get (4) Any G-submodule of a G-module M i s also a Dist(G)-submoduleof M .

Of course, on a factor module the structure as a Dist(G)-modulecoming from the G-structure is equal to the structure as a factor module for Dist(G). The Dist(G)-structure on a direct sum of G-modules is the one as a direct sum of Dist(G)-modules. We get from 2.10(2): (5)

If m E M G , then pm = p(1)m for all p E Dist(G).

More generally, 2.10(2') implies for each 1, E X(G) c k[G]:

(6) If m E Ma, then p m = p(1,)rnfor all p

E

Dist(G).

For any G-module M and any 1, E X(G) we can construct the G-module M Q k,, which we usually denote by M Q I , We can identify M Q 1,with M as a k-module. If AM@) = mi Q h, then AMs,(m) = mi Q A&. This implies (cf. 7.1 for the k[G]-module structure on Dist(G)):

xi

ci

(7) Any p E Dist(G) operates on M Q I as 1,p operates on M .

If G is infinitesimally flat, then any p E Dist(G) operates on a tensor product of two G-modules through AL(p) E Dist(G) Q Dist(G). Let M be a G-module that is finitely generated and projective over k. Then M * is a G-module in a canonical way, cf. 2.7(4). The operation of Dist(G) on M * is then given by

for all p E Dist(G),cp E M * and m E M . If G is flat, then 2.13(2) implies that each m E M is contained in a Dist(G)submodule of M finitely generated over k. In this sense M is a locally Jinite Dist(G)-module. If we restrict the action of Dist(G) on M to Lie(G) = Dist:(G), then we get a representation of Lie(G) as a Lie algebra. In case k is a field of characteristic p # 0, then M is in this way even a restricted module for the p-Lie algebra Lie(G). Consider k-submodules N' c N c M such that M / N ' is a projective k-module and recall the definition of the closed subgroup scheme GN,,Nfrom 2.12. We claim (9)

Lie(G,.,,) = { p E Lie(G)1 p N c N ' } .

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Algebras of Distributions

As in 2.12 we may assume that M I N ' is free and choose a basis ( e j ) j s Jof a complement to N ' in M . There are aj(m)E k and A,,,E k[G] for all m E M as in 2.12. As In = n for all n E N one has

hence k[G]Z + Z: = Z + I:. As shown in 2.12, we have G,,,, = V(k[G]Z), so any p E Lie(G) belongs to Lie(G,,,,) if and only if p(k[G]Z) = 0, so by the equality above if and only if p(Z) = 0. Using the definition of the fj," together with p(1) = 0, we easily see that p ( I ) = 0 is equivalent to pN c N' and thus get (9). Similarly, one has for any subset S c M , if M is projective over k (10)

LieZ,(S) = { p

E Lie(G)

1 prn = 0

for all rn E S ) ,

and if N is a k-submodule of M such that M I N is projective, (11)

Lie Stab,(N) = { p E Lie(G) I pN c N } .

The reader should be aware that we always regard Z,(S) and Stab,(N) as schemes, even if we deal with a reduced algebraic group G over an algebraically closed field. 7.12 (The Case G = G,) Let us use the basis (y,JncN of Dist(G,) as in 7.3 and 7.8. As k[G,] = kCT] is free with basis ( T i ) i 2 owe , can write uniquely AM(m) = rn, 8 T i for any G,-module M and rn E M with almost all mi = 0. Then ( y p ) 0 T", So the structure obviously y,rn = m, for all n, i.e., A&) = Ento as a Dist(G,)-module determines the comodule map uniquely, hence also the structure as a Go-module. This implies for G = G, that there is equality in 7.11(3) and that the converse holds in 7.11(4), ( 5 ) . In general, not all locally finite Dist(G,)-modules arise from Go-modules.If, e.g., k is a field of characteristic 0, then one can define for each b E k a structure as a Dist(G,)-module on k, where each yr operates as multiplication with b'/(r!). For b # 0 this module does not come from a G,-module. If k is a field of characteristic p # 0, then we can make k 2 into a Dist(G,)-module letting each

ciz0

yi operate as

(g t)

if iis of the formp'with r E N, r > 0, as 1 if i = 0, and as 0

otherwise. This structure does not come from G,. 7.13 (The Case G = G,,,Let ) us use the basis (dr),> of Dist(G,) as in 7.8. If M is mi 0 T iwith uniquely determined a G-module and rn E M , then AM@) =

Cisz

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mi E M , almost all zero. Then

6,m =

c (:>mi

ioZ

for all n E N.

Remember that M = OiPzMi where Mi = {m’E MI AM(m')= m' 0 T i } ,and that m iE Mi in the situation above, cf. 2.11. For a,,a,, ..., a,€ Z pairwise different, there is f E Q [ T ] with f ( a l ) = 1, f ( a z )= * * * = f (a,) = 0 and f(Z) c Z. There are integers bj E Z with f =

x j 2 0 b j ( T ) , cf. [St 13, p. 16. Denote then by ?the element x j 2 0 b j 6 j E Dist(G,l,' If we apply this construction to { a l , . . .,a,} = { i E Z I mi # 0}, then we get fi E Dist(G,) with x m = mi. This shows for any Dist(G)-submodule N of M that N = 0 ( N n Mi),hence that N is also a G-submodule, i.e., the converse of 7.1 l(4). Also the converse of 7.1 1(5),(6) is true, ie., for all j E Z: for all n E N]. Indeed, consider any m as on the right hand side. Take the mi as above. Then

(:)mi = (:)mi for all n E N. For i # j we take f as above with f(i)

=

1 and

f ( j ) = 0 and get mi = fmi = 0. Hence m E M j . Note that (2) implies that the Dist(G,)-structure determines the G,structure, especially that we have equality in 7.1 l(3) for G = G,. In general, not every locally finite Dist(G,)-module arises from a G,module. If k is a field of characteristic 0 and if a E k, then we make k into a Dist(G,)-module letting any 6, operate as

(3

. For a 4 Z this structure does

not come from G,. If k is a field of characteristic p # 0, one can make a similar construction with any p-adic integer a. 7.14 Lemma: Let G be an injinitesimally flat, noetherian, and integral group scheme over k. If M is a G-module that is projective over k, then for all iE X ( G ) :

MA = { m E MI prn = p ( i ) m

for all p E Dist(G)}

Proof: Observe that there is for each x E M 0 k [ G ] with x 4 M 0 1;" somep E Dist,(G) with (idM0 p)x # 0. (Use embeddings of M and k [ G ] / I ; + ' into free modules.)

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Algebras of Distributions

Now if pm = p ( i ) mfor all p E Dist(G), then (idM0 p)(A,(m) - m O 3,) = 0 for all p, hence AM(m)- m 0 il E M 0 I;+’ for all n by the argument above, hence A M ( m ) - m ~ I ~ ~ n ~ O ( M M ~@ Z (~n n+, 0l 1) 2=+ 1 ) .(Use a split embedding of M into a free module for the last equality.) Now Krull’s intersection theorem shows that the last term is 0, hence AM(m) = m 6I and m E MA.

Let G be an infinitesimally J u t , noetherian, and integral group scheme over k. Let M be a G-module and M‘ a k-submodule of M such that M/M’ is projective over k. Then M i is a G-submodule of M i f and only i f it is a Dist(G)-submodule.

7.15 Lemma:

Proof: As MIM’ is projective, the k-submodule M’ is a direct summand of M and we can identify M ’ O k [ G ] with the kernel of M 6 k [ G ] + (M/M’)Ok[G]. We have to show: If M’ is a Dist(G)-submodule, then AM(M’)c M‘ 0 k[G], i.e., the image N of AM(M’)in (M/M’) 0 k [ G ] is zero. Now Dist(G)M’ c M‘ is equivalent to (id, 6p)AM(M’)c M‘ for all p E Dist(G), hence implies (id,,M, 6p ) N = 0. As in the last proof this yields

N c

n (M/M’)o I;+’

= (M/M‘)

n>O

o

n

1:+1=

0,

n>O

hence the lemma.

Let G be an infinitesimally p a t , noetherian, and integral group scheme over k. Then one has for all G-modules M, M‘ such that M‘ is projective over k:

7.16 Lemma:

Proof: For any cp E Hom(M, M’) set v(cp) = AM’

cp - (cp @

AM’

So v: Hom(M, M’) --* Hom(M, M‘ 0 k [ G ] ) is linear with kernel equal to M’), then Hom,(M, M’) by 2.8(4). On the other hand, if cp E HomDiStcG,(M, (idM($0 p)v(cp) = 0 for all p E Dist(G), hence (as in the proof of 7.14) for all mEM

7.17 (The Case of a Ground Field) Let us assume in this section that k is a field and that G is an algebraic k-group. One knows (cf. [DG], 11, $5,2.1): (1) G smooth o dim G = dim Lie(G)

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Representations of Algebraic Groups

and (cf. [DG], 11, $6, 1.1): (2) Zf char(k) = 0, then G is smooth. In the situation of 7.11(10), (11) we get therefore (using the notations as there)

(3) &(S) smooth -=-

I

dim ZG(S)= dim{p E Lie(G) pm = 0 (4) StabG(N) smooth g

for all m E S > ,

dim StabG(N) = dim Stab,i,(G,(N).

If G acts on a scheme X and if x E X(k), then the morphism cp: G + x, satisfies cp-‘(cp(l)) = Stab,(x). So 7.2(7) yields:

H gx

( 5 ) StabG(x)smooth o dim Stab,(x) = dim ker(dcp),. Recall that an affine scheme Y is called irreducible if and only if $d is a prime ideal in k[ Y]. It is integral if and only if it is irreducible and reduced. In case char(k) = 0, then (2)implies that G is integral if and only if it is irreducible, as any smooth scheme is reduced. Suppose now that k is a perfect field of characteristic p . If G is irreducible, then there is by [DG], 111, §3, 6.4 an isomorphism G N X x Y of affine schemes with Y integral, and where k[X] is a finite dimensional local k-algebra. The only maximal ideal of k[X] is nilpotent. This shows that we have (7n,o 1;’‘ = 0 in k[G]. It was for this property that we needed G to be integral in the last proofs. So we see: (6) Suppose that k is a perfect field. Then the results of 7.14-7.16 hold for any irreducible algebraic k-group. We can use the same argument with respect to 7.5. (7) Suppose that k is a perfect field. Let G be an algebraic k-group and H, H’ closed subgroups of G. If H is irreducible, then H c H’ 0 Dist(H) c Dist(H’). Let K be an algebraic closure of k. If we no longer assume that H is irreducible, then we can still say:

(8) H c H’ o H(K) c H’(K)and Dist(H) c Dist(H’). Indeed, we have to prove only one direction (‘‘+”), Suppose H = V(Z) and H‘ = V(1’).As I 2 I if and only if I @ K 3, I’ @ K (and similarly for Dist(H), Dist(H’)) we may assume k = K . Decompose I = lj such that the V ( l j )

ni=o

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Algebras of Distributions

are the irreducible components of H and 1 E V(Zo)(k). If Dist(H) c Dist(H’), then (7) yields Zo 3 Z . For any j there we can choose g j E V(Zj)(k) as k = K. Then pl(gj)Zo = Zj. If H(k) c H’(k), then 1’ = P d g j ) z ’ c PdgjlZo = Ij

for all j , hence I’ c

n;=oZj

= Z and

H’ 3 H.

7.18 (Distributionsas Differential Operators) Let G be a group scheme over k. Any operation of G on an affine scheme X leads (cf. 2.7) to a representation of G on k[X], hence makes k[X] into a Dist(G)-module. When dealing with a right operation a: X x G + X (resp. a left operation B: G x X + X), then the operation of ,D E Dist(G) on k[X] is given by (id,,,, 0 p ) a* (resp. 0

0 id,,,,) B*). O

There is a general notion of diflerential operators on a scheme, cf. [DG], 11, $4, 5.3. In the case of an affine scheme X they can be described as follows ([DG], 11, $4,5.7): Each f E k[X] defines ad(f): End(k[X]) + End(k[X]) through (ad(f)cp)(f,) = fcp(fl) - cp(ff,), i.e., ad(f)cp is the commutator of the left multiplication by f and of cp. Then a differential operator on X of order ~n is some D E End(k[X]) with ad(f,)ad(f,)...ad(f,)D = 0 for all fo, , , , ,f,E k[X]. A differential operator on X is then defined as a differential operator of order In for some n E N.The differential operators form a subalgebra of End(k[X]). For G operating on X as above, any p E Dist,(G) operates on k[X] as a differential operator of order ~n as an elementary argument shows, cf. [DG], II,$4, 6.3. When dealing with the operation of G on itself by left resp. right translation, then we get an operation of any ,D E Dist(G) as a differential operator on G that commutes with the operation of G by multiplication from the other side. This construction turns out to yield an isomorphism of Dist(G) onto the algebra of all differential operators on G that are right resp. left invariant (i.e., that commute with the action of G by right resp. left translation), cf. [DG], 11, $4, 6.5. The conjugation action of G on itself yields a representation of G on k[G] that stabilizes I , , hence also all I!”. We get thus G-structures on all G is k[G]/Z;+’, hence also on all Dist,(G) = (k[G]/Z!+’)*-provided infinitesimally flat. If so, then we also get a representation of G on the direct limit Dist(G). The representation of G on Lie(G) = Dist:(G) constructed thus is the adjoint representation of G. We use the notation Ad for the representation of G on Dist(G) and all Dist,(G), Dist;(G), and the notation ad for the corresponding operations of Dist(G) on itself or its submodules.

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Representations of Algebraic Groups

Suppose that G is infinitesimally flat. An elementary calculation shows that the adjoint representation on Dist(G) and the action of Dist(G) on any Gmodule M are related by the formula

(Ad(g)cp)m= g(cp(g-'m)),

(1)

for any cp E Dist(G) 0 A 'v Dist(G,), m E M 8 A , g E G ( A )and any A . Let us write down explicitly how any p E Dist(G) operates on k [ G ] and Dist(G) under the conjugation resp. adjoint action (for G infinitesimally flat). Suppose Ab(p) = z i p L @ i pi. Then, because of 2.8(7), the conjugation action of p is given by

As A; ob = (ah 8 ab) A;, the adjoint action is given by (using 7.1 l(8) and 7.7(1)) 0

0

(3)

ad(pc)p' =

z i

(pi0 p’

0 aG(pi))

(idk[G]

@ AG)

AG

Suppose now that k is a field and that G is algebraic. For any closed subgroup scheme H of G, its normalizer N G ( H ) and its centralizer CG(H) are closed subgroup schemes of G, cf. 2.6(8), (9). One has obviously N,(H) c Stab,(Lie H ) and CG(H)c Cent,(Lie H ) , hence also (cf. 7.1 1(lo), (11)) Lie N G ( H )c Stab,,,,,,(Lie H ) = normalizer of Lie(H) in Lie(G) and Lie C G ( H )c Cent,,,(,,(Lie H ) . So 7.17(7) implies: (4) If dim N G ( H )= dim StabLi,(,,(Lie H ) , then N G ( H )is smooth.

( 5 ) If dim CG(H)= dim CentLieo,(LieH ) , then CG(H)is smooth.

Of course, one should expect this condition to work only if H is irreducible. One can show (cf. [ D G ] , 11,§5, 5.7) that Lie(CG(H))= Lie(G)Hand that Lie(N,(H))/Lie(H) = (Lie(G)/Lie(H))H.

(Xj)jE of subfunctors of a group scheme G there is a smallest closed subgroup scheme H of G containing all Xj.(Take the intersection of all closed subgroup schemes containing all Xj.) We call H the closed subgroup of G generated by all Xj.

7.19 For any family

Proposition: Suppose that k is an algebraically closed field. Let G be an algebraic k-group and let ( H j ) j G jbe a family of integral closed subgroups of G.

Algebras of Distributions

127

Let H be the closed subgroup of G generated by all (Hj)jEj. Then H is integral and Dist(H) is the subalgebra of Dist(G) generated by all Dist(Hj). Proofi The reduced subgroup of G defined by H ( k ) contains all Hj, hence H is reduced. We can assume (by [DG], 11, 55, 4.6 or [Bo], 2.2) that (Hj)jE = {H1,H2,..., H,}, and that the multiplication map cr:H, x H , x x H,+ H is surjective on points over k. This implies that H is irreducible, hence integral. Furthermore, the theorem of generic flatness ([DG], I, $3, 3.7) provides us with a point over k where c1 is flat, hence dcr, by 7.6, is surjective on the distributions with support in that point. As do! in (1,1,. . . , 1) is multiplication, the same argument as in [Bo], 7.5 yields

(1)

Dist(H) = (Ad(h,)Dist(H,))(Ad(h,)Dist(H* , * (Ad(h,)Dist(H,)) ))

for suitable h , , . . . ,h, E H ( k ) . Let R be the subalgebra of Dist(G) generated by all Dist(Hi). As Hi c H for all i, also R c Dist(H). Because of (1) we have to show that R is stable under Ad@)for all h E H(k), or by the surjectivity of a(k),that R is an H,-submodule of Dist(G) for each i. By 7.15 it is enough to show stability under each Dist(H,) for the adjoint action. This is now clear from 7.18(2) as Ab(Dist(Hi))c Dist(Hi) 0 Dist(Hi) and ab(Dist(Hi))= Dist(Hi) for all i . Indeed AL resp. crb restrict to A t i and a t i on Dist(Hi). Remarks: 1) There is another proof in [Y], 10.10. The proof above follows the one in [Bo], 7.6 that Lie(H) is generated as a Lie algebra by all Lie(Hi) provided char@) = 0. 2) Drop the assumption that k is algebraically closed. Let K be an algebraic closure of k. If each (Hj)Kis still integral, then the claim of the proposition is still satisfied: We get from [Bo], 2.2 that H K is the closed subgroup generated . R as in the proof we get R 0 K = Dist(HK)= Dist(H)0 K, by all ( H j ) K With hence R = Dist(H) using 7.4(1). Now (Hj), is integral if and only if it is reduced, cf. [DG], 11,§5,1.1.This will certainly be satisfied if k is perfect, cf. [Bo], AG 2.2.

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Representations of Finite Algebraic Groups

Let us suppose throughout this chapter that k is a field. A k-group scheme G is called a finite algebraic group if dim k[G] < 03. We have met already some examples (p(,,),Go,r).One can associate to each finite abstract group a finite algebraic group in a natural way (8.5.a). The examples that are most important for us will be introduced in Chapter 9 (the Frobenius kernels). In this chapter we look at some special features of the representation theory of such finite G. Let me mention right away that one can find in [Voigt2] many more results that we do not look at here. One of these special features is that injective G-modules are also projective as in the representation theory of abstract finite groups. Whereas in that case (abstract finite groups) the injective hull of a simple module is also its projective cover this is no longer true in our situation (in general). Here the simple head and the simple socle of an injective indecomposable module differ by a character of G that we call the modular function of G (8.13). Another special feature is seen when dealing with a closed subgroup H of G. We have not only the right adjoint indg to the restriction functor resg but also a left adjoint coindg (the coinduction). Both functors are exact and they are related by dualizing (8.14-8.16). In fact, one can get one from the other by first tensoring with a character depending on the modular functions of H and G (8.17).

129

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Representations of Algebraic Groups

One main ingredient in the proofs of these results is the fact that k[G] and k[G] * are isomorphic as G-modules (8.7 and 8.12). This is a special case of a more general theorem of Larson and Sweedler (cf. [Sw]). As a source for the other nontrivial results let me mention [Oberst and Schneider] and [Voigt 11. When working not over a field but over an arbitrary commutative ring (say R), then one should define a finite algebraic group over R as an R-group scheme G such that R[G] is finitely generated and projective as an R-module. It is elementary how to generalize 8.1-8.6 to this more general situation. For an extension of 8.12 and 8.17 to this situation one may consult [Voigtl], cf. also [Oberst and Schneider].

8.1 (Finite Algebraic Groups) A k-group scheme G is called finite (hence: a finite algebraic k-group) if dim k[G] < 00. It is called injinitesimal if it is finite and if the ideal I, = {f E k[G] 1 f(1) = 0) is nilpotent. If k' is an extension field of k, then obviously G is finite (resp. infinitesimal) if and only if Gk,is so. The closed subgroups of the additive group (introduced at the end of 2.2) are infinitesimal groups. They are examples of Frobenius kernels, the (for us) most important class of infinitesimal groups, which will be introduced in Chapter 9. The groups p(,)for each n E N are finite (cf. 2.2). If char(k) = p # 0 and if n is a power of p , then ,qn) is infinitesimal. 8.2 Lemma: Let G be an algebraic k-group. a) G is finite if and only if G ( K )is finite f o r each extension K of k. b) G is injinitesimal if and only i f G ( K )= 1 for each extension K of k.

Proofi a) If dim k[G] < 03, then each element in k[G] is algebraic over k, hence has only a finite number of possible images in any K (under an element of G ( K ) = Hom,.,,,(k[G], K)). As any g E G ( K )is given by its values on the basis of k[G], there are only finitely many possibilities for g. Consider on the other hand an algebraic closure K of k, and suppose that G ( K )is finite. We can replace G by G,, hence suppose k = K . We can write k[G] in the form k[T,, . . .,TJI for some ideal I. Any prime ideal containing I has to be a maximal ideal. The same is true for any associated prime ideal of I. This easily implies that dim k[G] = dim k[T,, . . . , q ] / I < 03. b) If I, is nilpotent, then it has to be annihilated by any homomorphism of k-algebras k[G] --* K into a field extension. As k[G] = kl 0 I , there is only one such homomorphism, hence G ( K ) = 1.

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131

Suppose on the other hand G ( K ) = 1 for an algebraic closure K of k. We may assume k = K and can identify k[G]/fi with all functions from G ( K )to K . This implies I, = 40,hence that I , is nilpotent. 8.3 (Duality of Finite Dimensional Hopf Algebras) For any finite dimensional vector space V (over k) the canonical map V + (V*)* is an isomorphism. Mapping any linear map q : V, + V, between two finite dimensional vector spaces to its transposed map q*: V y + Vf is therefore a bijection Hom( V,, V,) 7 Hom( V y , VT). Let R be a finite dimensional vector space over k. We get from above isomorphisms Hom(k, R) 3 Hom(R*, k) and End(R) 3 End(R*) and

Hom(R 0 R, R) 3 Hom(R*, R* 0 R*) using the isomorphism R* 0 R* 3 (R 0 R)*. So multiplications on R (i.e., bilinear maps R x R + R or, equivalently, linear maps m: R 0 R + R) correspond bijectively to comultiplications on R* (i.e., linear maps m*: R* -+ R* 0 R*). Similarly, comultiplications A: R + R 0 R on R correspond bijectively to multiplications A*: R* 0 R* + R* on R*. Furthermore, m is associative (resp. A is coassociative, i.e., satisfies 2.3(1)) if and only if m* is coassociative (resp. A* is associative). An element a E R is a 1 for the multiplication m if and only if the map E,:R* -+ k, q ~ q ( ais) a counit for m* (i.e., satisfies 2.3(2) with the appropriate modifications in the notation). Similarly, E E R* is a counit for A if and only if it is a 1 for A*. If we have on R both a multiplication m and a comultiplication A, then A is a homomorphism of algebras (with respect to m) if and only if m* is a homomorphism of algebras (with respect to A*). If so, then some a E End(R) is an antipode for A and m (ie., satisfies 2.3(3) and a(ab) = a(b)a(a) for all a, b E R) if and only if a* is an antipode for m* and A*. This shows: If R is a Hopf algebra, then so is R* in a natural way. For two such finite dimensional Hopf algebras R,, R, a linear map $: R, + R, is a homomorphism of Hopf algebras if and only if $*: R --+ Rf is a homomorphism of Hopf algebras. Thus we get: (1) The functor R H R*, $I+$* is a self-duality on the category of all finite dimensional Hopf algebras.

This anti-equivalence obviously has the property that R is commutative if and only if R* is cocommutative (cf. 2.3). 8.4 (Finite Algebraic Groups and Hopf Algebras) We have by 2.3 an antiequivalence of categories { group schemes over k} + (commutative Hopf

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algebras over k ) . Combining this with 8.3(1) we get an equivalence of categories: (1) { Jinite algebraic k-groups} + {finite dimensional cocommutative Hopf algebras over k } . Each finite algebraic k-group G is mapped to k[G]*. We denote this Hopf algebra by M ( G ) and call it the algebra of all measures on G. We usually denote its comultiplication by A;, its antipode by 0 ; = a;, and its counit by EL:p H p(1). We have an obvious embedding G(k) = Hom,-,,,(k[G], k ) 4 M ( G ) = Hom(k[G], k): To each g E G(k) there corresponds the (Dirac) measure S,:f Hf(g). An element p E M ( G ) = k[G]* is a homomorphism of = 1. The multiplication algebras if and only if A;@) = p 6p and E'&) on G(k) is just the multiplication in M(G). More generally, one can identify G ( A ) = Horn,,,,(k[G],

A ) c Hom(k[G], A ) = k[G]* 6A = M ( G ) 6 A

for any k-algebra A with { p E J w G ) 6 A I (A; 6 idA)(P) = p

6 p,

E m = 11.

In Chapter 7 we have associated to each group scheme G the algebra Dist(G), cf. 7.1 and 7.7. If G is finite, then obviously Dist(G) is a subalgebra of M ( G ) and G is infinitesimal if and only if M ( G ) = Dist(G). One easily checks that

(2)

Lie(G) = Dist:(G) = { p E M ( G )I Ab(p) = p

6 1 + 1 6 p}.

8.5 Examples: a) If r is an abstract finite group, then its group algebra kT is a cocommutative Hopf algebra in a natural way. Considered as a vector space kT has a basis that we can identify with r. These basis elements multiply as in r and we define the comultiplication via y H y 6y, the counit via y H 1, and the antipode via y H y-l for each y E r. Hence there is a finite algebraic k-group G with M ( G ) N kr. For any k-algebra A the group G ( A ) can be identified with the set of all ayy E AT = kT 6 A with ~ y E r a y6 ( yy) = ~ Y , Y , E r a y a6 y .y') ( y andx,,,a, = 1. If A is an integral domain (or, more generally, has no idempotents # 0, l), then G ( A ) = r. This construction can obviously be carried out over any ring, not only over a field. b) Suppose that char k = p # 0 and let g be a finite dimensional p-Lie algebra, cf. 7.10. Then its restricted enveloping algebra UEP1(g) is a cocommutative Hopf algebra. Any x E g is mapped to x 6 1 1 6x under the comultiplica-

cyor

+

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133

tion, to 0 under the counit, and to --x under the antipode. So there is a finite algebraic group G with M(G) 2: U[PI(g). One obviously gets g c Lie(G) from 8.4(2).The embedding of UIP1(LieG)into Dist(G) c M(G) E UIp1(g)therefore has to be an isomorphism. We get Lie(G) = g and M ( G ) = Dist(G) so that G is infinitesimal . See [DG], 11, $7, 3.9-3.12 for more details. 8.6 (Modules for G and M ( G ) ) .Let R be a finite dimensional Hopf algebra. If M is an R-module, then M is an R*-comodule in a natural way: Define the comodule map M --* M 0 R* N Hom(R, M) by mapping m to a H am. If M is an R-comodule, then M is an R*-module in a natural way: Define the action of any p E R* as ( i d M p ) 0 AMif A M is the comodule map M --* M 0 R. For two such comodules MI, M, a linear map +: MI -+ M, is a homomorphism of Rcomodules if and only if it is a homomorphism of R*-modules. In this way we get an equivalence of categories

6

(1) {R-comodules}r { R*-modules}. Let G be a finite algebraic k-group. Then the categories of G-modules and k[G]-comodules are equivalent by 2.8. Combining this with (1) we get an equivalence of categories

(2) { G-modules} 3 {M(G)-modules}. Here to any G-module M there corresponds the M(G)-module M with 6p ) 0 A M . We recover the action of G(k) via the embedding G(k) c M(G)" and, more generally, the action of any G(A) via the embedding G(A) c (M(G) 0 A)" and the operation of M(G) 0 A on M 0 A . It is clear that we get on Dist(G) c M(G) the same operation as in 7.11. Furthermore, all the statements in 7.1 1 generalize to M(G). The claims in 7.147.17 obviously hold for any finite algebraic group G with Dist(G) replaced by M(G). The representations of G on k[G] via p l and p , lead to two (contragredient) representations of G on M(G), hence to two structures of an M(G)-module on M(G). Using the generalization of 7.11(8) one checks that any p E M(G) operates on M(G) as left multiplication by p when we deal with p l , and as right multiplication with ab(p) when we deal with p r . For G corresponding to a finite abstract group r as in 8.5.a the theory of Gmodules is the same as that of kr-modules, hence equal to the representation theory of r over k. For G corresponding to a p-Lie algebra as in 8.5.b the theory of G-modules is the same as that of UIP1(g)-modules,hence equal to the representation theory of g considered as a p-Lie algebra.

p E M(G) operating as (id,

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8.7 From now until the end of this chapter let G be a finite algebraic k-group. When we regard k [ G ] resp. M ( G ) as a G-module it will be with respect to p I or pr resp. the contragredient representation, which we also call the left or right regular representation of G . In case we want to distinguish in our notations between p 1 or pr we add an index ‘‘I” or “r” to the modules, e.g., k [ G ] , and M(G),. Lemma: The G-modules M ( G ) and k [ G ] are isomorphic. We have dimM(G)G = 1.

Proof: By the tensor identity we have M ( G )0 k [ G ] N k [ G ] ” where n = dim k [ G ] . On the other hand M ( G )0 k [ G ] = k [ G ] * 0 k [ G ] is self-dual as a G-module, hence also isomorphic to (kcGI*)”. The Krull-Schmidt theorem about unique decomposition into (finite dimensional) indecomposable modules implies that k [ G ] .v k [ G ] * has to hold as k [ G ] “ N (k[G]*)” for some n > 0. The last equality now follows from 2.10(5).

8.8 (Invariant Measures) We call an element in M ( G ) f (resp. M(G)f) a left (resp. right) invariant measure on G. (In [Sw] such elements are called “integrals”, in [Haboush 31 “norm forms”.) The description of the left and right regular representations of M ( G ) on itself in 8.6 implies (1)

M(G)lG = { P O E M ( G )1 PPO = P(~)PO

for all P E M(G)J,

M(G),G = { P O E M ( G ) I POP = P(~)PO

for all P E M ( G ) } ,

and

(2)

as ab(p)(l)= ~ ( 1for ) all p E M ( G ) . Furthermore, we have

(3)

ab(M(G),G) = M(G)P,

as a; intertwines the left and the right regular representations (or, using (l), (2), as it is an anti-automorphism of M ( G ) considered as an algebra). Obviously M ( G f is stable under right multiplication by elements of M ( G ) , hence an M ( G ) - and G-submodule of M ( G ) with respect to the right regular representation. (This can also be seen directly.) As dim M(G)F = 1 the representation of G on M(G)P is given by some dG E X ( G ) c k [ G ] . So for all g E G ( A ) and any A

(4)

pr(g)(PO 8 l ) = P O 0 dG(g)

for

PO

M(c)?,

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Representations of Finite Algebraic Groups

and, equivalently, for all p E M ( G ) (5)

POP = gk(P)(6G)P0 =

p(6E1)P0

PO

for

M(c)?*

(Observe that o,(x) = x-’ for all x E X ( G ) . ) This character 6, is called the modular function of G. We call G unimodular if 6, = 1. (In the examples in 8.9 each G will be unimodular. We shall meet a case where 6, # 1 later on in 11.3.4.) We could have defined 6, also via M(G)f as (3) implies for all p E M ( G ) (6)

PPO = P ( 6 G ) P 0

PO

for

M(c)f

or equivalently, for all g E G ( A )and all A PI(S)(PO Q 1) = POQ 6G(g)

(7)

for all POE M ( G ) f .

8.9 (Examples) If G corresponds to an abstract finite group r as in 8.5.a, then M(G)? = M(G)f = k

1 y.

?Ti-

Consider as another example G = G,,' with r E N,r > 0 assuming char@)= p # 0. Set q = p'. As Ga,r is a subgroup of G, = S p , k [ T ] , we can identify M(G,,,) = Dist(G,,,) with the subalgebra of Dist(G,) spanned by all p with p ( T 4 + ' )= 0 for all i 2 0. Using the basis (y,JneN of Dist(G,) as in 7.8 we get

As yo(l)

=

1 and y,(l) = 0 for n > 0, as yay,- = yq-

and ynyq-

=

that y q - l is an invariant measure on Ga,r. Using dimM(G), = 1 or some additional computations we get

(2)

M ( G ) f = M(G)f = k y , - ,

for G = Ga.r.

Assume again char k = p # 0, let r E N,r > 0, and set q = p'. Let us consider G = p(,) and determine M(G),. As p(,) is an infinitesimal and closed subgroup of G, we can identify M(P(~)) = Dist(p,,,) with a subalgebra of Dist(G,). Let us use the notations of 7.8. Then M(P(~,) consists of all v E Dist(G,) with v(Ti(T4- 1)) = 0 for all i E Z. Obviously, 6 , ( T ' ( P - 1)) = ( q

;

i) -

(3

for all i E Z. The standard formula for binomial coefficients mod p (cf., e.g.,

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[Haboush 31, 5.1) shows dn(Ti(Tq- 1)) = 0 for all i E Z if 0 In < q. As dim i'vf(p(,J = dim k[p(,,] = q, we get 1

n-

We claim q-1

(3)

M(G),G = M(G)[; = k

IS:,’

,Z ( -

for G = p(q).

l)'di

1=0

Set po = ( -l)iC5i. As 6, is the 1 in M(G) and 6,(1) = 1 and dn(l) = 0 for n > 0, we have to show dnp0 = 0 for all n with 0 c n < q. We have by 7.8 (3):

If n

+ i -j

> q - 1, then

("f';')

=0

and we can delete the corre-

sponding summand. Substituting s = i - j we get

1

4- 1

dnpo = i = O

I,

s=rnax(O ' - n )

c ( 1 (-I)'(.

q-1-n

=

min(i.q- 1 -n)

s=o

(

s

)(irS)b.,s

n+s

n+s 1=s

+

1-s

))(

)dn+s=o*

8.10 (Projective and Injective Modules) We call a projective object in the category of all G-modules simply a projective G-module. They correspond under the equivalence of categories to the projective M(G)-modules. This shows that each G-module is a homomorphic image of a projective G-module, hence that projective resolutions exist in the category of G-modules. (This is not true for arbitrary group schemes.) The representation theory of finite dimensional algebras shows that the indecomposable projective G-modules are (up to isomorphism) the indecomposable direct summands of M(G). For each simple G-module E there is a unique (up to isomorphism) projective G-module Q with Q/rad(Q) = E . It is called the projective cover of E . In this way one gets a bijection between the isomorphism classes of simple G-modules and of indecomposable projective G-modules. Now the isomorphism M(G) N k[G] from 8.7 together with 3.10 shows that a finite dimensional G-module is projective if and only if it is injective. The

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Representations of Finite Algebraic Groups

indecomposable injective indecomposable G-modules are exactly the indecomposable projective G-modules. There is a bijection E H E' on the set of isomorphism classes of simple G-modules such that the injective hull QEof E (cf. 3.16) is the projective cover of E', i.e.,

QE/~WQE) 1: E'.

(1)

We intend to describe this bijection and have to be more precise about the isomorphism M(G) N k[G] at first. 8.11 ( M ( G ) as a Module over k [ G ] ) . There is a natural structure as a k[G]module on M(G): For any f E k[G] and p E M ( G ) we define f p through (1)

(fW-1) =

P(f.71)

for all

fl E MGI.

The following properties follow from straightforward computations that are left to the reader. for all f E k[G],

(2)

fEG

(3)

ob(fp) = oG(f)ob(p)

=f(l)&G

for all

f E k[G1,p

M(G),

(4) If p l , p2 EM(G) and f ~ k [ G ] with AG(f) = x:=lAC3fi, f(PlP2) = xi(APl)(f;PJ.

We have A G ( x ) = x 0 x and ~ ( 1= ) 1 for all and (4) imply: ( 5 ) If x E X(G), then p inverse is p H x-'p.

and

x E X(G) c k[G]. Therefore (2)

~ is an ~ algebra p endomorphism of M(G). Its

We claim, furthermore, for any f (6)

then

E

k[G], p E M(G), and g E G(A) (for all A ) :

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Representations of Algebraic Groups

8.12 If M is a G-module, then we denote by M' the (G x G)-module that is

equal to M as a vector space and where the first factor G operates as on M and the second factor operates trivially. Similarly M' is defined. For I E X(G) we shall usually write I' and I' instead of (ka)' and (ka)'. We regard k[G] and M(G) as (G x G)-modules with the first factor operating via p l and the second one via p,. Proposition: Let po E M(G)p, po # 0. Then f k[G]-rnodules and of (G x G)-modules:

Hfpo

is an isomorphism of

k[Gl@ (&)' q M(G). Prooji It is obvious from the definitions and from 8.11(6),(7) that the map considered is a homomorphism of k[G]- and of (G x G)-modules. We have to prove only its bijectivity. As both sides have the same dimension it is enough to prove its injectivity. Consider the endomorphism y of M(G) @ k[G] that is the composite of the map idMM(G) @ A G : M(G) @ k[G] + M(G) @ k[G] @ k[G] with the map M(G) 0 k [ G l @ kCGl-+ M(G) 0 4 G 1 , p 0 fl @ f2 H flp @ f2. We can identify M(G) @ k[G] with Mor(G, M(G),) associating to each p @ f the map g H f(g)p. Then y(p 0 f ) is easily checked to be the map g H (p,(g)f)p. Let us fix now f E k[G] and consider F E Mor(G, M(G),) with F ( g ) = 9' If ' G ( f ) = Xi=1 1;. @ fi, (P[(g)f)piJ= Pi(g)(fpO) for 9 G(A) and then p r ( g ) f = h(g-')f;, hence F corresponds to (f;po)@ ac(fi) E M(G) @ k[G]. Its image under y is therefore the morphism 9 1 (pr(g)aG(.fi))f;)pO’ Now 1 (p,(g)’G(f;))f: maps any 9' to ~ ~ = l l ; . ( ( g ' g ) - l ) f :)( = g f(g-’g’-’g’) = f(g-'). This implies y ( F ) = po 0 a&). If fpo = 0, then F = 0, hence po @ uG(f) = 0. As po # 0, this implies f = 0. So the map considered is injective.

ci=

ci=

1;s

(c;=

Remarks: 1) If we take po E M(G)f, po # 0, then f phism of k[G]- and (G x G)-modules

k[Gl@

(&I'

H fpo

is an isomor-

3 M(G).

2) The affine and finite scheme G is also a projective scheme of dimension 0. It has therefore a dualizing sheaf, cf. [Ha], p. 241. This is easily seen to be the coherent sheaf with global sections equal to M(G) = k[G]*: We have for each finite dimensional k[G]-module M a non-degenerate pairing k[G] *) x M -+ k[G] * + k mapping at first (cp, m) to cp(m) and HomkIGl(M, then p to p(1). (Use HomkIGl(M, k[G]*) N HOm,I~l(k[G],M ) 1: M * with the obvious structure as a k[G]-module on M * . )

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In [Kempf 51, 5.1 the proposition is proved using the interpretation of M ( G ) as the dualizing sheaf. 8.13 Proposition: Let E be a simple G-module and Q a projective cover of E .

Then

Proof.- Choose a complete set el,. . . ,er of primitive, orthogonal idempotents in M ( G ) ,hence a decomposition

M ( G ) = @ M(G)ei i= 1

into indecomposable (projective and injective) modules. There are simple Gmodules Ei and E:( I Ii I r) with M(G)ei/rad M(G)e, 1: Ei and soc M(G)ei N E: for all i. We have to show E : = Ei Q dG for all i. For any G-module M the map (PI+ cp(ei) is an isomorphism HOmG(M(G)ei,M ) 3 eiM. If M is simple, then M N Ei if and only if eiM # 0. Any ,u E M ( G ) operates on M * through pcp = cp 0 ob(p) and on M 8 x for x E X ( G ) as xp operates on M . Therefore (for M simple) (1)

M

N

Ei * eiM # 0 o ab(ei)M*# 0

* (xdJei))(M* Q x- )

# 0.

Because of 8.11(5), xob(el), . . .,xob(e,) is also a complete orthogonal set of primitive idempotents in M ( G ) . We get from (1)

(2)

ET Q x-

N

M(G)Zab(e,)/rad M(G)xob(ei)

for all i. Choose po as in 8.12 and let $: M ( G ) -,k [ G ] be inverse to the map f H fpo from 8.12. The (G x G)-homomorphism property of 8.12 implies for all P, P‘ E M ( G ) $(PP’) = $(Pl(P)P’) = PI(P)$(P’) = $(Pr(ab(P’))P) = Pr(dGab(P’))$(P)o

Therefore each $(M(G)e,) = p,(dGab(ei))$(M(G)) is orthogonal to each M(G)dGab(ej)with j # i. As $ is an isomorphism for p 1 we get for all i M(C)ei

1:

$(M(G)ei)

(M(G)dGab(ei))*,

hence

(3)

SOC M(G)ei 1: (M(G)dGab(ei)/radM(G)dGab(ei))*.

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Representations of Algebraic Groups

Now (2) and (3) imply

Remark: If 6, = 1 (i.e., if G is unimodular), then the projective cover and the injective hull of every simple G-module coincide. If we apply the proposition to the trivial G-module k, then we get that the converse also holds. One can show for unimodular G that M(G) is a symmetric algebra in the sense of [CR], ch. IX,cf. [Humphreys 91. In general M(G) is only a Frobenius algebra. 8.14 (Coinduced Modules) Any closed subgroup H of G is itself a finite algebraic k-group. We can identify M(H) with the subalgebra { ~ L M(G) I p(Z(H)) = 0) where Z(H) c k[G] is the ideal of H, cf. the corresponding result for Dist(H) in 7.2(3). The equivalence of categories 8.6(2) enables us to define a functor coindg from { H-modules} to { G-modules} through (1)

coindgM = M(G)

M

for any H-module M . We call this functor the coinduction from H to G. (When comparing this to what is done for Lie algebras, e.g., in [Dix], ch. 5 one has to observe that there the terms induction and coinduction have just the opposite meanings. Also in [Voigt2] our coindgM is called an induced module.) We have obviously: (2) The functor coindg is right exact. For any H-module the map i M : M + coindgM with iM(m)= 1 8 m is a homomorphism of H-modules. The universal property of the tensor product implies that for each G-module V we get an isomorphism (3)

Hom,(coindgM, V )2 Hom,(M, resg V ) ,cp H cp o i,.

Hence: (4) The functor coindg is left adjoint to resg. Furthermore: ( 5 ) The functor coindg maps projective H-modules to projective G-modules. 8.15 Lemma:

Let H be a closed subgroup of G and M a finite dimensional Hmodule. Then there is an isomorphism of G-modules coindgM'= (indg(M*))*.

E

Representations of Finite Algebraic Groups

141

Proof: For all finite dimensional G-modules V,, V2 the bijection Hom(Vl, V2)'v Hom(V1, V T ) mapping each cp to its transpose cp* induces a bijection Hom,(V,, V,) N Hom,(Vt, VT). Using this and 8.14(3) we get for each finite dimensional G-module V canonical isomorphisms Horn,( V,(coindgM)*)3 HOm,(COindgM, V * ) 3 HOmH(M, V * )3 Horn,( V,M * )

mapping any $ to (iM)* 0 $. This generalizes to all V by taking direct limits. Therefore (coindzM)*has the universal property of indg(M*) as in 3.5, hence is isomorphic to indg(M*).

8.16 (Exactness of Induction) Let H be a closed subgroup of G. As H is a finite algebraic k-group 5.13.b implies: (1) indg is exact. We get now from 4.12: (2) k[G] is an injective H-module. Hence:

(3) M ( G ) is a projective left and right M(H)-module, and: (4) coindg is exact.

Of course (4) follows also directly from (1) and 8.15. One can improve (3) and show that M ( G ) is a free module over M ( H ) , cf. [Oberst and Schneider], 2.4. We do not have to go into this. If M' is a projective and finite dimensional right M(H)-module then we have for each H-module M an isomorphism (5)

M'

@M(H)

M 3 HomM(H)(HomM(H)(M', M(H)),

mapping each m' @ m with m' E M' and m E M to the map cp H cp(m')m.Here we form Hom,,,,(M',M(H)) via the operation of M ( H ) on itself by right multiplication and we consider it as an M(H)-module via the left multiplication on M ( H ) . In order to prove bijectivity in ( 5 ) one restricts to the case M' = M(H)"for some n where both sides are isomorphic to M". Because of (3) we can apply this to M(G) considered as an M(H)-module under right multiplication. The map in ( 5 ) is now easily checked to be an

Representations of Algebraic Groups

142

isomorphism of G-modules (6)

coindzM % Hom,(Hom,(M(G), M(H)), M),

where the operation of G on the right hand side is derived from the left regular representation on M(G). 8.17 Proposition: Let H be a closed subgroup of G. Then we have for each Hmodule M an isomorphism:

coindzM % indz(M Q ( ~ 5 ~ ) , , , 6 i ~ ) .

Proof: We have isomorphisms of (G x H)-modules Hom,(M(G), M(H)) 2: (M(G)*0M(H)IH = (k[G] Q M(H))" = indi(k[G]

06,)

2:

(k[G] 0k[H] 0S,),

N

k [ G l Q 6,.

This is regarded as an H-module via the right regular representation on k[G] and via 8, and as a G-module via the left regular representation on k [ G ] . We get now from 8.16(6) isomorphisms of G-modules coindgM

N

Hom,(k[G] Q 6,, M) N (M(G) Q 6;'

N

( k [ G ]Q (dG)IH6i1Q M), = ind:(M

0 M), 0 (6G)l,6i1).

8.18 Corollary: Let H be a closed subgroup of G and M a finite dimensional

H-module. Then: (indgM)* N indz(M* 0 (6G)lH6i1).

Proofi This follows from 8.17 and 8.15. Remark:

One can interpret this as Serre duality for the sheaf cohomology of 5.12.

9G,H(M),Cf.

8.19 Proposition:

Let G' be a k-group scheme operating on G through group automorphisms. Then G' operates naturally on k[G] and M(G). The space M(G)p is a GI-submodule of M(G) and the operation of G' on M(G)f is given by some x E X(G'). If p o E M(G)f,po # 0, then the map f H fpo is an isomorphism k [ G ] 0 x r M(G) of G'-modules. I f G is a closed normal subgroup of G' and i f we take the action of G' by conjugation on G, then xIG = dG.

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Representations of Finite Algebraic Groups

Proof: We can form the semi-direct product G >Q G’ and make it operate on G such that G acts through left multiplication and G’ as given. This yields representations of G SQ G on k [ G ] and M(G) = k [ G ] * that yield the operations considered in the proposition when restricted to G and that yield the left regular representations when restricted to G. Hence M(C)f are the fixed points of the normal subgroup G of G >Q G ,hence a G’-submodule by 3.2. It is now obvious that G’ operates through some x E X(G’) on M(G)f and that f H fpo is an isomorphism k [ G ] 0 x 2 M(G) of G’-modules. Suppose finally that G is a normal subgroup of G’ and that we consider the conjugation action of G on G. Then each g E G ( A ) c G ’ ( A )acts through the composition of p l ( g ) and p,(g) on M(G) 0 A, hence through p,(g) on po 0 1. Therefore the definitions show x ( g ) = S,(g).

8.20 Proposition: Let G be a k-group scheme containing G as a closed normal subgroup. Let H‘ be a closed subgroup of G’ and set H = H’ n G.Let M be an HI-module. Then there is a natural structure as an H‘G-module on coindgM extending the structure as a G-module. For each H‘G-module V there is a canonical isomorphism (1)

Hom,,,(coindgM,

V ) r Hom,.(M, V ) .

If x E X(G’) resp. x’ E X(H’) is the character through which G resp. H’ operates on M(G)I; resp. M(H)r, then we have an isomorphism of H‘G-modules (2)

coindgM 2 indf:G(M0 (xl,,)x’-’),

If dim M c co,then we have an isomorphism of H‘G-modules (3)

(indfiGM)*2 indf:G(M*0 (xlHf)x’-l).

Proof: Let us work with the description of coindgM as in 8.16(6).We make H‘ operate on M(G) and M(H) via the conjugation action on G and H. We get thus a representation of H’ on Hom(M(G),M(H)) that extends to H’ D< H if we let H operate through the two right regular representations. By 3.2 the subspace Hom,(M(G), M(H))is an H’-module. Together with the given action of H ‘ on M this makes Hom(Hom,(M(G), M(H),M) into an H’-module. This operation of H‘ can be extended to H‘ K H with H operating via pl on M(H) and through the restriction of the H‘-action on M. Again Hom,(Hom,(M(G), M(H)), M) is an H’-submodule. We can extend the operation of H’ to H‘ K G letting G act through p l on M(G), i.e., inducing the action of G on coindgM.

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For any h E H ( A ) for some A the element ( h , h - ’ ) E (H’ pc G ) ( A ) acts trivially. (This follows easily from the definitions.) Therefore we get a representation of (H’ M G ) / H N H’G, cf. 6.2(1), on coindgM extending the given one of G. Using this structure, the isomorphism in 8.14(3) is easily checked to be an isomorphism of H’-modules (provided V is an H’-module). It therefore has to induce an isomorphism of the H‘-fixed points. This implies (1). We get (2)by examining the proof of 8.17. After replacing 6 , by x and aH by x all isomorphisms there are also compatible with the H’-action, hence with the structure as H’G-module. Similarly 8,15 generalizes from G to H ’ G and together with (2) yields (3) as in 8.18.

Remark: We denote coindgM when considered as an H’G-module by coindi? M. Obviously coindZF is a functor from {H’-modules) to {H’Gmodules} and 8.14(2)-(4), 8.16(4) generalize to this. Note that we have by construction an isomorphism of functors 0

which is dual to 6.13.

coindiY

N

coindg 0 resi’,

9

Representations of Frobenius Kernels

Throughout this chapter let p be a prime number. We shall always assume that k is a perfect field with char(k) = p . Let G be an algebraic k-group. If k = F,, then the map f H f on k [ G ] is an endomorphism of k-algebras and defines a morphism FG:G + G that is a group endomorphism. The kernels G, = ker(F‘,) are called the Frobenius kernels of G. They are infinitesimal algebraic k-groups. One can generalize this to all k by replacing F‘, as above by some group homomorphism G -,G )into a suitable k-group G@). In this chapter we give first the definitions and elementary properties (9.19.6,9.8). We compute the modular functions of the G, in the case of reduced groups (9.7 combined with 8.19) and prove that H‘(G,M) N @ H ‘ ( G , , M ) under special assumptions on G and M (9.9). The representation theory of the first Frobenius kernel G1 of G is equivalent to that of Lie(G) as a p-Lie algebra. Therefore each cohomology group H’(Gl M ) is equal to the corresponding “restricted Lie algebra cohomology group” in the sense of [Hochschild 31. In that paper these groups are compared to the ordinary Lie algebra cohomology groups (cf. 9.17), especially in low degrees. One of his main results can now be interpreted as a “six term exact sequence” arising from a spectral sequence (9.19/20).This spectral sequence was found for

145

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Representations of Algebraic Groups

p # 2, 3 and G reductive in [Friedlander and Parshall 13. (But also compare

the remark at the end of section 3 in [Hochschild 31.) Their results were generalized somewhat in [Andersen and Jantzen]. The present version of Proposition 9.19 is the same as in my lectures in Shanghai and was also proved in [Friedlander and Parshall 41. 9.1 (The Frobenius Morphism on an Affine Variety) Before defining Frobenius morphisms in general we want to motivate the definitions by an example. In this section let us assume k to be algebraically closed. Let X be an affine variety over k (as in 1.1). We can embed X as a Zariski closed subset into some k". The map F: k"+k", (a1, a 2 , .. . ,u,)H (a!, a $ ,. . . , a % ) is a bijective morphism of varieties. It is also a closed map. (Using that f p E im(F*) for all f E k[Tl,. . ., T,] one shows Jk[T1,. . . , T,]F*(F*)-'Z = &for each ideal Z c k[T,, ...,T,].) Therefore each F'(X) with r E N is a closed subset of k" and F' induces a bijective morphism X + F'(X). We want to show that the pair (F'(X), F': X --t F'(X)) has an intrinsic meaning, i.e., is independent (up to isomorphism) of the embedding of X into k". Define for each f e k[X] a map qr(f):F'(X) + k through q r ( f ) ( x ' ) = f(F-’(x’))P’ for all x’ E F'(X). Obviously qr is an injective ring homomorphism from k[X] to the algebra of all functions from F'(X) to k and satisfies qr(af) = aP’qr(f) for all a E k and f E k[X]. If f is the ith coordinate function on k" restricted to X, then q , ( f ) is the ith coordinate function restricted to F'(X). Therefore qr induces a bijection from k[X] to k[F'(X)]. Denote by k[X](') the k-algebra that coincides as a ring with k[X] but where each a E k operates as a(P-r)does on k[X]. Then we can regard q, as an isomorphism of k-algebras k[X](') 3 k[F'(X)]. This shows that F'(X) as a variety has an intrinsic meaning. If we identify k[F'(X)] with k[X](') via qr, then the comorphism of F'is the map k[X](') + k[X], f I-+ f" for allf, hence also F' has a description independent of the embedding of X into k". 9.2 (The Frobenius Morphism on a Scheme) From now on let k again be an arbitrary perfect field of characteristic p. For each k-algebra A and each m E Z we define A("') as the k-algebra that coincides with A as a ring but where each b E k operates as bP-" does on A. Trivially A(') = A. One obviously has isomorphisms (1)

(A("'))(") =

for all m,n E Z,

and (for all k-algebras A, A') (2)

HOmk-alg(A(-m), A ' ) 1: HOmk.a,g(A,A''"'))

for all m E z.

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Representations of Frobenius Kernels

(This is the identity map.) For each k-algebra A , each m E Z and r E N the map (3)

a H UP'

7': A ( m ) --* A'""',

is a homomorphism of k-algebras. We now define for any k-functor X and any r E N a new k-functor X(') through (4)

X(')(A)= X(A(-'))

for all k-algebras A .

Furthermore, we define a morphism F',: X (5)

-+

X(') through

F',(A) = X(yr):X ( A ) --f x ( A ( - ' ) )= X("(A)

for all A . We call F', the rthFrobenius morphism on X . Obviously X faithful functor from {k-functors} to itself. One gets from (1) for all r, s E N and all X (6)

H

X(')is a

(X(r))(s) = X('+@ and FSy(,, o F', = F',+s.

If we consider an affine scheme X = Sp,R for some k-algebra R, then (2) implies for all r E N (7)

(SpkR)"'

3:

Spk(R"')

Furthermore, F', has as comorphism R(') + R , f H f"'. So the construction of X") and F', generalizes the situation considered in 9.1. We can interpret the definition (4) as saying that X(') arises from X through base change from k to k(-') which then is identified with k as a ring. We can therefore apply the general remarks about base change in 1.10. So the functor X H X") maps subfunctors to subfunctors, commutes with taking intersections and inverse images of subfunctors and with taking direct and fibre products. It maps local functors to local functors, schemes to schemes, and faisceaux to faisceaux (cf. 5.3(9)). If X is an affine scheme and I an ideal in k [ X ] , then V(I)(')= V(I('))and D(Z)(') = D(Z(')),where I(') c k [ X ] ( ' )is just I with the new operation of k. If k = F,, then obviously X(') = X for all r and any k-functor X . If X is affine and if Fx is the endomorphism of X with F X ( f ) = f p for all f E k [ X ] , then obviously F', = (FX)'.More generally, if k is again arbitrary, but if X has an then we can F,-structure (i.e., there is some F,-functor X ' with X = (x')k), identify X(') with X . In the affine case one has k [ X ] = F p [ X ' ] OF,k, and the map f @ a ~f @ a"' (for all f E Fp[X'] and a E k ) induces an isomorphism k[X(')] = k [ X ] ( ' )3 k [ X ] . (For r = 1 this map is called the arithmetic Frobenius endomorphism of k [ X ] . )Taking this identification F', is the endomorphism of X with comorphism f 0 a H fpr 0 a (for all f,a as above.) For r = 1 this map is called the geometric Frobenius endomorphism of k [ X ] .

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Representations of Algebraic Groups

Remark: If we replace k by an arbitrary Fp-algebra,then we can still define A(') for all r 5 0, hence X(') as in (4)for all T 2 0. We can also take the interpretation via base change in that situation. It is left to the reader to find out later on how much generalizes to this case. 9.3 (Fibres of the Frobenius Morphism) Let X be an affine scheme over k. Consider a point x E X(k) and let us denote its ideal by I, = {f E k[X] I f(x) = O}. Then the ideal of F',(x) E X(')(k) in k[X(')] = k[X](') is I:) (i.e., I, with the new scalar operation) as f ( F : ( x ) ) = f ( ~ ) for ~ ' all f. This implies (for all r E N)

(Fi)-'(Fk(x))= V

1 k[X]fPr ).

(f€L

So the (Fi)-'(Fi(x)) form an ascending chain of closed subschemes of X. Suppose now that X is algebraic. Then I, is a finitely generated ideal, say I, = k[X]&. Then

1’’’

for all r. The ideal defining (Fi)-'(Fi(x)) is contained in I!' and contains I,"". This implies (cf. 7.1,7.2(2)) (2)

Dist(X, x) =

u

Dist((Fi)-'(Fi(x)), x).

r>O

We can choose the fi such that the fi + I : (1 I i I m) form a basis of I,/I;. If x is a simple point of X, then m = dim,X and the fi(1 I i I m) are alge* * * f :") braically independent. Therefore the residue classes of all f!(')j$2) with all n(i) c p' form a basis of k[(Fi)-'Fi(x)]. This shows (3) If x is a simple point of X, then dim k[(Fi)-'Fi(x)] = p'" where m = dim,X. Let me add that (1) generalizes to (4)

for all ideals I in k[X] (and any affine X), whereas (5)

(Use that

( F i ) - ' D ( I ( ' ) ) = D(I).

f i = ,/cf

k[X]fp' and 1.5(5), (lo).)

9.4 (Frobenius Kernels) Let G be a k-group functor. Then obviously each G(’) is also a k-group functor and FL: G --f G(’) is a homomorphism of k-group

149

Representationsof Frobenius Kernels

functors. Its kernel G, = ker(FL) is a normal subgroup functor of G that we call the rthFrobenius kernel of G . The factorization in 9.2(6)implies that we get an ascending chain GI c G, c G j c

(1)

*

*

a

of normal subgroup functors of G. If H is a subgroup functor of G, then H(') is a subgroup functor of G(') and F L is the restriction of F L to H. This implies (2) especially for all r, r'

(3)

H, = H n G,, EN

':{

(G,),, =

for I' for r' 2 r. 7"

If k = F, or if G is defined over F,, then we can identify each G(,)with G and interpret FL as the rth power of some Frobenius endomorphism FG:G -+ G (which is F A after the identification G N G(l)). This is true, e.g., for G = G, and G = G,. In these cases (FG)*(T) = TP in the notations of 2.2. Therefore G,,, = ,qPr, for all r, and the Go,, from 2.2 are the Frobenius kernels of G,. (So our new notation is compatible with the old one.) 9.5 Let G be a k-group scheme. The image faisceau (cf. 5.5) of Fb in G(') is isomorphic to G/G, (by 6.1)as G, = ker(Fb). For each subgroup scheme H of G

we can identify (1)

FL(H) N F',(H)

N

HIH,

by 9.4(2) and (FL)-'FL(H) with G,H, cf. 6.2(4). The factorization F$ = F$,'FL yields (2)

G,. = (Fk)-l((G(r))r,-r)

for all r' 2 r. Proposition: If C is a reduced algebraic k-group, then each FL induces isomorphisms G/G, 1: G(,)and G,
FL(G) = V({f E k[G]

If ,'= O}’,)).

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Representations of Algebraic Groups

If G is reduced, i.e., if k [ G ] does not contain nilpotent elements ZO,then obviously F,(G) = G"). As we have shown F b to be an epimorphism of faisceaux, each subfaisceau Y of G(') is equal to the image faisceau &((&)-' Y ) .Therefore the last claim follows from (1) and (2). Remark:

G,./G,

N

If G is defined over F,, we can express the results as G/G, G,. -

N

G and

9.6 (The G, and Lie(G)) Let G be an algebraic k-group scheme and I , the ideal in k [ G ] defining 1. Keep this assumption and convention until the end of this

chapter. Obviously G, is the closed subscheme of G defined by x f , I , k [ G ] f P ' . Therefore k[G,] is finite dimensional and the ideal of 1 in k[G,] is nilpotent. Hence (cf. 8.1):

(1) Each G, is an infinitesimal k-group. Choose f l , . .,fm E I , such that the f;. + 1: form a basis of Zl/Z:. Then m = dimLie(G) and the fi generate I, as an ideal. One has obviously dimk[G,] 5 p'" for all r, and equality holds if 1 is a simple point of G (cf. 9.3(3)). So we get (e.g., by [DG], I1,& 2.1/3)

(2) Zf G is reduced, then dim k[G,] = prdim(')for all r E N. We obviously have for all r E N (and any G)

(3)

Lie(G,) = Lie G.

The subalgebra UtP1(Lie(G))= UtPJ(Lie(Gl))of Dist(G,) c Dist(G), cf. 7.10(2), has dimension p", whereas dim Dist(G,) = dim k [ C , ] 5 p". This implies (4)

U["I(Lie(G))N Dist(G,).

This shows that G1 is the infinitesimal k-group corresponding to the p-Lie algebra Lie(G) as in 8.5.b and that the representation theory of G1 is equivalent to that of Lie(G) as a p-Lie algebra (cf. 8.6).

9.7 Proposition: Let G be a reduced algebraic k-group and r E N. Then G operates on Dist(G,)Fr through the character g H det(Ad(g))p"-'

where Ad denotes the adjoint representation of G on Lie(G).

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Representations of Frobenius Kernels

Proof: Recall from 8.19 that the conjugation action of G on G, leads to representations of G on k [ G , ] and M(G,) = Dist(G,) and that M(G,)Fris a one dimensional submodule on which G has to operate through some character i( E X ( G ) . Set q = p' and choose f l , . . .,f, E Zl such that the fi + I f form a basis of l l / l f .Let be the image of fi in k [ G , ] . As G is reduced, hence 1 a * ~ ~0 mI ) a ( i ) < q for all i simple point, the monomials ~ ~ ( ’ ~ $ ( 2 ) * .with form a basis of k[Gr]. We can identify k [ G , ] with the factor ring k[Tl,. . . ,T,]/(T4,,. , . , T4,) of the polynomial ring k [ T , , . . , , T,]. It is therefore a graded ring in a natural induces an endoway. Any endomorphism cp of the vector space f:-' is the only basis morphism of the graded algebra k [ G , ] . As F = element of degree m(q - l), it has to be mapped under cp into a multiple c(cp)F of itself. Obviously cp H c(cp) has to be multiplicative. This implies c ( q ) = det(cp)4-1 for all cp, as this is obviously true for cp in upper or lower triangular form (with respect to the J),hence for all cp by multiplicativity. This extends easily to any k-algebra A and any endomorphism of k x 0A as c(cp) is obviously a polynomial in the matrix coefficients of cp. This can be applied especially to the operation of any g E G ( A ) for any k-algebra A on k [ G , ] 0 A derived from the conjugation action on G,. Then k x 0 A N ( Z l / Z t ) 0 A N Lie(G)* 0 A is dual to the the action of g on adjoint action on Lie(G) 0 A, hence has determinant equal to det(Ad(g))-'. So this implies

fi

c'''iHx

nyZl

zy=

Z’’=

g F = det(Ad(g))-(4-')F. Consider now po E Dist(G,)p, po # 0. If p o ( F ) = 0, then p o ( k [ G , ] F ) = 0 as k [ G , ] F = kF, hence ( k [ G , ] p , ) ( F ) = 0 by the definition of the k [ G , ] module structure on Dist(Gr) in 8.1 1, hence Dist(G,)(F) = 0 by 8.12. This is a contradiction, so we must have p o ( F ) # 0. Then X(S)Po(F) = (Wo)(F) = Po(9-W = det(Ad(9))q-'Po(F) implies x(g) = det(Ad(g))q-' as p o ( F ) is a unit in A . Remark: The same proof works for any algebraic k-group G and for r = 1 because of 9.6(4). So we can take any p-Lie algebra g over k and consider the infinitesimal k-group G corresponding to g as in 8.5.b. Then G = G, and Dist(G) = UfP1(g). Then the proposition implies that the modular function 6, is given by 6&) = det(Ad(g))P-'. The representation of g on Dist(G)f is then given by the differential, i.e., by ( p - l)tr(ad(?))= - tr(ad(?)).As the operation of g determines that of G in this case, we see that G is unimodular if and only if

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Representations of Algebraic Group

tr(ad(x)) = 0 for all x E 9. This is a theorem of Larson and Sweedler, cf. the discussion in [Humphreys 91. 9.8 (G and the G, for irreducible G) Because of 9.4(1) the Dist(Gr) form an ascending chain of subalgebras of G, and one has by 9.3(2):

Dist(G) =

u Dist(G,).

r>O

Therefore 7.14-7.17 imply if G is irreducible:

nr10

(2) If M is a G-module, then M G= MGr. HomJM, M’). (3) Zf M, M‘ are G-modules, then HomG(M,M’) = (4) Let M be a G-module and N a subspace of M. Then N is a G-submodule i f and only i f it is a G,-submodule for all r E N .

or>,,

In (2) and (3) we have descending chains M G 1=I M G 23 MG33 -.. and HOm,,(M,M’)~Hom,,(M,M’) 3 HOmG,(M,M’) 3 .... If dimM < 00 resp. if dim M @ M’ < co, then these chains have to stabilize. So we get (still for G irreducible): ( 5 ) Zf M is a G-module with dimM < 00, then there is an n E N with M G = MGrfor all r > n. ( 6 ) Zf M, M’ are G-modules with dim(M @ M’) < 00, then there is an n E N with HOmG(M,M’) = HOmGr(M,M’) for all r > n.

9.9 We have for any G-module M natural restriction maps Hj(G, M) + H’(G,, M) and (for r’ 2 r) Hj(G,.,M) + Hj(G,, M ) that induce a natural map H’(G, M) + l@Hj(Gr, M). For j = 0 and for G irreducible this is an isomorphism by 9.8(2). For arbitrary j such a result has been proved only under additional assumptions. We shall see in 11.4.12 that they are satisfied in an important case.

Proposition: Suppose that G is irreducible and reduced, that H’(G,k ) = 0 for all i > 0, and that dim Hj(G, M) < co for any G-module M with dim M < co and for any j E N. Then the natui’al map Hj(G, V )+ l@Hj(G,, V ) is an isomorphism for all j n i t e dimensional G-modules V. Proof: Consider for each r 2 1 the Hochschild-Serre spectral sequence Ef?j(r)converging to H’(G, V ) with E,-terms Ei’(r) = H’(G/G,,Hj(G,, V ) ) . We have dim k[G,] < co, hence also dimHj(G,, V ) < co for all j . So G/G, N G and our assumptions on G imply that all Ef.j(r) have finite dimension. For all s 2 r the natural maps G, c+ G, and G / G , + G/G, induce

153

Representations of Frobenius Kernels

a morphism of spectral sequences Efqj(s)+ Ef*j(r)that is compatible with the identity on the abutment, cf. the remark to 6.6. We get thus a directed system of spectral sequences where all terms are finite dimensional vector spaces. Now lim is exact on directed systems of c finite dimensional vector spaces, cf. [Roos], prop. 7, so we get a spectral . . sequence with El" = Efvj(r) converging still to H'(G, V ) . For s 2 r the map on the E,-level factors as follows:

&I

E ; ~ ( s= ) H'(G/G,, H ~ ( G , v)) , -, H'(G/G,, H ~ ( G ,v)) , + H'(G/G,,Hj(G,,

V ) G s+ ) H'(G/G,, Hj(G,, V ) )= E i j ( r ) ,

One gets the first map from the restriction from G / G s to G/G,, and the other ones by observing that the restriction map H'(Gs, V )-,Hj(G,, V ) takes values in the Gs-fixed points of Hj(G,, V ) . For any r and j there is s o ( r , j ) E N such that H'(G,, V ) G s= Hj(G,, V)' for all s > so(r,j), cf. 9.8(5). If i > 0, then H'(G/G,,Hj(G,, V)') = H'(G, k ) 0 Hj(G,, V)' = 0 by our assumption. So E$j(s) + E $ j ( r ) is the zero map for all i > 0 if s > so(r,j ) . .. Therefore E';j = c lim E $ j ( r ) = 0 for all i > 0, the spectral sequence El,' degenerates and yields isomorphisms (1)

H'(G, V ) 2 5 E i s i ( r )=

9H'(G,, V ) G .

On the other hand, the restriction map H'(G,, V ) -, H'(G,, V ) takes values in H'(G,, V ) Gfor all s > so@, i). So the inverse limit of the H'(G,, V ) is equal to that of the subsystem of all H'(G,, V)'. So the claim follows from (1). Remark: This result and its proof have been communicated to me by W. van der Kallen. Cf. 11.4.12for more historical remarks. 9.10 (Frobenius Twists of Representations) For any vector space M over k and any r E N we denote by M(') the vector space that is equal to M as an abelian group and where any u E k operates as up-' does on M . If M is a G-module, then we have a natural structure as a G-module on each M(') with r 2 0, cf. 2.16. Suppose now that M has a fixed Fp-structure, i.e., an Fp-subspace M ' c M with M ' Q F , k = M . We get then a Frobenius endomorphism F, on M and on each M 0 A = M ' QF, A through Fdm' 0 a) = m' Q up. Each F L is an isomorphism of A-modules M 0 A -, M") Q A . Suppose that G is defined over Fp and denote the corresponding Frobenius endomorphism by FG: G -, G. If the representation of G on M is defined over Fp (i.e., if FG(g)F,(m) = F,(gm) for all m E M , g E G(A)), then we can define a new

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Representations of Algebraic Groups

representation of G on M by composing the given G -+ G L ( M ) with F',:G-+G. Then F U : M + M ( r ) is an isomorphism of G-modules if we take the new structure on M just defined and on M ( r ) as above. (This follows from an elementary computation.)

9.11 (The Associated Graded Group) The powers of I , define a filtration of k[G] and we can form the associated graded algebra gr k[G] = OnzO I ; / I ; + ' , There is obviously a surjection from the symmetric algebra S(I,/Z:) onto gr k[G] compatible with the grading. The formulas 2.4(l), (2) show that A G and also induce a comultiplication and an antipode on gr k [ G ] making (together with the obvious augmentation) gr k[G] into a (commutative and cocommutative) Hopf algebra. So there is a k-group scheme gr (G) with gr k[G] N k[gr(G)] (the associated graded group). We can interpret S(Zl/I:) as k[((I,/I;)*),] = k[(Lie G),]. Then the surjection S ( I l / I ; ) + gr k [ G ] = k[gr(G)] is compatible with the Hopf algebra structure (again because of 2.4(1), (2)).Thus:

(1) gr(G) is canonically isomorphic to a closed subgroup scheme of Lie(G),. The same arguments as in 9.6(2)imply

(2)

If G is reduced, then gr(G)

'v

Lie(G),

and

(3)

If G is reduced, then gr(Gr) = (Lie(G),), for all r E N.

9.12 (A Filtration of the Hochschild Complex) The filtration of k [ G ] as in 9.11 leads also to a filtration of the Hochschild complex C ' ( G , M ) for each G-module M . We set for all i, n E N (1)

C'(G,M),,) =

M @I I;(') @I I"," 0 .* * @ Zy"),

where we sum over all i-tuples (a(l),. . . ,a(i)) E N’ with x j a ( j )2 n. Because of 2.4(l), (2) and as AM(m) - m 0 1 E M 0 I , for all m E M the definition of the coboundary operators in 4.14 shows

(2)

a'C'(G, M),,, c C'+'(G,M),,,

for all i and n. Each quotient C'(G, M),,,)/C"(G, M),,+ ,) can be identified with the direct sum of all

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Representations of Frobenius Kernels

over all i-tuples (a(l), . . . ,a(i))with a(i) = n. We can on the other hand regard M as a trivial gr(G)-module and form C'(gr(G), M). The grading on k[gr(G)]leads in a natural way (cf. 4.20) to a grading on each C'(gr(G), M ) . We denote the homogeneous part of degree n by C'(gr(G), M),,. Then

(3)

C'(G, M)(n)/'Ci(G,MI(,+ 1) N Ci(gr(G),MIn

for all i, n. These identifications are easily checked to be compatible with the boundary operators so that the associated graded complex of C'(G, M) is isomorphic to the graded complex C'(gr(G), M ) = C'(gr(G), k ) 0 M. The general theory about filtered complexes (consult, e.g., [GI, 1.4) shows that there is a spectral sequence with E,-terms E i j = H'+j(gr G, k)j 0 M. If G is irreducible, then r),,,,Z;+l = 0, hence C'(G, M),,, = 0 for all i. Therefore in this case the spectral sequence converges to the cohomology of the original complex.

no>,

Suppose G is irreducible. Then there is f o r each G-module M

9.13 Proposition:

a spectral sequence with E i j = H'+j(gr(G),k)i 0 M

(1)

H'+j(G, M).

This is what we proved in the last section. Let me add that the spectral sequence is compatible with the cup product in case M = k resp. with the H'(G, k)-module structure on H'(G, M) in the general case. If some other group H operates on G through group automorphisms, then it operates on C'(G,k) preserving the filtration. Then we get a natural action of H on each term of the spectral sequence such that all differentials are homomorphisms of H-modules. Also the filtration on the abutment is compatible with the action of H. This generalizes to an arbitrary G-module M if we also have an operation of H on M compatible with the operation of G (i.e., defining a representation of G = H). 9.14 Proposition:

Let G be reduced and irreducible. Set g = Lie(G). a) There is for each G-module M a spectral sequence converging to H'(G, M ) with the following El-terms: If p # 2, then

where we sum over all finite sequences (a(n)),,*1, and (b(n)),,? in N with i +j

=

1 (2a(n)+ b(n))

na 1

and

i=

C (a(n)pn+ b(n)p"-').

na 1

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Representations of Algebraic Groups

I f p = 2, then

,

where we sum ouer all jinite sequences (a(n)),,? in N with i

+j

=

1 a(n)

n t1

and

i= flb

1

a(n)p"-'.

b) Let r E N and M be a G,-module. If we take above only r-tuples (a(n)),, n s r and (for p # 2)(b(n)),,,,,, then we get the EYj-terms in a spectral sequence converging to H'(G,, M ) . Proof: This follows from 9.13 and 4.27 using 9.11(2),(3). Remark: Again these spectral sequences are compatible with the operation of some group H on G or G, through automorphisms if H operates also on M in a compatible way (e.g., always for the trivial module M = k). This follows from the fact that H then operates on gr(G) = g. or gr(G,) = (go)?through a representation on g so that the isomorphisms in 4.27 are compatible with the action of H.(This applies especially to the operation of G on G, through conjugation.) The (r) denotes a twist of the operation of H as in 9.10. 9.15 The spectral sequence in 9.14.b is especially easy for r = 1.

Lemma: If p = 2, then we can compute H'(G,, M ) for any G,-module M as the cohomology of a complex O+ M

--+

M Q g* + M Q S2g* + M Q S39* + * * *

where g = Lie(G).

Proof: We have by 9.14 that M Q S'g = E t 0 whereas EYj = 0 for j # 0 or i c 0. So the only non-zero differentials in the spectral sequence are d';': E';' + EY'.'. They provide the maps in the complex and its cohomology groups E$' are equal to its abutment. Remark: Note that we do not have to assume G to be reduced and irreducible when dealing with G, (here and below.) The assumption of irreducibility is needed to make the spectral sequence in 9.13 converge to the G-cohomology. As each G, is irreducible we do not need the irreducibility of G in 9.14.b. The assumption of reducedness was needed to get 9.11(3). But we have gr(G,) = (Lie(G),), for any G by 9.6(4).

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Representations of Frobenius Kernels

Let M be a G,-module and set g = Lie(G). If p # 2, then there is a spectral sequence with

9.16 Lemma:

E$j = M

Q

(sig*)(l)Q Aj-lg*

(Gl, M )

+ - Hi+j

Proof: We have in 9.14.b as El-terms Ec;(p-1)+b*-(p-2)0 =M Q Q Ab-Og* for all b 2 a 2 0 and all other El;.' are 0. So E l J = 0 for (p - 2)4'j, hence df.j= 0 for r 1 mod(p - 2) as dr has bidegree (r, 1 - r). We can therefore re-index the spectral sequence by now calling Ef.j the old E(p-1)i+jg-(p-2)i. ( p - 2)r+ 1 This then gives Ebj as above.

+

9.17 (Lie Algebra Cohomology) In order to compute the El-terms of the

spectral sequence from 9.16 it will be necessary to deal with (ordinary) Lie algebra cohomology (cf., e.g., [B3], ch. I, 53, exerc. 12). If g is a finite dimensional Lie algebra over any field and if M is a g-module, then the Lie algebra cohomology H'(g, M ) of M can be computed using a complex M Q Ag* where we take the standard grading of Ag*. The map mi @ cpj E do: M + M Q g* maps any m E M to the unique element M @ g* with x m = cpj(x)mjfor all x E g. (It is something like a comodule map.) In general one has for any m E M and $ E A'g*

I;=

cs=

(1)

di(m 0 $) = c m j Q (Yj A j

$1 + m 0 di($),

with mj, ' p j as above and where d;:A'g* + Ai+'g* is the boundary operator in the case of the trivial module. This in turn is uniquely determined by d;: g* + A2g* N (A2g)* which is the transpose of A2g + g, x A y~ -[x,y], and by the derivation property (2)

d: +j(cp

A

$1 = ~ X V )A $ + ( - 1)'q A dJ($)

for all cp E Aig* and $ E Ajg*. 9.18 Lemma: Let M be a G,-module and set g = Lie(G). Suppose p # 2. Then one has in 9.16

EY.' = Hj(g, M )

for all j e N.

Proof: We have E$j = M Q Ajg* and dE*jmapsM Q Ajg* to M Q Aj" 9* for all j E N.So we have to show that the complex (E:,', d$') is the same as the one computing the Lie algebra cohomology. The compatibility of the spectral sequence with the cup product in the case k = M and with corresponding module structures in general implies that the

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Representations of Algebraic Groups

d$i have derivation properties analogous to 9.17(1),(2).It is therefore enough to prove that d$': M -r M Q g* and d $ ' : g* -+ A'S* in the case M = k are the same maps as in 9.17. In the original notation of 9.14 our present E:3i was called E';' and arose as a subquotient of Ci(G1,M)(i)/Ci(G,,M)(i+,). Any e E E:*i has a representative Z E Ci(G1,M)(i,with d i g € Ci+'(G1,M)(i+,)and d$'(e)is theclass of d i g in the subquotient E:3i+1of Ci+'(G1,M)(i+l)/Ci+l(G,, M)(i+2,. In the case i = 0 we have d o : M -r M Q k [ G , ] , m H A,(m) - m Q 1 . We can write AM(m)= m Q 1 + mi 0 & where J E I , = { f k ~[ G , ] I f(1) = O}. We have C'(G,,M)(,) = M and C'(G,, M),,, = M 01; for all n E N, hence E:*' = M Q l,/lf = C1(G1,M)(,,/C'(G,, M)(2).Therefore dg3'(m) = mi Q where 3 = fi + 1:. The operation of any x E g = (l,/l:)*is given by x m = f;.(x)mi.This shows that d;*' is the same map as in 9.17. Take now M = k and consider d:,'. It maps g* = I J I f = C1(G1,k)(,J For any f E I, we C1(Gl,k)(,)into a subquotient of C2(G1,k)(2)/C2(G1,k)(3). can write A G ( f )= 1 Q f+ f Q 1 + &Of: with 4, E I,, cf. 2.4(1). & Q So f = f + 1: E l , / l ; = E:,' is mapped to the Then 8 7 = in the subquotient H2(grG , , k ) , of C 2 ( G , , k ) ( , , / class of -Xi= J Q C2(G,,k),,, = C2(grG , , k ) , . By the definition of the cup product this is the sum of the products of f;. = fi + 1: and f:. = f ; 1: in H'(gr G,, k). It belongs to the subalgebra generated by H'(gr G,, k ) N g* which is identified with Ag*. So d:v'r = A As the Lie algebra structure on (11/1:)*= Dist:(G) is defined through [ x , y ] = ( x Q y - y Q x ) 0 AG we see that d:,' is transposed to x A y H [ x , y ] as claimed.

If= ,

If=, x

If=,

cI=,

fi. fi

I:=,

fi

+

-If=,x 7;.

Remark: Notice that this computation also gives the boundary maps in the complex of Lemma 9.15. 9.19 (Ordinary and Restricted Cohornology) If M , M' are G,-modules, then we can interpret each Ext&(M', M) resp. Exte(M', M ) as set of equivalence classes of exact sequences 0 - r M -+ M, + M, - + . * . + M i -r M ' + O

of homomorphisms of G,-modules (resp. g-modules). So we have a natural map Extb,(M', M) + Exti(M', M). Taking M' = k we get a natural map H i ( G 1 ,M ) -+ H'(g, M). Let us describe this explicitly for i = 1. Each 1-cocycle cp: g -+ M defines an extension of g-modules 0 -+ M -+ M(cp) -r k -+ 0 where M(cp) = M 0 k as vector space with x E g operating through x(m, a) = ( x m + acp(x),0 ) for all a E k and m E M . One checks easily that this is an extension of G,-modules if and only if cp(x["])= xp-'cp(x) for

159

Representations of Frobenius Kernels

all x E g. This equation is certainly satisfied if cp is a coboundary, i.e., of the form x H xm for some m E M . So we get an embedding H'(G1,M ) 4 H ' ( g , M ) . More precisely, the image is exactly the kernel of the map associating to the class of cp as above (in H'(g, M ) ) the map X H cp(xIP1)- xp-' cp(4 from g to M . This map is semilinear, i.e., it is additive and satisfies cp(ax) = apcp(x) for all a E k and x E g. Let Homs(g, M ) be the space of all such maps. We have so far constructed an exact sequence O-+H'(G,,M)-+H'(g,M)-+Homs(g,M). We can be more precise. An elementary computation using the cocycle property cp([x, y]) = xcp(y) - ycp(x) for all x, y E g shows ( P ( X [ ~] ) xp-'cp(x) E M g for all x E g. So we can replace Homs(g,M ) by Homs(g,M g ) .We can now go on and associate to any $ E Horns((, M g ) a p-Lie algebra g($) that is an extension 0 --* M

-+

g($)

-+

g -+ 0

of p-Lie algebras, where we regard M as a commutative p-Lie algebra with mrP1= 0 for all m E M . We take g($) = M @ g with Lie bracket [(m,x), ( m ' , ~ ' ) ]= (xm' - x'm, [x,x']) and pth power (m,x)[P1= (xP-lm + $(x),xIP1) for all m, m' E M and x, x' E g. (It is left to the reader to check that this is indeed a pthpower map on the semi-direct product.) Now g($) and g(0) are equivalent extensions if and only if there is an isomorphism g(0) -+ g($) of p-Lie algebras of the form (m,X)H (m + cp(x), x) for some cp E Hom(g, M ) . Such a map is a homomorphism of Lie algebras if and only if cp is a l-cocycle, and it is compatible with the pth power map if ) for all x E g. So g($), g(0) are equivalent and only if $(x) = ( P ( X [ ~-~xP-'cp(x) if and only if $ is in the image of H ' ( g , M ) -+ Homs(g, M ) . The set of all equivalence classes of all central extensions of p-Lie algebras (resp. of Lie algebras) 0 -+ M -+ b + g + 0, such that the adjoint operation of g z t)/M on M is the given operation, is a vector space in a natural way with g(0) as zero. One can identify this space with H2(G1,M ) resp. H Z ( g ,M ) and one can show that the map $ ~ g ( $ )induces a linear map Homs(g,Mg)-+ H2(Gl,M ) . Furthermore, one can show that the image is exactly the kernel of the forgetful map H 2 ( G , ,M ) -+ H 2 ( g ,M ) . In this way we get an exact sequence O--*H'(G1,M)-+H'(g,M)-+HomS(g,Mg) (1) -+

HZ(Gl,M)-+ H 2 ( g , M ) HomYg,H'(g, MI), -+

where I want to refer to the original proof in [Hochschild 1) (cf. p. 575) for the last map and the exactness at the last two places to be looked at. We shall construct an exact sequence in 9.20(1) that will contain the same terms as (1)

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Representations of Algebraic Groups

and ought to be isomorphic to (1). In order to prove that all terms are the same in both sequences, we need (1) in a special case: (2) If M is an injective G,-module, then the canonical map H'(g, M ) -+ Horns(g, M E )is an isomorphism. 9.20 Proposition:

The spectral sequence in 9.16 has the following E,-terms: Etj N

Hj-'(g, M ) 8 (S'g*)(').

Proof: The derivation property of the differential

0 (sig*)(l) -+ M 8 I \ j - i + ' g * 8 (sig*)(l)

dij:M 8

+

implies dkj(m 8 cp 8 +) = d;J-'(m 8 cp) 8 + ( m 0 cp 8 1)(1 8 d&(+)) for all m E M , cp E W i g * and 1,5 E (S'g)('), where d& is the differential in the case M = k. Therefore it is, by 9.18, enough to show d & = 0 for all i. Again the derivation property shows that it is enough to show d;;; = 0. We know from 9.16 that E:j # 0 implies j 2 i ' 2 0. As d, has bidegree ( I , 1 - r) this shows E i j = E Y for all ( i , j )E ((0, l),(O,2),(1, l)} and Ei-' = E:*'/im(dy*'), E;.' = ker(dY.') c Ey.', and E$iz= ker(dys2) c EysZ.We see also that E:' = H'(G,, M ) and that there is an exact sequence 0 -+ Ekl -+ H Z ( G , ,M ) + -P 0. Combining this with 9.18 we get a six term exact sequence 0 -+ H'(G,, M ) + H'(g, M ) + E:.' (1) -+

HZ(Gl,M)+ H Z ( g ,M ) + E:,'.

Here E;*' = ker(dA9') c E;.' = M 8 g*('). Take now an injective G,-module M , with M I = k, e.g., the injective hull of k or k[G,] with the left or right regular representation. Now 9.19(2) implies H1(g,M,) N Hom"(g, M ; ) = g*(l), and that (because of the naturality of the maps) the inclusion of k into M , induces a surjection H'(g, k) -P H'(g, M l ) , hence (by the naturality of (1)) a surjection of E:.' for M = k to Et-' for M = M,, hence to H'(g,M,) = g*(l). But E:,' for M = k is equal to ker(di;;) c g*(l),so dimensional considerations show d;;; = 0. 9.21 Using the spectral sequence from 9.16 and Proposition 9.20 one can

show that H'(G1, k) is a finitely generated k-algebra (under the cup product), cf. [Friedlander and Parshall 41. One can now define a cohomology variety of G , and support varieties for finite dimensional G,-modules. For some results in this direction one may consult [Friedlander and Parshall 5,6] and [Jantzen 12, 13, 141.

10

Reduction mod p

Let G be a flat group scheme over Z. If V is a finite dimensional GQ-mOdUle, then we can find a G-module V, with V, 0, Q 1: V,cf. 10.3. We can then form the Gk-mOdUle V, = V, 0, k for any ring k. If p is a prime number and k = F, (or k = an algebraic closure of F,), then we say that we get V, from V through reduction mod p . In general there will be more than one module (even up to isomorphism) that we can get from V through reduction mod p , as we can choose different V,. One can still show that they have the same composition factors. (One can express this in the form that the class of Vk in the Grothendieck group of Gk is uniquely determined by V.) This independence was proved in [Serre] generalizing the corresponding statement for abstract finite groups due to Brauer. One can even generalize Brauer’s lifting of idempotents if Z[G] is free. So every injective indecomposable Gk-module lifts to the p adic completion of Z. Furthermore, then Brauer’s reciprocity law holds in this situation. These results were proved in [Green 11, and we follow Green’s approach here. In the case where GQ is semi-simple, V, has usually been constructed as a Dist(G)-stable lattice in V. Such a lattice is indeed a G-module (10.13). This is proved using a property of Dist(G) and Z[G] which holds (as proved by Bruhat) for any “smooth” G, cf. 10.11.

161

162

Representations of Algebraic Groups

We can replace Z above by any Dedekind ring R and F, by any residue field of R . Then all the results will still hold and we do everything in this generality. (Therefore the term “mod p” occurs in the title and the introduction of this chapter.) 10.1 (Restriction of Scalars) Let k‘ be a k-algebra and G a k-group functor. We observed in 2.7(6) that any G-module M leads in a natural way to a Gk.-mOdUk M @ k’: For each k’-algebra A’ the group Gkt(A’)= G(A’) operates as given on M 6 A‘ = (M 6 k’) @k, A‘. There is a functor in the opposite direction: We can regard each Gk-module V in a natural way as a G-module. For any k-algebra A the map a H 1 0 a is a homomorphism of k-algebras A + k‘ @ A , hence induces a group homomorphism G(A) -+ G(k’ @ A ) = Gk,(k’6 A ) and thus an operation of G(A) on V@kr(k’@ A ) N V @ A . These operations are compatible with homomorphisms of k-algebras and lead therefore to a representation of G on V regarded as a k-module. In the case of a group scheme we get the comodule map of V as a G-module (i.e., V + V @ k[G]) from that as a GkdnOdUk (i.e., V + V @k! k’[Gk,]) using the identification V @k’k’[Gk,] = V @k, (k’ 0 k[G]) = V @ QGI. If M is a G-module, then the map i,: M + M @ k‘, rn H rn 6 1 is a homomorphism of G-modules, if we regard the Gk.-mOdUle M @ k‘ as a G-module as above. Indeed, the operation of any G(A) on M @ k’ @ A comes from the operation of G(k’ @ A ) on this module and the homomorphism j,:u H 1 @ u from A to &‘ @ A . We can regard ,i @ id,: M @ A + M 6 k’ 6 A also as id, @ j , and it is therefore compatible with the action. The universal property of the tensor product implies that cp H cp 0 i, is a bijection HOmk.(M @ k’, V )-+ HOmk(M, V ) for any k-module M and any k’-module V. We claim that it yields a bijection (1)

HOmG,,(M @ k‘, V )3 HOmG(M, V ) ,

when M is a G-module and V a Gkr-module.As ,i is a homomorphism of G-modules we have already proved one direction. Suppose now that cp 0 ,i is a homomorphism of G-modules and let us show that cp is a homomorphism of Gkt-mOdUleS. Consider any k’-algebra A’ and the map cp @ id,,: M @ k‘ @kfA’ + V @k, A’. We have to show that it Commutes with the action of Gk,(A’)= G(A’). We can identify M @ k’@k’A’ 2 M @ A’ and then factorize the map at first into (cp 0),i @ idA,:M 0 A’ V 6 A’ and then the canonical map V @ A‘ -+ V @kt A’. By assumption the first map is G(A’)-equivariant, where we get the action of G(A’) on V Q A’ --f

163

Reduction mod p

from that of Gk.(k’6 A’) on V Qk,(k’ @ A’) N V @ A’ via A’ + k‘ 6 A‘, a H 1 6 a. As k‘ 0 A’ + A’, b 0 a H ba is a homomorphism of k’-algebras, the corresponding map V @ A‘ N V @k,(k’6 A’) + V @k? A’ is compatible with Gkg(k’0 A ’ ) + Gk,(A’),hence with the action of Gk,(A’). This is what we had to prove. 10.2 Lemma: Let k be an integral domain, G a jlat k-group scheme, and M a Gmodule. a) For all a E k the k-submodule

ker(a1,) = { m E M I am = 0 } is a G-submodule of M. b) The torsion submodule

is a G-submodule of M.

Proof: Let k’ be the field of fractions of k. Then MI,, is the kernel of the homomorphism of G-modules i,: M + M @ k‘ as in 10.1, hence a Gsubmodule, cf. 2.9(3). In order to get a), we have to prove that A,(ker(aI,)) c ker(a1,) 6 k[G]. As A M is a homomorphism of k-modules, we have to know that ker(alM)6 k[G1 = ker(aIhf@k[G]). The exact sequence 0 + ker(a1,)

+ M + M,

where the last map is multiplication by a, yields another exact sequence 0 + ker(a1,) 6 k [ G ] + M 0 k [ G ] + M 6 k[G], as k [ G ] is flat. The last map is again multiplication by a, so the claim follows. 10.3 Lemma: Let k’ be a k-algebra that is finitely presented as a k-module. Let G be a jlat k-group scheme and let M, N be G-modules. If M is projective over k and if N is a direct summand of E 6 k [ G ] for some k-module E (regarded as a trivial G-module), then the canonical map (1) is bijective.

Hom,(M, N ) 8 k‘ + Hom,,,(M

@ k’,I N@ k’)I

164

Representaliom of Algebraic Groups

Proof: If the result is true for M and N , it follows for direct summands. So we can assume that M is free over k and that N = E 0 k[G], hence N 0 k‘ N (E 0 k’) 0,“k’[Gk.]. Then 3.7(1) and the Frobenius reciprocity yield isomorphisms HOmG(M, N ) ‘Y Homk(M,E ) and HomG,,(M6 k‘,N 8 k’) N Homk.(MO k‘,E 6 k‘).

So we have to show that the canonical map Hom,(M,E) 0 k ’ + HOmk,(M@ k’, E 0 k’) is an isomorphism. This is clear for M = k (both sides being isomorphic to E 0 k ’ ) whereas, in general, one has to use that tensoring with k‘ commutes with taking direct products, cf. [Bl], chap. I, $2, exerc. 9a. Remark: Let G be a k-group scheme such that k[G] is a projective k-module. Then we can apply the lemma to M = k[G] = N and get:

(2)

(EndGk[G]) O k’ 3 End~,,(k’[Gk,])

for each k-algebra k‘ that is finitely presented as a k-module. 10.4 (Lattices) Let R be a Dedekind ring and K its field of fractions. Let me remind you that a lattice in a finite dimensional vector space V over K is a finitely generated R-submodule M of V such that the canonical map M OR K + V is an isomorphism. This map is always injective, so we can weaken the condition to “V is generated by M over K”, cf. [BZ], ch. VII, $4, no 1, rem. 1. As R is a Dedekind ring, any such lattice is a projective R-module and its rank is equal to dim, V. If M is a lattice in V and V’ is a subspace of V, then M n V’ is a lattice in V’ and (M + V ’ ) / V ’is one in V/V’. If M, c V, and M, c V2 are lattices, then M, ORM, is one in V, OKV2.(For more details, consult [B2], ch. VII, $4,no 1.) Lemma: Let R be a Dedekind ring and G a flat group scheme over R. Let K be

the field of fractions of R and V a finite dimensional G-stable lattice in V.

GK-mOdUk.

Then there is a

Proof: Let v , , v,, . ..,0, be a basis of V. By 2.13(3) there is a G-submodule M of V containing all v i that is finitely generated over R. As M generates V over K , it is a lattice.

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10.5 Let us assume from now on (in this chapter) that R is a Dedekind ring that is not a field. Let K be the field of fractions of R and let m be a maximal ideal of R. Set k = R/m. Furthermore, let G be an R-group scheme such that R[G] is a projective R-module.

Proposition: Suppose that R is a complete discrete valuation ring. Then there is for each idempotent e E End,,(k[G,]) an idempotent e " End,(R[G]) ~ inducing e. Prooji

The canonical map from End,(R[G]) BRk

N

End,R[G]/mEnd,R[G]

to End~,(k[Gk])is an isomorphism by 10.3(2). We want to apply Proposition 3.15 to the ring End,R[G] and its chain of ideals mi = m'End,R[G]. If that is possible, then we get the proposition as an obvious consequence. So we have to prove that naturally End,R[G]

(1)

N

lim End,R[G]/m'End,R[G].

c

If M is a G-submodule of R[G] that is finitely generated over R, then it is a free R-module and we have an isomorphism Hom,(M, RCG]) N M * by Frobenius reciprocity. As R is complete, we get Hom,(M, R[G]) N&mHom,(M,

R[G])/m'Hom,(M, RCGI).

This implies (l), as RCG] is the direct limit of such M . 10.6 Corollary: Let R be as in 10.5. For each indecomposable and injective Gkmodule Q there is a direct summand 0 of R[G] with Q N 0 @ R k.

Proofi We may assume that Q is a direct summand of k[Gk]. (Combine 3.16 and 3.10!) Therefore we can find cp E End,,(k[Gk]) idempotent with Q N cp(k[Gk]). Let E End,(R[G]) be idempotent inducing cp. Then +(R[G]) is a direct summand of R[G] and

+

Q

cp(k[GkI) = (+ @ idk)kCGkl

$(RCG1) @ k*

Remark: For G = SL, and G = GL, one can find explicit decompositions of R[G] in [Winter] and [Sullivan 11.

10.7 (Reciprocity) For any finite dimensional G,-module V we can find by 10.4 a G-stable lattice VRin V and then form the Gk-moduleV, = VR@lRk. We have obviously dim

= i'kR(VR)= dim, I/.

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Representations of Algebraic Groups

The choice of a G-stable lattice is not unique and different choices will lead in general to non-isomorphic Gk-modules. We claim, however, that the composition factors of V, are uniquely determined by V. Let E be a simple G,-module and let QEbe an injective hull of E, cf. 3.16/ 17. The multiplicity [V,: E] , , of E as a composition factor of V, is then equal to (cf. 3.17(3)) [h:E],,

(1)

= dim Hom,,(I/,,Q,)/dim

End,,(E).

Let fi be the rn-adic completion of R and denote by d its field of fractions. We can identify k with the residue field of d . By 10.5there is a direct summand 0, of the Gk-modukfi[Gk] = R[G] B R f iwith O E @ l R kN QE. Now 4.18(1) implies (as V, N ( VR@, fi) @a k )

fi, &,I @R k Horn,,( V, QE). On the other hand (V, aR fi) @k k N V OKd and d is flat over fi, so already Horn,,( VR@R

(2)

N

9

2.10(7) implies

(3)

Horn,,(

v, @ R fi, 0,)

@fi

k

N

Horn,,( V @K

d,&,@R k).

A comparison of ranks and dimensions implies the “Brauer reciprocity formula” (4)

[ K: E ] G k = dimi Horn,,( I/ @ K

d ,0,

8 2 k)/dim End,,(E).

10.8 Assume in addition that each simple G,-module E (resp. each simple GK-

module V )satisfies End,,E = k (resp. End,,l/ = K ) and that each G,-module is semi-simple. (This is, e.g., satisfied for a split connected reductive group if char(K) = 0, see Part 11.) In order to simplify, let us assume that R = 8. Consider a simple G,-module V and a simple Gk-moduleE . Let us construct &, and V, as in 10.7. The semi-simplicity of &, @, K and End,,V = K imply that dim,Hom,,(V,& & K ) is equal to the multiplicity of V as a composition factor of &, ORK . So 10.7(4)yields (1)

=

@R

K : ‘]GK*

If we take an abstract finite group r and carry out the construction of 8.5.a over R, then we get Brauer’s original theorem. 10.9 (Grothendieck Groups) Return to the more general situation of 10.7. We

can interpret the result as a statement about Grothendieck groups. Recall that one can associate a Grothendieck group to each abelian category. One starts with the free abelian group generated by the objects of the

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Reduction mod p

category (let [MI denote the generator corresponding to an object M ) and one then divides out the subgroup generated by all [MI - [M’] - [M”] for all short exact sequences 0 + M ’ + M -,M“ + 0 in the category. Let us denote by W ( G )the Grothendieck group of all those G-modules that are finitely generated over R. Define W(GK)and W(Gk)by analogy. Then W(GK) and W(Gk)are free abelian groups with the classes [E’] resp. [El of all simple GK-modulesE‘ resp. simple Gk-modules E as a basis. For any finite dimensional Gk-module M one has

where E runs through a system of representatives of isomorphism classes of simple Gk-mOdUleS. (Similarly for GK.) In these cases (over a field) the Grothendieck groups have a natural ring structure induced by the tensor product, i.e., with [ M @ M ’ ] = [MI [M’]. We can now deduce from 10.7(4) that the class [V,] of V, is uniquely determined by V and does not depend on the choice of V,. In this way one easily gets a homomorphism of rings W(G,) + W ( G k )with [V] H [V,]. 10.10 Let me mention some results about W ( G )proved in [Serre]. The map M H M 8 K induces a homomorphism W ( G )+ W(GK).Its kernel is equal to the subgroup Wlor(G) of W ( G )generated by all [MI such that M is a (finitely generated) torsion module (and a G-module). Lemma 10.4 implies that the map is surjective, i.e., that we get an exact sequence of the form

(1)

0 + Wtor(G) + W ( G )+ W(GK)+ 0.

Consider the category of all G-modules that are finitely generated and projective over R and let Wpr(G) be its Grothendieck group. The inclusion of categories induces a homomorphism from Wpr(G) to W ( G )that turns out to be an isomorphism (2)

gpr(G)

=SW ( G )

The reduction mod rn (i.e., M H M @ k = M/mM) defines a homomorphism Wpr(G) + W(Gk),by (2) also W ( G )+ W(Gk).One checks that Wlor(G) is mapped to 0 and gets W(GK)+ W(G,) by (1). This is the same map as the one constructed using 10.7(4). If R is a principal ideal domain, then Wlor(G)= 0. (If M is a G-stable lattice in a finite dimensional GK-moduleV, then [M/mM] = 0 in R(G)as M N mM. One can show that W,,,(G) is generated by such [M/mM] for all possible m.) Let me point out that in [Serre] the R-module R[G] is not assumed to be projective, only to be flat.

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Representations of Algebraic Groups

10.11 (Smooth Group Schemes) Set n(R)equal to the set of all maximal ideals

of R . Let us drop the assumption that R [ G ] is projective. We shall assume instead that G is smooth (cf. [EGA], IV, $17 or [DG], I, $4, no 4). Then R[G] is a flat R-module and a finitely presented R-algebra. So the natural map from R[G] to K[G,] = R[G] m R K is injective and we shall always identify R[G] with its image. Furthermore, (cf. [B2], ch. 11, $3, cor. 4 du th. 1)

Note that we can regard each R[G], as R,[GR,]. Furthermore, if I = I , = {f E R[G] I f(1) = 0}, then all Zn/Zn+’ are finitely generated and projective R-modules (cf. [EGA], O,”, 19.5.4). So G is infinitesimally flat and Dist(G) resp. each Dist,(G) is naturally embedded into Dist(G,) N Dist(G) @IRK resp. Dist,(G,) N Dist,(G) BRK . The smoothness of G implies (by definition) that GK and all GRIP with p E n(R)are smooth. If these schemes are in addition irreducible, then they are even integral (cf. [DG], 11, $5, 2.l), so K[GK] and all (R/p)[G,,,] = R[G]/pR[G] are integral domains. Especially R[G] c K [ G , ] is integral. So, in this case 7.14-7.16 can be applied to G. 10.12 Proposition: Suppose that G is smooth. Zf GK and all GRIPwith p E n(R) are irreducible, then (1)

R [ G ] = {f E K[G,] lp(f) E R

for all p E Dist(G)}.

Proof: Of course, we only have to prove one inclusion ( I )Because . of 10.1l(1) we can restrict ourselves to the corresponding result for all R[G], = RP[GR,]. So we can assume that R is a discrete valuation ring with a unique maximal ideal p = R x . Let d be the p-adic completion of R . For any f E K[GK], f 4 R[G] there is n E N with x”f 4 R[G], but xn+’f E R[G]. So the claim will follow if we can prove: (2)

xR[G] = { f R~[ G ] I p(f) E Rx

for all p E Dist(G)}.

Let I = {f E R[G] If(1) = 0} and J = Z + nR[G] = I @ xR1. So R[G]/J R/p and J is a maximal ideal of R[G]. It contains the ideal nR[G] which is prime as R[G]/xR[G] N (R/p)[GRIP].So, if we localize at J, we get N

(3)

xR[G] = R[G] n nR[G],.

Let A be the J-adic completion of the local ring R[G],. Then A is a faithfully

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Reduction mod p

flat R[G]~-modUk(cf. [B2], ch. 111, @3,prop. 9), so [B2], chap. I, 43, prop. 9 together with (3) yields:

(4)

nR[G] = R[G] n nA.

Because of (4)it will be enough to show:

(5)

{f E R[G] I p ( f ) E Rn

for all p E Dist(G)} c nA.

Then (2) will follow. Choose T,, T 2 , .. . , T, E J R [G] such that their images in J R [G] J/ ((JR[G],)' + pR[G],) form a basis of this vector space over R/p. As R[G]j = R + JR[G]j we have JRCG] J = IRCG] J

+ pR[G]j

=I

+ (JR[G] jI2 + pR[G)j.

So we can and shall choose T,, T2,. . . ,T, E 1.Now the smoothness of G implies (cf. [DG], I, @4,5.9):

ff "Tl ,T2 ,. . .,TI1 A. For any a = (al, a2,.. . ,a,) E N’ let c,: A + R be the map associating to each

(6)

nr=,

f E A the coefficient of T" = T"' in the power series development of f arising from (6). Each f E I satisfies co(f) = 0 (as 1 c JR[G]J), so c, annihilates I l U l + l where [(a,,. . . ,a,)( = c a i . So (cf. [B2], ch. I, $2, prop. 10): C , I R [ ~E]

HOmR(RIG]/ll"l+',R) Z Ho~,(R[G]/I~"~+',R) @RR N Dist,,,(G) @ R R.

So, if f~ R[G] with p ( f ) E Rlr for all p E Dist(G), then c u ( f )E riff for all a E N’, hence f E nA. This implies (5) and the proposition. Remark: This proposition was announced in [BrT], 3.5.3.1. The proof has been communicated to me by F. Bruhat. 10.13 Proposition: Assume that G is smooth and that G K and all GR,p with p E ll(R) are irreducible.Let V be a finite dimensional G,-module and let M be an A-lattice in V. Then M is G-stable i f and only i f Dist(G)M = M.

Proofi As R is a Dedekind ring, the R-module M is projective of finite rank. We can add an R-module M' to M and M' ORK to V(with trivial action of GK). So we may assume that M is free over R. Choose a basis m,, m 2 , ,. .,m, of M as an R-module. Then this is also a basis of V as a vector space over K . There are (uniquely determined) fjE K[GK]

170

Representations of Algebraic Groups

with

for all i. Now M is G-stable if and only if J j E R [ G ] for all i, j . On the other hand pm, = p ( f t i ) m jfor all p E Dist(G), so M = Dist(G)M if and only if p ( J j ) E R for all p and i, j . Now the claim follows from 10.12(1). Remark: Of course this result generalizes to all group schemes satisfying 10.12(1).

Part 11

Representations of Reductive Groups

This Page Intentionally Left Blank

1

Reductive Groups

The purpose of this first chapter is to introduce split reductive group schemes and their most important subgroups, to fix a lot of notations (which will be used throughout part II), and to mention (without proof) the main properties of these objects. Furthermore, the algebra of distributions on such a group scheme is described and the relationship between the representation theories of the group and of its algebra of distributions is discussed. In the case of an algebraically closed field the reader ought to be familiar with the notions and results described here from [Bo], [ H u ~ ] ,or [Sp2]. (Except for the part on the algebra of distributions, of course.) So the reader is asked to believe that everything she or he knows (about these groups) generalizes nicely to the case of an arbitrary ground ring, which we assume to be integral in order to simplify a few technical details. The existence of these group schemes over Z (hence over any ring k) was first proved in [Chevalley]. They were then characterized by Demazure in [SGA 31. There one can find proofs of most of the results mentioned below and one can use Demazure’s thesis [D] as a guide to where to find them. One can also find many results in [Borel 11. The algebras of distributions were first determined in [Haboush 31. At least in the semi-simple and simply connected case they had been used before as they coincide in this case (for k = Z) with Kostant’s Z-form of the enveloping

I73

174

Representations of Algebraic Groups

algebra of a complex semi-simple Lie algebra with the same root system. (See [St 13, [Kostant 21, and [Borel 13 for more details.) More or less the only result proved in this chapter is the isomorphism theorem: A connected reductive group (over an algebraically closed field) is determined uniquely (up to isomorphism) by its root datum. The proof given here (1.14) goes back to Takeuchi. It uses the algebra of distributions and avoids case-by-case considerations. Let me remind you that throughout this chapter k denotes an integral domain. 1.1 (Split Reductive Groups) Let GZ be a split and connected reductive algebraic Z-group. Set GA = (Gz)Afor any ring A and G = GL. Then G K is for any algebraically closed field K a reduced K-group, and it is a connected and reductive K-group. The ring Z[Gz] is a free Z-module, so k [ G ] is free over k and G is flat. Furthermore, Lie(G,) is a free Z-module of finite rank and we have

(1)

Lie G = Lie(G,) Ozk.

Let T, c GZ be a split maximal torus of GZ and set TA= (TZ)A for any ring A and T= 5.Then T, is isomorphic to a direct product of, say, r copies of the multiplicative group over Z. The integer r is uniquely determined as the rank of the free abelian group X(T,) and it is called the rank of G (denoted by rk G). For any algebraically closed field K the group TK is reduced and it is a maximal torus in G K . The k-group T is isomorphic to (GJ and X(T) N X(T,) is isomorphic to Z'. (This uses the integrality of k, cf. 1.2.5(1).)Any T-module M has a direct decomposition into weight spaces (cf. 1.2.11(3))

The A with Ma # 0 are called the weights of M . If we apply this to the adjoint representation on Lie(G), then the corresponding decomposition has the form (3)

Lie G = Lie T 0

0(Lie G),.

ueR

Here R is the set of non-zero weights of Lie(G). So (3) amounts to (4)

(Lie G), = Lie T.

The elements of R are called the roots of G with respect to T, and the set R is called the root system of G with respect to T. For any a E R the root subspace (Lie G), is a free k-module of rank 1.

175

Reductive Groups

Note that, using (l), R can be identified with the root system of GZ with respect to T,. One has for all a E R (Lie G), = (Lie G,), Ozk.

(5) One also has

R=-R.

(6)

1.2 (Root Subgroups) For each a E R there is a root homomorphism xu: G, + G,

tx,(a)t-'

for any k-algebra A and all induces an isomorphism

tE

= x,(a(t)a)

T ( A ) ,a E A , such that the tangent map dx,

dx,: Lie(G,) 3 (Lie G),.

(3)

Such a root homomorphism is uniquely determined up to a unit in k. We shall always assume that xu arises from a similar homomorphism over Z making it unique up to a sign change. The functor A H x,(G,(A)) = x , ( A ) is a closed subgroup of G denoted by U,. It is called the root subgroup of G corresponding to a. So xu is an isomorphism G, 3 U, and we have

(4)

Lie(U,) = (Lie G),.

We denote the corresponding groups over Z by U,,,. So U, = (Uu,z)k. For two roots a, /.3 with a + /.3 # 0, there are integers cij for all i, j > 0 with ia + j f i E R such that the commutator ( x u ( 4 xp(b)) = x,(a)xp(b)x,(

-a ) x & - 4

is given (for all A and all a, b E A ) by

Here the product has to be taken in a fixed (but arbitrarily chosen) order. The cij will depend on that choice. (We have cij E Z as we take the xu over Z.) 1.3 (Coroots) The set (1)

Y ( T )= Hom(G,, T )

has a natural structure as an abelian group. We usually write the group

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Represeatatiomr of Algebraic Groups

law additively (as we do for X(T) = Hom(T, G,)). Now T N (GJ implies Y ( T )N End(G,,,)' N Z'. For any A E X(T) and cp E Y ( T )we have A 0 cp E End(G,,,) N Z, so there is a unique integer (A,cp) such that I 0 cp is the map a H a<".'P) on each G,(A) = A '. This pairing <, > on X(T ) x Y ( T )is easily seen to be bilinear and to induce an isomorphism Y(T ) N Hom,(X(T), Z). The definition of Y ( T ) generalizes obviously to any commutative group scheme. So does the definition of the pairing. Note that Y(T,) 2: Y ( T )and this isomorphism is compatible with the pairing and with X( T,) N X(T ) . For any a E R there is a homomorphism (2)

c p ~ SL2 : +G

such that for a suitable normalization of x, and x - ~ : (3)

for any A and a E A. If so, then

and

for any a E A" and any A. Obviously av E Y ( T ) .It is called the coroot or dual root corresponding to a. It is uniquely determined by a. We shall usually choose cp-, such that

for all

( ) in a b c d

SL,. Note that

(t :)=(

-1 0 a b 0 l)(c d )

-' -1

(

0 0 1)

-'

for all matrices in SL,. This normalization is compatible with (3) and (9,i.e., (- a)v is indeed the inverse of a". If cp, is not injective, then its kernel consists of all

(:

:)witha'

= 1.This

occurs if and only if I(av(a))= 1 for all such a and all I E X ( T ) .So: (7)

ker(cp,)

N

p(2,o av E 2Y(T),

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Reductive Groups

The image of cp, is normalized by T and intersects T i n a'(G,) as any element in cp;'(T) centralizes all diagonal matrices in SL,, hence is a diagonal matrix. So we have inside G the product

G, = W , ( S L , ) = (T

(8)

cpu(SL2))/aV(Gm)*

Each G, is a split and connected reductive group with maximal torus T and root system {a, - a } . One has G, = G-, and G, = CG(ker(a)).

(9)

Let G’ be a flat group scheme over k and let cp,cp' be homomorphisms G -+ G’.Then: (10) If

(PIG.

= (P'IG.

f i r dl

a E S,

then cp = cp'.

Indeed, let K be an algebraic closure of the field of fractions of k. As k [ G ' ] is a flat k-module, the natural map k [ G ' ] -P K [ G h ] is injective. Therefore cp is uniquely determined by c p K , hence also by cp as G K is reduced. The group G ( K )is generated by T ( K )and all U,(K)with a E S u ( - S ) , hence by all G,(K) with a E S . So cp is uniquely determined by all cp IG, with a E S .

IG(K)

1.4 (The Weyl Group) The set R together with the map a H a' is a root system (as in [ B 3 ] , ch. VI, $1, no 1) in the space generated by R in X(T)&R. Let us denote by s, for each a the corresponding reflection on X ( T )

(1)

s,A

=A

- (A,a')a,

which we extend to the whole of X ( T )OzR by extending a' E Y ( T )N X ( T ) * to X ( T )OZR. so,

(2)

W = (s, I a E R )

is the Weyl group of R . Any g E N G ( T ) ( Afor ) some A acts through conjugation on TA,hence also (linearly) on the Z-modules X ( T A )and Y(TA).If A is integral, then X(T’) N X ( T ) and Y(T , )N Y ( T ) . We get thus operations on X ( T ) and Y ( T ) for which the pairing <, > on X ( T ) x Y ( T )is invariant. The operation of any n,(a) (with a E R , a E A, and A integral) on X ( T )is the same as that of s,. We get for any integral A isomorphisms (3)

(NG(T)/T)(A)

NG(T)(A)/T(A)*

The last equality follows from the fact that each generator s, of W has a representative n,(l) in NG(T)(A),hence so has any w E W. (More generally, NG(T ) / T is isomorphic to the finite k-group associated to W as in I.8.5.a.)

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Representations of Algebraic Groups

Let us choose for any w E W a representative w E NG(T)(k).Then there is for any a E R a constant c, E k" with

(4)

wxu(u)w-l = X,(,)(C,U)

w E NGz(Tz)(Z), then c, E { i-l}.) This

for all u E A and all A. (If we choose implies

wu,w-l = UW(,).

(5)

1.5 (Simple Reflections)Choose a positive system R+ c R and denote by S the N a and (R')' c z e S N a V . corresponding set of simple roots. Then R+ c Cues We can define an order relation 5 on X ( T )(or on X(T )OzR)by

A ~ p o p - A ~ x N a = Np. PER+

UES

(So R + = { a E R I a > O ) and - R + = { a E R I a < O ) . ) O b v i o u s l y A l p - p 5-1. Now the Weyl group is generated already by the simple rejections with respect to R+,i.e., by all s, with a E S. The length l(w)of any w E W is defined as the smallest m such that there exist pl, p2,. ..,p,,, E S with w = Sp1sfl2* - * spm. So I(w) = 0 if and only if w = 1, and I(w) = 1 if and only if w = s, for some a E S. One has obviously l(w) = l(w-') for all w E W. One has for all w E W and a E S l(ws,) =

I(w) + 1 l(w) - 1

if w(a) > 0, if w(a) < 0,

and, symmetrically,

I(s,w) =

(3)

I(w) + 1 l(w) - 1

if w-l(a) > 0, if w-'(a) < 0.

One can show

(4)

I(w) = / { aE R+ 1 w(a) < 0)l = I{a E R+ I w-"(a) < 0)l.

As W permutes the positive systems simply transitively, there is a unique w, E W with w,(R+) = - R'. Then w,"R+ = R+,hence w," = 1 and

(5)

l(w,) = IR+I.

We have I(ww,) = I(w,w) = IR+I - l(w) for any w E W and l(w) < IR+I for w # w,. Obviously l Ip o w,p Iw,l for all A, p E X ( T ) . Set p = jxaER+ a E X(T)S z Q . Then 2p E Z R c X ( T )and (6)

(p,B') = 1

for all p E S.

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Reductive Groups

This implies ssp - p E Z R , hence w p - p E Z R for any w E W. Therefore the “dot” action w * 1 = w(1

(7)

+ p) - p

of W on X(T) 0, R maps X(T)into itself. 1.6 (Semi-Simple Groups) The centre Z ( G ) of G is equal to the intersection Z(G)= ueR

ker(a) c T.

This group scheme is not necessarily reduced, i.e., it can happen (even if k is a field) that k [ Z ( G ) ] contains nilpotent elements ZO. Obviously Z ( G ) is isomorphic to the diagonalizable group scheme Diag(X(T ) / Z R ) ,using the notation from 1.2.5, We call G semi-simple if Z ( G ) is a finite group scheme. By the remark above we get (1) G semi-simple o (X(T):Z R ) < 00.

In the extreme case where Z ( G ) = 1 we call G adjoint. So we have (2) G adjoint o X(T)= Z R .

If G is semi-simple, then S is a basis of the vector space X ( T )0, Q. Therefore {a” I a E S } is a basis of Y ( T )@,Q N (X(T) @, Q)* so there are ( o , ) , , with ~ (o,,/?’)= d,, (the Kronecker delta) for all a,/? E S. These o, are called the fundamental weights. They are, in general, only elements in X(T) @,Q. We call G semi-simple and simply connected if wu E X(T)for all a E S . Then ( w , ) , ~is~ a basis of X(T)and has to be the dual basis of Y ( T ) .so

(3) G simply connected o Y ( T )= ZR’. Observe that 1.5(6) implies (4)

p =

1 o,

(for G semi-simple).

uss

1.7 (Regular Subgroups) A subset R’ c R is called closed if (Na + N/?) nR c R’ for any a,/?E R‘. It is called unipotent (resp. symmetric) if R‘ n( - R ’ ) = /21 (resp. R’ = - R’). For any R’ c R unipotent and closed we denote by U ( R ’ ) the closed subgroup generated by all U,with a E R‘. Using 1.2(5) one can show that the

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Representations of Algebraic Groups

multiplication induces (for any ordering of R ' ) an isomorphism of schemes over k

Obviously (2)

Lie U ( R ' ) = @ (Lie G),. UER'

Each U ( R ' )is connected and unipotent. It is isomorphic to A" with n = I R'I as a scheme (hence reduced). It is normalized by T. Using (l), one can identify k [ U ( R ' ) ] with a polynomial ring k [ Y , / aE R ' ] . For the conjugation action of T one has Y,E k [ U ( R ' ) ] - , for all a E R'. If R' c R is symmetric and closed, then let G(R')be the closed subgroup of G generated by T and by all U, with g E R'. Then

(3)

Lie G(R') = Lie T 0 @ (Lie G ) , . ueR'

The k-group G(R')is split, reductive, and connected. It contains T as a maximal torus. Its root system is exactly R', and we can identify its Weyl group with (s, 1 a E R ' ) c W. We can take especially some Z c S and set R , = R n ZZ. Then R, is closed and symmetric. Set L, = G(R,).Then L, is split and reductive with Weyl group isomorphic to W, = (s, I a E I ) . 1.8 (Borel Subgroups and Parabolics) Both R' and - R' are unipotent and closed subsets of R. Applying 1.7 to these sets, we get

(1)

U' = U(R')

and

U = U(-R').

Then

(2)

B+ = TU' = T MU'

and B = T U = T

DC

U

are Borel subgroups, with B n B' = T . Note that B corresponds to the negative roots. If w, E N,(T)(k) is a representative of w,, then w,U14;,' = U' and 6,BW,' = B'. For any 1 c S the subsets R' - R, and (- R') - R, of R are closed and unipotent, hence (3)

U: = U ( R t - R,) and

U, = U ( ( - R ' ) - R,)

are closed unipotent subgroups of G . The commutator relations 1.2(5)imply

181

Reductive Group

that LI normalizes U: and U,. One has U: n LI = 1 = U, n L,, so we get semi-direct products inside G :

(4)

PI = LIU, = LI DC U, and P: = L I U : = L,

DC

U:.

The PI (resp. P : ) with I c S are called the standard parabolic subgroups (containing B resp. B ), and LI is called the standard Leui factor of PI (and of P : ) containing T. Furthermore, U, resp. U: is the unipotent radical of PI resp. P: . Note that P , = B and Ps = G . For I c J c S one has PI c PJ and the quotient PJ/PI is an irreducible and projective scheme over k. The inclusions LJ c PJ c G induce an isomorphism and an embedding (5)

LJ/(LJ

4

pJ/pI

If k is a field, then G/PI is a projective and irreducible variety over k. This implies k N k [ G / P , ] = ind&(k). Using I.4.18.b we get for any k: k[G/PI] = k

(6)

1.9 (Big Cells and the Bruhat Decomposition) For any k-algebra k’ that is a field, G(k’) is decomposed as the disjoint union (cf. [ B o ] , 14.11 for k’ algebraically closed or [BoT], 2.1 1 for the general case): (1)

G(k’) =

u B(k’)wB(k’) u U(k’)wB(k’).

wsw

=

wew

(Note that B(k’)wB(k‘)= U(k’)wB(k’)as B(k’) = U ( k ’ )M T(k’) and as w normalizes T(k’).)This is the Bruhat decomposition of G(k’).As left multiplication with w, is a bijection on G(k’),and as w,B(k’)w;’ = B+(k’) one also has (again disjointly): (2)

G(k’) =

u

WE

w

B+(k’)WB(k’)=

u U+(k’)wB(k’).

WE

w

Using the IR’lth exterior power of the adjoint representation, one constructs (as a suitable matrix coefficient) a function d E k [ G ] with d(ultu2)= (2p)(t)-’ and d(u,wtu,) = 0 for all w # 1, all u1 E U + ( A ) ,t E T(A),u2 E U ( A ) , and all A. So G,is an open subscheme of G with G,(k‘) = U+(k’)B(k’) for any kalgebra k’ that is a field. One can show more precisely that G, is dense in G, and that the multiplication induces an isomorphism of U + x B onto Gd.Therefore we usually denote G, by U’B or U’TU. Any subscheme like U + Bis called a big cell in G . (Another choice of T and B will lead to other big cells. For example, we get U T U + by keeping T and replacing B by B .)

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Representations of Algebraic Groups

Using 1.7( l), we get an isomorphism of schemes (given by multiplication, for any ordering of the roots): (3) Therefore k[U+B] and its subalgebra k[G] are integral domains. We get especially:

(4) G is integral. If K is an extension field of the field of fractions of k, then k[G] = K[GK] n k[U+B].

(5)

(One easily reduces this to the case where K is the field of fractions of k. One then has to show that the k-module k[U+B]/k[G] is torsion free. In the case k = Z one can use the argument in [Borel 11, Lemma 4.9.The general case then follows, as d comes from Z[G], hence U + B = ( U i f 3 z ) k . ) Let K be a k-algebra that is an algebraically closed field. Let us regard G ( K ) as a variety over K . For any w E W one has B+(K)WB(K)= U+(K)GB(K)= W(G-' U+(K)W)B(K) Set R' = { c t E R+ I w-'(a) > 0) and R" = { a E R+ I w-'(a) c O } . Then R' and R" are closed and unipotent subsets of R+ with R' with R' v R" = R + , R' n k" = fa. Therefore the multiplication induces an isomorphism U ( R ' ) x U ( R " )+ U + . Obviously W-'U(R')G c U + and G-'U(R'')G c U, hence B + ( K ) w B ( K )= wU(W-' R ' ) ( K ) B ( K )c w U + ( K ) B ( K ) .

As JR'I= l ( w ) and as the multiplication induces an isomorphism U ( R ' )we get:

nasR, Ua+

( 6 ) Each B + ( K ) w B ( K )is a closed subvariety (isomorphic to K'(") x B ( K ) ) of WU+(K)B(K).

This implies, of course, that B+(K)GB(K)is a locally closed subvariety of G ( K ) of dimension equal to l(w) + dim B(K). Furthermore the Bruhat decomposition implies: (7) The G U + B with w E W form an open covering of G .

The only B + ( K ) G B ( K )of dimension equal to dim G ( K )is B+(K)w,B(K)= w,,U+(K)B(K),and the only ones of codimension one are the B+(K)W,i,B(K) c G,i,U+(K)B(K) with

ct

E S (cf. 1.5). So the complement of U + ( K ) B ( Ku) U a e s i , U + ( K ) B ( K )

183

Reductive Groups

in G ( K )has codimension at least two. (Observe that multiplication by w, is an isomorphism of varieties.) As G ( K )is smooth, hence normal, this implies: (8) Any regular function on U + ( K ) B ( K u ) U a E s I a U ' ( K ) B ( K )can be extended (uniquely) to G ( K ) . 1.10 (Local Triviality) Let x : G -+ G / B be the canonical map. Each n(GU+B) is open in G / B , cf. I.5.7(3). For any k-algebra K that is an algebraically closed field, z ( K ) is surjective, so 1.9(7) implies ( G / B ) ( K )= U,,,n(WU+B)(K). So we get, cf. 1.1.7: (1)

The x(WU+B)with w E W form an open covering of G/B.

Because of 1.9(3) the map (u,b) H wub is an isomorphism of schemes U + x B -+ WU'B. It is compatible with the right multiplication by B. Therefore u H x(wu) is an isomorphism from U + onto the image functor of w U + B in G / B . So this image functor is a faisceau and a scheme, hence equal to its ) associated faisceau x(WU+B). Therefore x i : U + -+ n(WU+B),u H ~ ( w t u is an isomorphism of schemes, and q:x(IU'B) -+ G, x H wxf'(x) is a local section, cf. 1.5.16. Hence:

(2) The canonical map G -+ G / B is locally trivial. We get also (3) For any k-algebra k' that is a Jield, the canonical map G(k')-+ ( G / B ) ( k ' ) is surjective. For each I c S there are similar results for G/PI. To start with, the results in 1.9 generalize. For example, the multiplication induces an isomorphism of UT x PI onto an open and dense subscheme of G denoted by U’P, and containing U'B. More precisely, we have (for any ordering of the roots) an isomorphism of schemes (4)

n

aER+ -RI

Ua x LI x

n

aER+ - R r

U-,% U'PI.

As U’P, 3 U + B the W’P, (resp. their images in G/PI) with w E W also form an open covering of G (resp. of G/PI). Furthermore:

(5) The canonical map G -+ G/PI is locally trivial. Again, G(k') maps onto (G/PI)(k')for any k-algebra k' that is a field. 1.11 (The Lie Algebra) Set for any tl E R (1)

X a = (dxa)(l)E (Lie Gz),,

where we regard x, as a homomorphism

-+

GZ. Choose a basis ql,. . .,cp,

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Representations of Algebraic Groups

of Y(T,)

N

Y ( T )and set for each i(l Ii Ir) Hi = ( d q i ) ( l E) Lie(T,).

(2)

Lie(G,). ThenH,,.. .,H,isa basisof Lie(T,)and(Hi)l~i~,,(Xa)aERisabasisof By 1.1(1)we get for any ring A that (Hi @ l)i, ( X , @ l), is a basis of Lie(G,). For any a E R the element H, = da'(1) E Lie(?"') is equal to [ X a , X - , ] and satisfies a(H,) = 2. (Compute in SL, as in 1.3.) This implies that Lie(T,) is an admissible lattice (in the sense of [B3], ch. VIII, 812, no 6) in Lie(Tc) = Lie(T,) €3, C. 1.12 (The Algebra of Distributions) The origin of G is contained in the open subscheme U +TU that is a product of copies of G, and G,. So 1.7.4(2)and 1.7.8 imply: ( 1 ) G is infinitesimally Jut,

and: (2) The multiplication induces an isomorphism of k-modules

Dist(U,) @ Dist(T) @ @ Dist(U-,) 3 Dist(G). asR

U E R ~

(We can take any ordering of the roots.) Furthermore, we have for any k-algebra Dist(G,)

(3)

z

Dist(G) @ A.

Applying this to (G,, Z) instead of (G,k) we get that Dist(G,) is a lattice in Dist(G,) @, C

N

Dist(Gc) N U(Lie Gc),

Take Xu and Hias in 1.11. The computations of Dist(G,,) and DistfG,) in 1.7.8 show that the X z / ( n ! )with n E N form a basis of Dist(Uu,z) for each ct E R, and that all

(H,>( H2)

(3

m(1)

m(2)

...(m(r) Hr)

form a basis of Dist(T,). Set Xu,n= X z / ( n ! )8 1

E Dist(Ua)

and Hi,, =

8 1 E Dist(T). Then (2), (3), and the result over Z imply

n;=

(4) ,411 n a r R + Xa,,,(.) form a basis of Dist(G).

1 Hi,,,,(i) nasR+

X-a,n'(a) with n(a), m(i), n'(a)E N

Similarly for any R' c R unipotent and closed the algebra Dist U(R’)has n a s R ' Xa,n(a) with n(a) E N as a basis. (Use 1.7(1).)If I c S and if we take

185

Reductive Groups

in (4)only a E R , , then we get a basis for Dist(L,). If we take in (4)all u E R + in the last product, but only a E R + n R , in the first one, then we get a basis for Dist(P,). The description of Dist(GZ) shows that Dist(G,) is the Z-subalgebra of U(Lie G,) generated by all X ! / ( n ! )with a E R and n E N,and by all

H E Lie(T,) and m E N.We have Dist(G,)

(3

with

n Lie(Gc) = Lie(G,), so th. 2 in [B3], ch. VIII, §12 implies that the (Xa)aeRform a Chevalley system and that Lie(G,) is a Chevalley order (as defined there) in Lie(Gc). Furthermore Dist(G,) is the corresponding biorder in U(Lie GJ. In the case where G is semi-simple and simply connected, this shows that Dist(G,) is Kostant's Z-form of U(Lie Gc).

1.13 (Homomorphisms of Root Data) The root datum of G is the quadruple ( X ( T ) R, , Y ( T ) ,R') together with the pairing of X ( T )and Y ( T ) ,cf. [Sp2], 9.1.6. Let G' be another connected and split reductive k-group, let T' c G' be a split maximal torus, and let ( X ( T ' ) ,R', Y(T'),R") be the corresponding root datum. We denote by B', Vk,,, . the objects for G' analogous to ByV,, . . .. A homomorphism of root data (from that of G to that of G') is a group homomorphism f:X(T ' ) + X(T ) that maps R' bijectively to R and such that the dual homomorphism f ': Y ( T )-P Y(T') maps f(P)' to 8' for each fl E R'. Then f 8 id, maps ZR' 8, Q bijectively to Z R 8, Q and induces an isomorphism of root systems. There is an isomorphism of Weyl groups W + W', w H w' such that always f 0 w' = w 0 f and sf(,) H s, for all a E R'. We say that a homomorphism $: T -+ T' is compatible with the root data if the induced map $*: X(T’)+ X(T)is a homomorphism of root data. Of course, $ H $ * is a bijection from {homomorphisms T + T' compatible with the root data} to the set of all homomorphisms of root data. Assume from now on that $: T + T' is a homomorphism compatible with the root data. If a(t) # 1 for some a E R and t E T ( A ) ,then P($(t)) = $*(P)(t)# 1 for the unique /IE R' with $*(P) = a, hence $(t) # 1. This implies, cf. 1.6(1): (1)

ker($) c Z ( G ) .

On the other hand, ker($*) n ZR' = 0 implies: (2)

$ ( T ) Z ( G ' )= T'.

Consider again a E R and fl E R' with $*(P) = a. We get from 1.3(7)

186

Representations of Algebraic Groups

Therefore we get a homomorphism $,: cpu(SL2)-+ cpb(SL,) with $, 0 cp, = cpb. It induces isomorphisms U,-+ U; and U-,+ U L p . We have $, 0 av = /3’ = $ o aV.So $, and ) I coincide on (p,(SL,)n T = aV(Gm).Therefore $, can be extended to a homomorphism (3) with

$,IT

$,: G, = $ and $,

0

-+

Gb,

cp, = cpb.

1.14 Keep all the assumptions from 1.13.

Proposition: Suppose that k is an algebraically closed field. There is a homomorphism cp: G + G’ with cp = $ and ker(cp) = ker($) c Z ( G ) and im(cp(k))Z(G’)(k)= G‘(k).

IT

Proof: We can assume that $*(S’) = S. Otherwise we can replace $ by a suitable $ 0 Int(G) with w E W, similarly for cp. In order to simplify notations we shall identify R and R‘ via $*. So we have $*(a) = a and $*(p’) = j for all a, /3. Set and (for all a E S ) (1)

Ho = {(t,$(t))I t E T ( k ) }c G ( k ) x G’(k)

Hu

= ( ( 9 ,$u‘a(gN 19 E G U W

= G(k) x

G’(k)*,

These are closed subgroups of G ( k ) x G’(k)(with Ho c H,for all a) isomorphic to T(k) resp. G,(k) via the first projection, hence irreducible. Therefore, the subgroup H of G(k) x G’(k)generated by all H, with a E S is also closed and irreducible. Let x : H + G(k)and x ‘ : H -+ G‘(k)be the two projections. We want to prove that x is an isomorphism of linear algebraic groups. Then x‘ 0 x-’ is a homomorphism G(k) + G’(k)of algebraic groups inducing $&) on each G,(k) with a E S. As G is reduced there is a unique homomorphism cp: G -+ G’ of group schemes with ~ ( k=) x’ x-l. As each G, is reduced, cp(k)IGeck,= $,(k) implies cp I G m = I), for all a E S, hence cp = $, IT = $. We shall need several steps to get our claim. 0

IT

(2) x is surjectioe. This is clear as G ( k ) is generated by T ( k )and all U,(k) with a E S u ( - S). (3)

G’(k)= im(x’)Z(G‘)(k).

The group G’(k)is generated by T ’ ( k ) and all Ub(k)with a E S u ( - S ) . Both

187

Reductive Groups

U # ) and U’&) are (for any a E S ) contained in the image of cpb = $, 0 cp, hence in n’(H,) c n’(H). Finally T’(k)= im($(k))Z(G’)(k)by 1.13(2).

(4) H is reductive.

The unipotent radical R J H ) of H is mapped under n resp. n’ to a unipotent subgroup of G(k) resp. G’(k) resp. G’(k) which is normal by (2) resp. (3),hence equal to { l}. This implies R J H ) = {I}. We shall apply the notation Lie(H’), Dist(H’) to linear algebraic groups H’ by regarding H‘ as a reduced group scheme.

( 5 ) dn: Lie(H)

--f

Lie(G) is surjective.

As the image of dn contains Lie(G,) = Lie T a E S this follows from Lie G = Lie T

+

+ (Lie G), + (Lie G)-,

for each

Ad(~)(Lie G),, WEWUES

and the compatibility of dn with the adjoint action. Set V, = {(g, $,(g)) I g E Ua(k)}and define similarly V-, for each a E S . The isomorphism g H (9, $,(g)) maps the dense and open subset (a big cell) U,(k)T(k)U-,(k)ofG,(k) to the dense and open subset V,H,V-, of H,. So: (6)

Dist(H,) = Dist( V,)Dist(H,)Dist(V-,).

The algebra Dist(H) is generated by all Dist(Ha)with c1 E S, cf. 1.7.19. Let Dist(H)+ resp. Dist(H)- be the subalgebra generated by all Dist(V,) resp. Dist(V-,) with a E S . If a, fl E S with a # /3, then V , and V.+ commute, hence so do Dist( V,) and Dist( V-&. Therefore we get using (6): (7)

Dist(H) = Dist(H)+Dist(H,)Dist(H)-.

On the other hand, 1.12(2)applied to G x G’ yields (8)

Dist(G x G’) = Dist(G x GI)+ 0 Dist(T x T‘) @ Dist(G x G’)-,

where Dist(G x G’)’ resp. Dist(G x G’)- is generated by all Dist(UBx 1) and Dist(1 x U b ) with fl E R+ resp. fl E -R+. Now V, c U, x Uh for all a implies Dist(H)+ c Dist(G x GI)+ and Dist(H)- c Dist(G x G ’ ) - . So we get from (7) and (8): (9)

Dist(H) n Dist(T x T‘) = Dist(H,).

As H, N T ( k ) is a torus, it is contained in a maximal torus H, of H. Necessarily H, c CG(k)xG,(k)(Ho). As no root of G x G’ with respect to T x T’ (Lea,no a x 1 or 1 x a with a E R ) vanishes on H,, this centralizer

188

Representations of Algebraic Groups

is equal to T ( k ) x T’(k). Therefore Dist(H,) c Dist(T x T’) n Dist(H) = Dist(H,) by (9),hence HI c Ho as H , is irreducible. This implies: (10) H , is a maximal torus of H.

We already know H to be reductive by (4). Therefore each nontrivial normal subgroup intersects Ho nontrivially. As n is injective on H , we get: (11) n is injective. Combining this with (2) and ( 5 ) we see that n is an isomorphism. Therefore we get cp: G + G’ as mentioned before with cp I G a = I),for all a E S. This implies hence for any w E W especially that ( ~ ( 3 , )is a representative for s, in NG,(T )(k), that cp(6)is a representative in NG,(T’)(k)of the corresponding element in W‘. This implies that cp induces an isomorphism U, + U &for all a E R. It does so for all a E S as cp I u = = I)z/orlu, and we get it for all a by conjugating with suitable w. This implies ker(cp) n U + B = ker (I)), hence Dist(ker(cp))= Dist(ker(I))). The Bruhat decompositions of G(k) and G’(k) yield ker(cp(k))= ker(I)(k)). So ker(cp) = ker(I)) follows from 1.7.17(8). Finally, (3) implies G’(k)= im(cp(k))Z(G’)(k). Remarks: 1) The homomorphisms cp as constructed in the proposition are called central isogenies. There is a larger class of isogenies if char(k) # 0. They can be constructed in the same way as above. For the necessary changes one may consult the original paper [Takeuchi]. (They no longer satisfy ker(cp) c T.) 2) Any homomorphism cp: G + G’ extending I) satisfies ker(cp) = ker(I)) and G’(k) = im(cp(k))Z(G’)(k).(Using cp 0 a’ = $ 0 a’ one can show that cp maps any U, isomorphically to U & .Then one can argue as above.) 1.15 Keep all the assumptions from 1.13.

Proposition: There is a homomorphism cp: G + G‘ with cp ker(I)). If I) is an isomorphism, then so is cp.

IT

= I) and ker(cp) =

Pro08 Let K be an algebraic closure of the quotient field of k. By 1.14 we on have a homomorphism cp‘: G K + GL restricting to I)K on TK and to It therefore induces (for each a E S ) an isomorphism (Ua)K+ ( U & each (Ga)K. with cp’*k[U&]= k[U,]. Furthermore, it maps any 3, (which is in NG(T)(k)) to a representative of s, in NG,(T’)(k),hence any w E N,(T)(k) to a representative of w again over k. Therefore cp‘ induces for all a E R an isomorphism (Ua)K+ (U&)Kwith cp’*k[U&]= k[U,]. As cp‘ induces I)K on T K we get that cp‘

189

Reductive Groups

maps U i B , to U;+B; and satisfies cp’*k[U’+B’] c k[U’B]. Now 1.9(5) implies cp’*k[G’] c k[G]. Therefore there is amorphism cp: G -+ G’ with cp’ = cp., The injectivity of the maps k[G] -+ KCG,], k[G,] -+ K[(G,),], etc. implies that cp is a homomorphism with cp I G a = $, for all a E S, especially with cp = $. By construction, cp induces isomorphisms U, -+ Uh for all a E R. Therefore the same argument as in 1.14 yields ker(cp) = ker($), If $ is an isomorphism, then so are all $, with a E S. We can apply the same and get cpl: G’ + G inducing $;’ on each GL with argument to $-’ and all all a E S. Then ‘pi cp is the identity on each G, with a E S , hence the identity on G by 1.3(10). Similarly cp cpl = id. (One could also go back to the proof of 1.14 and show that 7t is an isomorphism if $ is so.)

IT

0

0

There is an antiautomorphism z of G with t 2 = idGand 7 id, and z(U,) = U - , for all a E R.

1.16 Corollary:

IT

=

Proof: We want to apply 1.15 to G = G‘, T = T’, and $: T -+ T with $(t) = t-’ for any t. Then $ is compatible with the root data as $*(a) = - a and $*(a’) = -av for each a E R. The homomorphisms $,: G, -+ G-, = G, for all a E S satisfy $, o (pa = cp-,, hence $: = id,. for all a E S . The extension cp: G -+ G with cpIG, = $, for all a E S satisfies cp(t) = t-’ for all t in T and cp(U,) = U-,for all a E R and cp’, , I = $: = id for all a E S, hence cp’ = id by 1.3(10). So 7:G -+ G, g I+ cp(g-’) has the desired properties. Remark: In the special case where G = SL, and T = {diagonal matrices} we usually take z(g) = ‘g for all g. 1.17 (Quotients and Covering Groups) The proposition 1.15 implies that G is determined up to isomorphism by its root datum. (This is known as the isomorphism theorem). On the other hand, one has an existence theorem: To each “possible” root datum there corresponds a group. More precisely, if X , Y are free abelian groups of finite rank with a pairing inducing an isomorphism Y 2 X * and if R‘ c X,R” c Y are finite subsets together with a bijection a H a’ from R‘ to R”, such that R’ and R” induce a root system in ZR‘ @= Q in the sense, say, of [B3], ch. VI, 1.1, then there is a group (as in 1.1) having a root datum isomorphic to ( X ,R‘, Y, R’”). In the case where (X: ZR’) < 00 one can find a construction in [Borel 11. This yields all semi-simple groups. In general one can take a quotient of a direct product of a semi-simple group with a torus. Let me describe in general what happens if we take a quotient of G by a group contained in Z(G).

190

Representations of Algebraic Groups

nlEX,

If X’ is a subgroup of X(T) with Z R c X’, then Z = ker(%)is a closed subgroup scheme of G contained in Z(G). The quotient T/Z is a torus (isomorphic to Diag(X’)) and G/Z is a connected and split reductive group with split maximal torus T/Z. One has k[G/Z] = @i.ox,k[G],, where we take weight spaces with respect to the left (or right) regular representation. So k[G/Z] is a direct summand of k[G] as a k-module. The root system of G / Z is just R c X’ and the canonical map G + G/Z induces an isomorphism of U, onto the corresponding group in G / Z (for any a E R). The coroots of G/Z are just the homomorphic images of the coroots of G in Y(T/Z) = Y(T)/Y(Z). In the case where (X(T):X’) < co (or, equivalently,where 2 is a finite group scheme) we call G a covering group of G/Z. On the other hand, we can take a finitely generated subgroup X’ of X(T) OZQ with X(T) c X’ and (I,a’) E Z for all l E X’ and a E R. Then we can identify the dual lattice Y = (X’)*with a subgroup of Y(T)containing R’, and (X’,R, Y , R’) is a possible root datum. Let G’ be a reductive group having this root datum. Then the inclusion of X(T)into X’ yields a homomorphism G’ --t G inducing an isomorphism G ’ / ( ker R) 3 G. So we can regard G’ as a covering group of G , and we get thus all covering groups of G . For example we can choose 0: (a E S ) with (o:,p‘) = &., for all a, p E S and then take X’ = X(T) + x u s S Z ~ hIn. this way we get a covering (G’,T’)of (G, T ) such that each I E X(T) has for each m E N a decomposition l = I , + mIl with l o I, l E X(T ) and 0 I ( A o , a’) c m for all a E S .

nAEx(T)

1.18 (The Derived Group and Characters) Set

for all a E R )

X,(T) = { I E X(T)\(I,av) = 0 and Xb(T) =

C QanX(T).

asR

Then (1)

Tl =

n

ker(I) and

T2 =

0

ker(l)

i.E X b ( T )

i.E X o ( T )

are closed subgroups of T. In fact, as X(Tl) N X(T)/X,(T) and X(T,) N X(T)/Xb(T) are obviously torsion free, both TI and T2 are tori. Obviously T2 is contained in the centre Z(G) = ker(a). More precisely, it is the largest subtorus contained in the centre. (If k is a field, then T2 is the reduced part of the connected component Z(C)O of the identity.) The definitions yield

nueR

Y(Tl) N

1Qa’n

UER

Y(T)

191

Reductive Groups

and ~ ( ~ , ) - { c p ~ Y ( T ) ( ( a , c p ) = forall o aER}.

This implies easily that (X(Tl),{aITII a E R } , Y(T,),R') is a root datum. Let G1 be a corresponding split, reductive, and connected group containing Tl as a maximal torus. Obviously (Y(T,):ZR') < co,hence also (X(Tl):Z R ) < co. So G1 is semi-simple. The inclusion of Tl into T leads to a homomorphism of root data, hence to an injective homomorphism G1 + G. Its image contains all U, with a E R . For any algebraically closed field K that is a k-algebra Gl(K) is equal to its derived group (being semi-simple). Its image in G ( K ) contains the derived group of G ( K ) which is generated by all U,(K), hence is equal to the derived group of G(K).We shall call the image of G1 in G the derived group of G and denote it by BG. (In case k is a field this is compatible with the general definition in [DG], 11, §5,4.8.) As T, is central in G we get a homomorphism (2)

9 G x T,+G

induced by the multiplication. O n the maximal tori the comorphism of Tl x T, + T has kernel Xo(T)n X b ( T ) = 0. Therefore (2) is a quotient map, i.e.,

= (BG x W(T1 n T,),

G

with Tl n T, embedded via t H (t,t-'). Any character ,u E X ( G ) has to be trivial on 9 G . So it is well determined by its restriction to T, c Z ( G ) c T and this restriction has to vanish on T, n T,, or (equivalently) ,u has to vanish on Tl. Using the quotient map (2) we also get the converse, i.e.,

IT

(3)

X ( G ) N Xo(T)

We can also apply this to each LI with I c S. As any character of PI has to vanish on each U, c 9 P I , hence on U,, we also get X ( P I )= X ( L , )

(4)

-

{ A E X(T)I(A,@’) = 0

for all

o!

EI}.

1.19 (Weight Spaces of G-modules) Any T-module M has a weight space decomposition M = @ l e X ( T ) M A , cf. 1.1(2). The operation of Dist(T) on M can be described as follows: Any Hi,m acts on M A as multiplication by

('";n""). (A,

cpi)Ih

(Recall that Hi = (dcpi)(l), so A restricts to cpi(Gm)N G, as the power map.)

192

Representations of Algebraic Groups

Recall that we associated (in 1.2.11(6)) to any T-module M, which is projective and of finite rank over k, a formal character ch M E Z[X(T)]. There are some elementary properties mentioned in 1.2.11(7), (8) where we simply have to replace A by X(T). If M is a G-module and if w E N,(T)(k) is a representative of some w E W, then an elementary calculation shows (1)

WM, = Mw(A) for all

L E X(T),

So rk(M,) = rk(Mw(LJin case M is projective of finite rank over k, hence (2)

ch(M) E Z[X(T)Iw

in this case. (Note that W acts on Z[X(T)] via we@) = e(wL)for all w and L.) We can regard Dist(G) as a G-module under the adjoint action, cf. 1.7.18, hence also as a T-module. Then we have for all a E R and all n E N and (4)

Dist(T) c Dist(G),.

Indeed, we can identify k[UJ with a polynomial ring in one variable Y,, which is a weight vector of weight - a for the adjoint action of T as k Y , N Lie(&)*. Then Y: has weight -na and the "dual" vector Xu," weight na. Of course, (4) is an immediate consequence of the commutativity of T. If M is a TUa-modulefor some a E R, then (3) and 1.7.18(1)imply for all L and all n (5)

c MA+,,*

Recall from 1.7.12 that the Xa," determine the operation of Ua = G, on any Ua-moduleM as follows: If m E M and a E A for some k-algebra A , then Let M be a B+-module (or even a G-module). Suppose A E X(T)is maximal among all weights of M (with respect to I as in 1.5). So L + na is not a weight of M for all a E R + and n > 0, hence X,,,M, = 0 by (5). Therefore (6) implies: (7) If

A is maximal among the weights of M, then MAc MU'.

1.20 (G-modules and Dist(G)-modules) The lemmata 1.7.14-7.16 can be proved in a more direct way for our G (as in the case of G, and G,,,). For

193

Reductive Groups

example, any Dist(G)-submodule M ' of a G-module M is also a submodule for Dist(T) and all Dist(U,), hence a T-submodule and a U,-submodule for all a (using the cases G, and G,). In the case where k is a field one can now use the fact that G ( K ) is generated as a group by T(K) and the U,(K) for an algebraic closure K of k. In general, one can use the density of U'B in G. The inclusion k[G] c k[U+B] remains injective after tensoring with M / M ' (assumed to be projective over k). As M' is stable under all of U'B, the composed map M' + M 6 k[G] + ( M / M ' )6 k [ G ] + ( M / M ' )6 k[U'B] is zero. This implies 1.7.15for our G. For the other properties one can argue similarly. Let M be a Dist(G)-module. Suppose that there is also a structure as a T-module on M such that: (1) The structures on M as a Dist(T)-module induced from the actions of Dist(G) and of T coincide,

Dist(G), t E T(A), and any A )

and (for all m E M , p

E

(2)

t(p(t-'(m 6 1))) = (Ad(t)p)(m6 1).

This implies X,,,MA c MA+,, for all a E R and n E N as in 1.19(5).Let us suppose that this "Dist(G)-T-module" M is locally finite, i.e., that for each m E M there is a Dist(G)-T-submodule M' c M with m E M' and M ' finitely ; , it is a T-submodule, and only generated over k. Then M' = O A e X ( T ) M as finitely many M i can be nonzero, as M' is finitely generated. This implies that there is for each m E M and a E R an integer n,(m) with X,,.m = 0 for all n > n,(m). Then the action of Dist(U,) on M is induced by a structure as X" 6 Xor,"m,where X is U,-module. The comodule map sends m to the function x,(a) H a on U,. So any x,(a) acts on m 6 1 E M 6 A via x&)m = 0 (X,,,m) 6 an* Now there is a description of G in terms of generators and relations ([D], 4.4) that tells us when homomorphisms T + H and U, --t H (for all a E R) into a group scheme H (or only a group faisceau H ) over k can be glued together to a homomorphism G + H.Consider especially H = G L ( M ) and the representations above. Then the computations in [St 11 show in the case of a ground field that these relations are satisfied. From this we can deduce the same for our integral domain k, e.g., for modules that are projective over k. So for a projective k-module M there is a one-to-one correspondence between possible structures as G-modules and as locally finite Dist(G)-Tmodules.

zn2

194

Representations of Algebraic Groups

If k is a field and if G is semi-simple and simply connected, then any locally finite Dist(G)-module is in a natural (and unique) way also a Dist(G)-Tmodule. So in that case there is an equivalence of categories between { G-modules) and {locally finite Dist(G)-modules}. Probably because of some remark in [Humphreys 81 this result became known as Verma's conjecture although Verma must have been aware of a proof along the lines sketched above when writing [Verma]. These arguments were used in special cases in [Borel I], 53 and [Wong 11, p. 46. When I needed the result for a larger class of modules I sketched a proof in [Jantzen 31, p. 119. There is a completely different proof in [Sullivan 41 following earlier partial results in [Sullivan 21. (These proofs work only if char&) # 0, but for char@) = 0 the result is classical.) In [Cline, Parshall, and Scott 61, 9.2/4 one can find another proof and a generalization to B-modules. 1.21 (The Case GL")The general linear groups (cf. 1.2.2)are (besides the tori) the simplest examples of reductive groups. Fix n E N, n 2 2 and take G = GL,. The conventions and notations introduced below will be used whenever we look at this example. For all i, j (1 < i, j In) let E, be the (n x n)-matrix over k with (i, j ) coefficient equal to 1 and all other coefficients equal to 0. The E, form a basis of M,(k). Let us denote the dual basis of M,(k)* by Xij (1 < i, j In). So the Xij are the matrix coefficients on M,(k) and k[G] is generated by the Xij and by det(Xij)-'. We choose T c GL, as the subfunctor such that T ( A ) consists of all diagonal matrices in GL,(A) for all A, i.e., T = V({Xij1 i # j ) ) . Then T is isomorphic to a direct product of n copies of G,. The ei = XiilT(lIi In) form a basis of X(T), and the eI(1 Ii In) with -$(a) = Ejj aEii form a basis of Y ( T ) .One has (ci, E ; ) = 6, (the Kronecker symbol). The root system has the form:

cj+i+

(1)

R = { E ~ - E ~I (i I, j I n , i # j } .

It is of type A,-l and our notations will be consistent with [B3], ch. VI, planche I (for 1 = n - 1). One has (Lie G)e,-e,= kEij(for i # j ) and (Lie G ) , = kE,,. We take (for i # j)

cy=,

xei...ej(a)= 1 + aEij

(2) and (3)

=

+

Ehh aEii

+ bE, + cEji + d E j j .

195

Reductive Group

One has ( E , - E ~ ) ' = E ; - E> for all i , j. The Weyl group permutes { E ~ e, 2 , . . . ,E , } and can be identified with the symmetric group S,: Map any w E W to the permutation CJ with w(E,) = E , ( ~ ) for all i. Then s,,-,, is mapped to the transposition (i, j ) . The composition of this isomorphism with the canonical map N,(T)+ W admits a section: Map any CT to the permutation matrix Eo(iJ,i.The element w, corresponds to the permutation oo with a,&) = n + 1 - i for all i. We choose as system of positive roots (4)

R + = { E , - e j l 1 I i < j I n>.

Then (5)

S = {a, = E , - E

, + ~11

I i

< n}.

The centre of G is isomorphic to G,: G, 3 Z ( G ) ,a H a

n

1Eii. i= 1

The Borel subgroup B (resp. B + ) is the functor associating to each A the group of lower (resp. upper) triangular matrices in GL,,(A).Furthermore, U ( A )(resp. U + ( A ) )consists of all matrices in B ( A ) (resp. B+(A))such that all diagonal entries are equal to 1. Let us identify GL, and GL(k")via the canonical basis e l , e , , . . . ,en of k". (This is the natural representation of GL,.) Set r/; = ( e n ,en- l,. . . , e n + - i ) for 1 5 i I n. Then B is the stabilizer of the flag (V, c V, c c KA1),i.e., B= Stab,(Y). The stabilizer of any partial flag (l$ c Y2 c * . * c Yp) with 1 Iil < i2 c < i, is the parabolic subgroup P, with I = {ai 1 1 I i < n, i # n - ih for 1 I h I r } . One has especially

The subgroup X,(T) of X ( T ) is just Z(el+ E, + + E,,). The homomorphism det: GL, + G, restricts to el + e2 + ... + E, on T. Hence: (8)

X ( G ) = ((det)'I r E Z}.

The derived group 9 G is equal to G' = SL,. Furthermore, T' = T n G' is a maximal torus in G' with X(T’)= X(T)/Z(el + E,) and Y(T')= { ai&;I a, = O } . The canonical map X ( T ) + X(T’)induces an isomorphism of root systems. The standard Borel and parabolic subgroups of G' are just B n G' and P, n G'. The homomorphisms xu and cpcl from (2), (3) already have values in G' and yield thus the corresponding homomorphisms for G'.

c;=

+

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2

Simple G-Modules

Let k be a field throughout this chapter. The first aim of this chapter is the classification of all simple G-modules. It turns out (2.7) that Cartan's classification for compact Lie groups carries over to this situation: The simple modules (up to isomorphism) correspond bijectively to the dominant weights of T. This was first proved by Chevalley (see [SC]) and there are other accounts of this fact in [Hu 21, [St 13, [St 21, and [Borel 11. Having established the classification we prove some elementary properties (2.8-2.14). Most of them do not require a reference to an original paper. See, however, the remark to 2.1 1. The last sections (2.15-2.18) contain some examples. In 2.15 the exterior powers of the natural representation of GL, (or SL,) are shown to be simple and the corresponding dominant weights are determined. In 2.16 we look at some special induced representations of GL, (and SL,) and prove that they are isomorphic to the symmetric powers of the natural representations (or rather of its dual). In the case G = SL, we get from this an explicit description of all simple G-modules. In 2.17/18 this example is generalized from GL, to the symplectic groups and the special orthogonal groups. (In the last case we do not get the symmetric powers, but homomorphic images.) 2.1 As

U and U + are unipotent, 1.2.14(8)implies for any G-module V # 0: Vu#O

and

Vu'#O.

197

198

Representatioas of Algebraic Groups

As T normalizes U and U + ,these two subspaces are 7'-submodulesof V,hence direct sums of their weight spaces. For any A E X ( T )the I-weight space of V u is the sum of all simple B-submodules in V isomorphic to k , (similarly for U + and B'). We shall write A instead of k , whenever no confusion is possible. So we can also express (1) as follows:

(2) There are A,A’ E X ( T )with Horn&, V ) # 0 # HOmg+(A',V ) . If dim V < co,then applying (2) to V* we get (3) If dim V < co,then there are A, A’ E X ( T ) with Hom,(V,lw)# 0 # HOm,+(V,A'). Using Frobenius reciprocity (1.3.4)this implies (4) If dim V

-=

00,

then there are A, A’ E X ( T ) with

Horn,( V,indin) # 0 # Horn,( V,indg+A’). It will abbreviate many formulas to use the notation (5)

H'(M) = R'indgM

for any B-module M and any i E N.(This notation is, of course, inspired by the isomorphism R'indgM z H'(G/B, de(M)),cf. 1.5.12.)As announced above, we shall write Hi(I) = Hi&,). This could lead to some confusion in the case A = 0 as H'(0) may also refer to the zero-module. So in that case we shall usually write Hi@). Note that

(6)

H o ( k ) = k,

as indgk = k [ G / B ] and as G / B is projective and irreducible. The tensor identity and ( 6 ) imply Ho(M) N M for any G-module M. The isomorphism is given by the evaluation map f Hf(a). For any rn E M E the subspace km is a trivial B-submodule of M, hence ind;(krn) is a trivial Gsubmodule of indgM. The isomorphism ind,GM7 M maps indi(krn) to krn. This implies (for any G-module M): (7 1

MB = MG.

2.2 Proposition: Let A E X ( T )with Ho(A) # 0. a) We have dim H0(A)" = 1 and H0(A)" = Ho(A),. b) Each weight p of Ho(A)satisfies w,A I,u IA.

199

Simple G-Modules

Proof: Recall that

H0(4= {f E kCGl I f(gb) = A @ - 'f(d

for all g E G ( A ) , bE B(A) and all A } .

The operation of G is given by left translation. So any f E Ho(l)"' satisfies f ( u 1 tu2)

=

w -'fm

for all u, E U'(A), t E T ( A ) ,u2 E U ( A )and for all A. Thus f(1) determines the restriction of f to U'B, hence also f, as U'B is dense in G (1.9). This implies dim H0(A)" I 1, and equality follows from 2.1(1) and from our assumption HO(L) # 0. Furthermore, the evaluation map E : Ho(A)+ A, f Hf (1) is a homomorphism of B-modules and is injective on H0(A)". This implies HO(L)U'

c HO(A),.

If p is a maximal weight of Ho(A)(or of a finite dimensional submodule, if we do not yet want to apply 1.5.12.c),then Ho(A), c H0(A)"' by 1.19(7).This and the inclusion H0(A)'" c Ho(A), from above imply Ho(#" = Ho(A), and p I A for each weight p of Ho(A). If p is a weight of Ho(A), then so is w,p by 1,19(1)yhence w,p IA and w,A Ipycf. 1.5. Remark: We can generalize part of the result as follows. Let P = PI 2 B be a standard parabolic subgroup of G (cf. 1.8)and set = Pi 3 B'. Denote the unipotent radical of P by U ( P ) = Ui . Then U(P)Pis open and dense in G and the multiplication induces an isomorphism U ( P ) x P + U ( P ) P of schemes, cf. l.lO(4). Let M be a P-module and let cM: ind$M + M be the evaluation map f H f(1). The proof of the proposition shows: (1) E~ maps (ind$M)'(') ( P n P = L,)-modules.

injectively into M and is a homomorphism of

2.3 Corollary: If Ho(A) # 0, then socGH0(A)is simple. Proof: If L , , L 2 are two different simple submodules of Ho(A), then L , @ L 2 c Ho(A), hence Ly' @ Ly' c H0(L)" and dim H0(A)" 2 2 by 2.1(1) contradicting 2.2. Therefore soc,HO(A) has to be simple.

Representations of Algebraic Groups

200

2.4 Set

L(A) = soc,HO(I)

(1)

for any A E X(T ) with Ho(A) # 0. Proposition: a) Any simple G-module is isomorphic to exactly one L(1) with 1 E X(T)and H'(1) # 0. b) Let A E X ( T ) with Ho(A) # 0. Then L(1)" = L(A), and dim L(A)" = 1. Any weight p of L(1) satisjes w,1 Ip IA. The multiplicity of L(1) as a composition factor of HO(1) is equal to one. Proof;. b) As L(1)" # 0 the formulas L(l)u' = L(A), and dim L(1)" = 1 follow immediately from 2.2. The same is true for w,A < p < A. Finally, the multiplicity of L(A) in Ho(A) is at least one by construction, but cannot be strictly larger as dim L(A), = 1 = dim Ho(l), and as V H V' is an exact functor. a) The existence follows from 2.1(4), the uniqueness from the formula L(1)"' = L(1), in (b). Remarks: 1) This proposition shows that A is the largest weight of L(A)with respect to <.It is the custom to call it the highest weight of L(I)and to call L(A) the simple G-module with highest weight A. 2) Using t+,ut+i1 = U + we see:

(2)

dim L(A)U= 1 and L(A)' = L(A),o,.

2.5 Corollary: Let A E X(T)with Ho(A) # 0. The module dual to L(1) is L( - won). Proof;. For any finite dimensional T-module V the p-weight space of V* is naturally identified with (V-,,)*. So the weights of V* are exactly the - p with p a weight of V. In the case of L(1) this implies that the weights p of L(A)* satisfy w,A I- p < A, hence -1, I p I -woA, and that -w,A will occur. As L@)* is simple, it has to be isomorphic to L( - won). 2.6 It remains to determine the

A with HO(A)# 0. Set

X ( T ) += { A E X ( T ) I ( A , a v )2 0 =

(1E X(T)I( A , a v ) 2 0

forall a E S } for all a E R + } .

201

Simple G-Modules

The elements of X ( T ) + are called the dominant weights of T (with respect to R’). Proposition:

Let 1 E X ( T ) . The following are equivalent:

(i) 1is dominant. (ii) HO(1) # 0. (iii) There i s a G-module V with (V"'), # 0. Proofi (iii) =. (ii) Using a composition series of a suitable finite dimensional submodule of V we reduce to the case where V is simple. Now (ii) follows from 2.4. (ii)*(i) Suppose H'(1) # 0. By 1.19(2) any s u l with a E R + is a weight of Ho(A), hence s u l I 1by 2.2. Now s,1 = 1- ( l , a v ) a , so ( l , u v ) a 2 0, hence (A,av) 2 0. (i) (ii) Because of I.3.5(3) we can assume that k is algebraically closed. We can regard G(k) as a variety over k and have to find a regular function f: G(k) -P k, f # 0 such that f(gtu) = 1(t)-'f(g) for all g E G(k), t E T ( k ) , u E U(k). Consider the function f, on the open subvariety U + ( k ) T ( k ) U ( kc) G ( k ) given by f,(ultuz) = 1(t)-' for all u1 E U + ( k ) ,t E T ( k ) , u2 E U(k). Obviously f, #O. As the restriction map k[G]-rk[U+ T U ] is injective and B-equivariant we just have to show f, E k[G].Then automatically f, E Ho(A)and Ho(A) # 0. We want to show that f, can be extended (uniquely, of course) to a regular function on each 3,U+(k)B(k)u U+(k)B(k).Then it can be extended also to Y = U+(k)B(k)u U,,,i,U+(k)B(k), as for all a,p E S the extensions coincide on 3,U+(k)B(k) n S,U+(k)B(k)n U+(k)B(k),which is dense in G(k), hence also on S,U+(k)B(k)n 3,U'(k)B(k). If f, extends to Y, then it extends to the whole of G(k), cf. 1.9(8). Let us consider now a fixed simple root a E S . Set U: = (U, I p > 0, B # a). Then U + = U: U, N U : x U, (as schemes) and 3, normalizes U:. So

:

3, U (k)B(k ) = U (k)i, U,(k)B(k). +

The group U,(k) consists of all x,(a) with a E k. The map ( u l , a ,t , ~H ) u13,x,(a)tu is an isomorphism of varieties U:(k) x k x T ( k ) x U ( k ) - , i,U'(k)B(k). We may suppose 3, = n,(l), cf. 1.3. Then i,x,(a) = x u ( - a ~ l ) a v ( - a ~ l ) x ~ u ( a ~ l )

for all a # 0. Then uliux,(a)tu = ulx,(

- a-')av( - a-')tx-,(a(t)u-')u

E

U+(k)B(k)

202

Representations of Algebraic Groups

for all ul,t, u as above, and fa(u 1 i,x,(a)tu) = I(t)-' Il(aV(- U - '1)= L(t)-'( -a)(+

As ( A , a v ) 2 0 this function on U:(k) x ( k - (0)) x T(k) x U ( k ) can be uniquely extended to a regular function on U : ( k ) x k x T(k) x U(k),hence f A can be extended to i,U+(k)B(k)as a regular function. (ii) =.(iii) This is obvious.

Remark: There are other proofs of (i) =.(ii), cf. [St 11 and [Ha], 31.4. One can also reduce by 1.4.18.b to the case k = C and there apply a construction using the enveloping algebra of Lie(Gc). This construction can be imitated over any k using Dist(G). (There is for each 1 E X(T) a simple Dist(G)-module with highest weight 1,cf. [Haboush 31. It has finite dimension if A is dominant, cf. the proof of Lemma 1 in [Jantzen 31, and lifts to G for such A, cf. 1.20.) 2.7 Corollary: The L(A)with A E X(T)+are a system of representatives for the isomorphism classes of all simple G-modules. Remark: We have by 2.4 and 2.6 for any 1 E X(T)+ ch L(1) = e(1) +

p
dim L(l)fle(p).

This shows that the ch L(A)with 1 E X(T)+are linearly independent. As ch(?) is additive on short exact sequences, the map V H ch(V) induces a homomorphism from the Grothendieck group of {finite dimensional G-modules} to Z[X(T)] which is injective as a basis (the classes of the simple modules) is mapped to a linearly independent set. This implies that two finite dimensional G-modules have the same formal character if and only if the multiplicities of the simple G-modules as composition factors are the same in both cases. We usually denote these multiplicities by [V :L ( 4 ] (for any G-module V and any

1 E X(T)+).

2.8 Proposition: Let 1 E X(T)+. Then

Proofi

Using Frobenius reciprocity and 2.2 we get End,H'(A)

N

Hom,(H'(A), 2) c Hom,(HO(l), 2)

N

Hom(HO(l)n,l)= k.

203

Simple G-Modules

On the other hand, the identity map is a non-zero element in EndGHo(& so EndGHo(i)= k. Similarly, EndGL(1) C HOm,(L(il),Ho(2)) N Hom,(L(I),L) c Hom,(L(I),L)

N

Again EndGI,@)# 0, hence EndGL(d)= k.

Hom(L(I),,I) = k.

2.9 Corollary: For any jield extension k’ of k the G,,-module L(3,)0 k’ is the

simple G,,-module with highest weight 3,.

Proofi We know (by 1.7.15/16) that L(3,) is also a simple Dist(G)-module and that End,ist&n) = End&@) = k. Therefore Wedderburn’s theorem implies that the canonical map from Dist(G) to End,L(A) is surjective, hence so is that from DiSt(G,,) = Dist(G)@ k‘ to End,,(L(I) @ k ’ ) = (End,L(lb))@k‘. This implies the simplicity of L(3,)@ k’, hence the corollary. (Cf. also [Bl], ch. VIII, $13, no 5, cor. de la prop. 5.) Remark:

(1)

The corollary implies for any G-module I/: SOC~,,( V @ k’) = SOCG( V )@ k‘.

Here one inclusion (“3”)is a trivial consequence of the corollary. In order to get the other, suppose dim V < co.The multiplicity of any L(1) in SOC,(V) is equal to dim HomG(L(3,),V ) , that of L(3,)@ k in SOC,,,(V @ k‘) equal to dim,. Hom,,,(L(l,) @ k, V @ k’). Therefore equality follows from 1.2.10(7). Obviously (1) implies that V is a semi-simple G-module if and only if V 8 k’ is a semi-simple G,,-module. Of course (1) generalizes to the higher terms of the socle series, cf. 1.2.14. 2.10 It follows from 2.8 that the centre Z ( G ) of G acts on H’(3,) through scalars. As Z ( G ) c T and as H o ( I ) , # 0 this scalar has to be the restriction of I to Z ( G ) . This result is really a special case of the following statement:

(1) For each I E X ( T ) the group Z ( G ) acts on each H’(A) through the restriction of 3, t o Z ( G ) . This is shown by applying 1.6.11 to the natural map from G x Z ( G ) to G induced by the multiplication and by using the tensor identity. Let Z be a closed subgroup of Z(G). For any I E X(T)+with 3,1z = 0 the group Z operates trivially on Ho(I), hence also on L(I). Therefore L(3,) is a

204

Representatiom of Algebraic Groups

simple G/Z-module in a natural way and it is the simple module with highest weight A E X(T/Z) c X(T). Observe that G/Z is a split reductive group with split maximal torus T/Z, cf. 1.17. Conversely, any simple G/Z-module is a simple G-module in a natural way, hence isomorphic to some L(A) with = 0.

We can apply this argument to ( 9 G x T2,G) instead of (G, G/Z), using the notations from 1.18(1). As T2 c Z ( 9 G x T2) acts through scalars on any simple ( 9 G x T2)-modules one gets: (2) Regarded as a 9G-module any L(A) with A E X(T)+ is simple with highest weight 2.11 Recall the notations LI,

U:, U,from 1.8.

Proposition: Let I c S and A E X(T)+. Then: a) L ( W = @ v E z , L ( ~ ) l - vand , L(il)l-v is the simple L,-module with highest weight 1. b) OvaZI Proof: a) For any a E R + - RI, any n > 0 and any v E Z I the element I - v + na is no weight of L(1) as il - v + na $ A. (There is a simple root f l # I occurring in a, hence in na - v with a positive coefficient.) This implies Xa,nL(A)l-v= 0, hence L(il)A-vc M = L(I)': using 1.19(6). As L, normalizes U:, the subspace M is an L,-submodule. It is especially M,+, is an L,the sum of its weight spaces and for each p the sum OVEzI submodule. (It is a Dist(L,)-submodule as Dist(L,) is generated by Dist(T) and the Dist(U,) with a E R,.) If there is a p 4 il + Z I with M, # 0, then OvEZ, M,,, contains a non-zero vector invariant under the unipotent group U + n L,. As U + = (U+n L,) U: and as U: fixes M , this vector is invariant under U + ,hence contained in L(A)"' = L(A)A.This implies p - il E ZI, a L(&-,. contradiction. Therefore M = OvEzI b) Keep the notation M from above. The proof of a) implies that L(A), = Mi is the space of all (L, n U+)-invariantelements in M. This implies as in 2.3 that M has a simple socle as an L,-module and that this socle is the L,-submodule generated by any v E L(A)i, v # 0, i.e., socLrM= Dist(L,)v. On the other hand, the simplicity of L(A) implies L(A) = Dist(G)u. Using the basis of Dist(G) as in 1.12(4),but with R+ replaced by L(I) = Dist(U)u

- R + one gets

205

Simple G-Modules

and then (using 1.19(5))

hence the simplicity of 0,

Remarks:

21

L(1), - v .

1) The same proof as above shows

2) This proposition appears for the first time in [Smith]. Part b) was certainly known before, cf., e.g., [Jantzen 11, Satz 1.5, p. 15. 2.12 (Extensions) We can identify Extk(M,, M,) for all G-modules M,, M, with the set of equivalence classes of all short exact sequences 0 --t M1 M 1,M, + 0 of G-modules. Take, for example, M, = M, = L(1) for some 1 E X ( T ) + and consider an u # 0 and, u’ E M with Ju’) = u. exact sequence as above. Choose u E ~5(1)~, As 1 is the largest weight of M, we have u’ E MU+,cf. 1.19(7),hence

Dist(G)u’ = Dist(U)u’ as in the proof of 2.11.b, and (using 1.19(5)) (Dist(G)u’), = ku . This implies i(u) # Dist(G)u’, hence i(L(1)) n Dist(G)u’ = 0 by the simplicity of L(1). Therefore M = i(L(I))0 Dist(G)u’ and the sequence splits. We have thus proved: (1)

Extk(L(il),L(A))= 0

for all 1 E X ( T ) + .

In order to prove another elementary property of Extk let us use the antiautomorphism T from 1.16. For any finite dimensional G-module M we can define a new G-module ‘M as follows. Take M = M* as a vector space, but define the action of G on some cp E M* via gcp = cp 0 z(g) instead of the usual cp 0 9-l as in M*. One has ch ‘M = ch M as T = id and ‘CM) N M as T = 1. If M is simple, then so is ‘M. As any simple G-module is determined by its weights (2.4) we get

IT

(2)

‘L(1) 1: L(1)

for all 1 E X(T)+.

Any exact sequence 0 + M, + M + M, --t 0 of finite dimensional G-modules yields (taking the transposes) an exact sequence 0 + ‘M, --* ‘M + ‘M, + 0.

206

Representations of Algebraic Groups

This produces an isomorphism (3)

Exth(M,, M , ) N Ext;('M,, ( M 2 ) .

Let us apply this to simple modules. Using (2) we get:

(4)

Exth(L(1),L ( p ) )N ExtA(L(p),L(A))

for all 1,p

E X(T)+.

2.13 We have dim Ho(l)< EIfor all 1 by I.5.12.c. (There is an elementary proof of this, cf. [Donkin 91, 1.8.) So we can define for each A E X(T)+ a G-module via V(A)= HO(- w,ll)*.

(1)

stabilizes B and The automorphism cr of G with a(g) = T(w,g-'w;') induces - w, on X(T). So "H0(A)N Ho(-woA) by I.3.5(4) for all A E X(T)+. Twisting a module with Int(w,) produces an isomorphic module. Hence we also get

V(1) z 'HO(1).

(2)

Lemma: Let A E X(T)+. a) There are for each G-module M functorial isomorphisms

HOmG(V(A),M ) 3 Horn,+(& M ) N (kf"’)~. b) The G-module V(1)is generated by a B+-stable line of weight 1.Any Gmodule generated by a B+-stableline of weight 3, is a homomorphic image of V(1).

Proof;. a) Let E : Ho( -qJ)-+ - w o l be the evaluation map as in 1.3.4.We get canonical isomorphisms HOmG(V(A),M ) N HOmG(M*,H o ( - Won)) N

N

HOmB(M*, - W o n )

HOmB(W,1, M ) Z HOmB+(1,M )

mapping any cp at first to cp*, then to E o cp*, to cp o E * , and finally to wo o cp 0 E* = cp 0 wo 0 E * , where wo is a representative of w, in NG(T)(k)and simultaneously denotes the operation of this element on the G-module M tesp. V(1). So we get an isomorphism y: HomG(V(A),M ) 3 HomB+(l,M ) that is obviously functotial. Indeed, we have y(A) = cp 0 y(idv(AJ.(Note that the proof above only works for dim M < 00. The general case follows by taking direct limits.) b) Set u = y(idv(AJ(l).Obviously u is a B+-eigenvector of weight 1. If cp: V(1) -+ M ' is a homomorphism of G-modules with q ( u ) = 0, then y(cp) = q 0 y(idv(A)) = 0, hence cp = 0. This shows that V(1) is generated by u. For

207

Simple C-Modules

any G-module M and any B+-eigenvector m E M of weight A we have a B+homomorphism A + M with 1 H m, hence a G-homomorphism V(A)+ M with V H m. If m generates M , then M is a homomorphic image of V(A). Remark: Because of the universal property in b) we call V(A)the universal highest weight module of weight A. 2.14 We get from dualizing 2.4(1):

V(A)/rad,V(A)

(1)

N

L(A).

Proof: We get from the short exact sequence (cf. (1))

0 -+ rad, V(A)+ V(A)+ L(A)-+ 0 a long exact sequence

(2) 0

+

H o m d W ) , L(P)) HomdV(4, L(P)) Hom~(rad~V(A), L(P))

+ Ext;(L(A),

-+

+

L ( p ) )+ Exth(V (A), L(p)))+ . . . .

Any homomorphism from V(A) to the simple G-module L ( p ) has to factor through V(A)/rad,V(A). So we get from (1) that the first map in (2) is an isomorphism Hom,(L(A), L(p)) N Horn,( V(A),L(p)). So the proposition will follow from (2) as soon as we prove Exth(V(A),L(p)) = 0. Consider an exact sequence of G-modules (3)

0 -+ L ( p ) + M + V(A)+ 0.

Choose some u E MA that is mapped to a B+-eigenvector generating V(A). By our assumption A is a maximal weight of M, cf. 2.2. Therefore u is a B+eigenvector of weight A, cf. 1.19(7).So the G-submodule M' of M generated by u is a homomorphic image of V(A)by 2.13. On the other hand, it is mapped onto V(A)in (3). So M' has to be mapped isomorphically onto V(A)in (3) and has to be a complement to the kernel L(p).So the exact sequence splits, hence Exth( V(A),L ( p ) )= 0.

208

Representations of Algebraic Groups

2.15 Let I E X ( T ) + .Suppose M is a G-module with dim MA = 1 and with all its weights of the form WIwith w E W. By 1.19(2)all non-zero weight spaces have dimension one. If M' c M is a G-submodule with M' # 0, then there is a p E X(T)with M I # 0, hence with M ; = M#. Then p E WI and by 1.19(1)

all weight spaces are contained in M'. So M is simple, hence isomorphic to L(A). Let me give an example where this occurs. Suppose that G = GL, for some n 2 1 and set V = k". Let e l , e,, . . .,en be the canonical basis of k" and suppose that T is the diagonal subgroup with respect to this basis. (We can also work with SL, and T n SL,.) So each e, is a weight vector for T. Let us denote the weight of e, by c i . Then cl, E,, . . . ,E, form a basis of X(T ) .(In the case of SL, one has to drop E, because of the relation c1 + E, + ... E, = 0.) Let us consider the G-module AmV for some rn with 1 I rn I n. It has as a A e,, with 1 Iil < i, < < i, I n. Such a basis basis the e,, A eiz A vector has weight cil + eiZ+ + E,,. So all these weights are different, hence all non-zero weight spaces have dimension one. (This is true also for SL,.) The Weyl group in this case can be identified with the symmetric group S,, which we may regard as the permutation group of {el, E,, . . .,E " } . Therefore obviously all weights are conjugate to om= c1 + E , + E,. If we choose R+ = {ti - cJ I 1 I i < j I n3, then cumE X ( T ) + .Therefore AmV is the simple G-module with highest weight cum (with respect to this choice of R + ) . In characteristic 0 the only A # 0 that can occur in such a situation (again for arbitrary G, say semi-simple with R indecomposable) are the minuscule fundamental weights, cf. [B3], ch. VI, 41, exerc. 24 and & exerc. I, 15. There are more cases in positive characteristic.

+

+

2.16 In this and the following sections we want to describe Ho(A) = indgA in some special cases. Let us take at first G = GL, with n 2 2, let us use the ,,.,,a n - 2 ) be the notations introduced in 1.21 and set I/ = k". Let P = P{al,a2 stabilizer in G of the line ken c k" = V.There is a character o E X ( P )such that

for all g E P ( A ) and any A . We have for any r E N (1) H o ( r o

IT)

= indi(ru)

N

ind;(indgrw)

N

indg(ro 8 indgk) N indg(rw),

using the transitivity of induction (1,3.5(2)), the tensor identity (1.3.6), and finally the equality indik = k [ P / B ] = k, cf. 1.8(5),(6).

209

Simple G-Modules

Let k be an algebraically closed extension of k. As all groups concerned are reduced we have

(2)

ind,G(ro) = {f E kCGl1 f(gs1) = w(gl)-'f(g)

for all 9 E G ( h 91 E

m}.

Set H equal to the kernel of w in P. By (2) any f E indg(rw) is invariant under right translation by any element of H(k). The map g H ge, induces an isomorphism of varieties G ( k ) / H ( k )3 G(k)e, = (V 6 k) - 0 as the tangent map is obviously surjective (cf. [Bo], 6.7). So E [ G ( k ) / H ( k ) ]= k[(Y 0 k ) - 01. As V 6 k is a smooth, hence normal variety of dimension at least two, any regular function on ( V 0 k ) - 0 extends (uniquely, of course) to V 6 k: So k [ G ( i ) / H ( k ) ] N k[ V 0 k] = S( V * )6 k. (As usual S(?) denotes the symmetric algebra.) On the other hand, k [ G ( k ) / H ( k ) ] = k [ G / H ] 0 k and there is a natural map S(V*) -,k [ G / H ] . As it becomes an isomorphism after tensoring with k, it was one already before. Hence S(V*)N k [ G / H ] . E S ( V * ) there corresponds g Hf(ge,). If g1 E P(E), then glen = w(g,)-'e,, hence f(ggle,) = f(w(g,)-'ge,). Therefore (2) implies

To any f (3)

indg(rw) =

{f E S ( V * ) I f(au) = a'f(u)

for all a E k and u E V 6 E}

This space is obviously equal to S'(V*), the homogeneous part of degree r in S(V*):

(4)

indg(ro) E Ho(rw)E S'(V*).

(It is left as an exercise to realize S r ( V )in the form Ho(A).) If we denote the basis of V* dual to e l , . . .,e, by XI,, . .,X,, then we can identify S ( V * ) with the polynomial ring in X1, ...,Xn and S'(V*) with the subspace of homogeneous polynomials of degree r. The weight of each e, is E , , so each Xi has weight - E , and any monomial X","l...Xt(") has weight a(i)E,. Therefore all different monomials in S'(V*) have a different weight, each weight space in Sr(V*) has dimension one and is spanned by one monomial X",")..*X;(") with x a ( i ) = r. We have o = - E , , so X; is the unique monomial of weight rw that has to span H0(rm),, = Ho(rw)"'.So,

-I;=

(5)

L(rw) = Dist(U)X;.

The representation of G on S ( V * ) and on any S'(V*) can be constructed via

210

Representations of Algebraic Group

base change from a representation of G, = GL,,, on S( V g )resp. S'( Vg),where V, is the natural representation of G,. These G,-modules are lattices in the corresponding representations of GQ on S( V t ) resp. Sr( V t ) .So we can compute the action of any X,,,(a E R,m E N) on S r ( V * )from the action of the corresponding element in Dist(GQ). We have Lie(G) = MJk) and X , = E, if a = E~ - e j . Then Xa,"= (Eij)"/n! in DiSt(GQ). From Eijel = ajlei (with 6 , the Kronecker symbol) we get EijXl = -diIXj, hence

In the case n = 2 this implies

(7)

L(ro)=

C

m2O

kE21,mX$ =

m=o

k( r ) X y X 5 - m m

So, if char(k) = p # 0, then L(rw)is spanned by the XYX$-"' with p $

(3.

If

char k = 0, then L(ro) = S'(V*). In this simple case (for any n) one can determine the whole submodule structure of Ho(rZ) by elementary methods. This was first done for n = 2 in [Carter and Cline], cf. also [Cline 21 and [Deriziotis 2,3]. For arbitrary n one has to look at [Doty]. For arbitrary 1 there are so far only partial results, especially for SL,, cf. [Doty and Sullivan 13, [Irving], and [Kuhne-Hausmann]. The proofs there require more advanced methods. 2.17 Let G = Sp,, with n 2 1 and set V = k2". Choose e E V, e # 0, and set P equal to the stabilizer of the line ke. Then P is a parabolic subgroup in G and we may assume P 3 B. There is again o E X ( P ) with ge = o ( g ) - ' e for all g E P ( A ) and any A. Then the same arguments as in 2.16 yield isomorphisms (1)

H o ( r o ) N indg(ro) N S'( V * )

The symplectic form on V defines an isomorphism of G-modules V z V * , so the module in (1) is isomorphic also to Sr( V ) .

+ 1 2 3 or 2 n 2 4 with a basis ( e i ) - n s i s ndropping e , if dim V = 2n. Let q be the quadratic form q ( C i x i e i )= Cr(=,x i x + + x i dropping the term x t in case dim V = 2n. Let G be the corresponding special orthogonal group. (Its

2.18 Let us consider now a vector space V over k of dimension 2n

211

Simple G-Modules

definition requires special care if char&) = 2, but I do not want to go into this.) The diagonal matrices in G with respect to this basis form a split maximal torus T, the lower triangular matrices form a Bore1 subgroup B, and with respect to the order e l , . . . ,en,(e0),e-,,, .. , ,e-l the stabilizer of ke-, is a parabolic subgroup P 3 B. Again let o E X ( P ) such that P operates via -a on ke- We have as before

,.

Ho(rw) = ind,G(ro),

(1)

where we use the notation o also for its restriction to B or T. Let k be an algebraic closure of k and set Y = { u E V 6 kl q(u) = 0}, using the obvious extension of q to V 6 k. The differential of q at any point u E V 6 k, u # 0 is non-zero except for the case char(k) = 2, dim V = 2n + 1, and u E ke,,. As Eeo n Y = 0 we can always deduce from Serre's normality criterion ([M2], III,§8, prop. 2, p. 391) that Y is a normal variety and that q generates in S( V * 6 k ) the ideal of functions vanishing on Y, hence (2)

k [ Y - 01 = E[Y] = S(V* 6 k)/qS(V* 6 k).

If we denote by H t P the kernel of o,then G(E)/H(&)N Y - 0 as a variety (the tangent map being surjective), hence k [ G / H ] 6k

N

k [ G ( E ) / H ( k ) ]N (S(V*)/qS(V*))6 k

and (3)

k[G/H]

N

S(V*)/qS(V*).

As in 2.14 we pick up in ind;(ro) the elements homogeneous of degree r in this ring, i.e., we get an exact sequence of G-modules (4)

0 -+ S r - 2 ( V * ) -+ Sr( V * ) + indg(rw) -+ 0,

where the first map is multiplication with q. Using the bilinear form associated to q we can replace V * by V in this formula, except for the case where at the same time char&) = 2 and dim V is odd.

This Page Intentionally Left Blank

3

Irreducible Representations of the Frobenius Kernels

In this chapter let p be a prime number and assume that k is a perfect field of characteristic p . (The assumption of perfectness is not really necessary and is only made for the sake of convenience as in chapter 1.9.) In this chapter we want to describe the simple modules of the Frobenius kernels Gr of G. Note that we get in the special case Y = 1 the irreducible representations of Lie(G) as a p-Lie algebra. The classification of the simple Gr-modules parallels that of the simple G-modules. The latter ones have a one dimensional B+-socle. If I is the weight of T on this socle, then the module is isomorphic to the G-socle of indgA, hence uniquely determined by 1. Now the simple Gr-modules have a simple B:-socle, and if A is the weight of T, on this socle, then the module is isomorphic to the G,-socle of indg;I, hence uniquely determined by I (3.10). In contrast to the case for G now all characters of T, can occur as the weight of the B:-socle of some simple module. We now have a natural realization of our simple modules not only as socles but also in the form M/radG,M for a suitable Gr-module M,which can be chosen as coinduced from a one dimensional representation of B:. Furthermore all these modules, induced and coinduced from characters of Br resp. B:, have a natural interpretation when regarded as B:- resp. Brmodules. They yield then the indecomposable injective (= projective, cf. 1.8) modules (3.8).This was first observed in [Humphreys 1) (for Y = 1).

213

214

Representations of Algebraic Groups

Now there is an intimate relationship between simple G-modules and simple Gr-modules. Let us describe this in this introduction only in the case where G is semi-simple and simply connected. If we take a simple G-module L(I) with highest weight I having coordinates (with respect to the fundamental weights) strictly less than p' (for short: I E Xr(T)),then L ( I ) stays irreducible under restriction to G, and we get thus all simple Gr-modules. This was first proved in [Curtis 11 for r = 1. When the G, for r > 1 were first looked at, it was obvious that Curtis' proof carried over to this more general situation, cf. [Humphreys 81. Looking at arbitrary simple G-modules and at their structure as Grmodules one gets an easy proof (due to [Cline, Parshall, and Scott 71) of Steinberg's tensor product theorem (first proved in [Steinberg 21): If I = ~ ~ = o p with i l i all I i E X , ( T ) ,then (3.17) L(I) N L(&) 6 L(I,)"l@

* *

@ L(I,)["', I

where the exponent [i] denotes a twist of the representations with the i I h power of the Frobenius endomorphism. This is a very important result that reduces the detailed determination of all L(A) (e.g., that of their formal characters) to the finite set of those L ( I ) with I E X,(T). Finally, we introduce the Steinberg module St, (3.18). This is a simple Gmodule which is simple also as a Gr-module, but which is also isomorphic to a suitable indi;I. Indeed we could show now (but do only in 11.8) that this is the only simple G,-module with this property. This is done in order to prove a formula about G-modules of the form R'ind,G(M)r'l 6 St, for some B-module M , which will play an important r61e in the next chapter and which is due to [Andersen 51 and [Haboush 41 independently. If k = F, and if we regard St, as a G(F,)-module where q = p', then we get the "original" Steinberg module from [Steinberg 11, which is responsible for this terminology. 3.1 (The Frobenius Endomorphism on G) As G arises from GZ through base change we also get G from GF, through base change. Therefore (cf. 1.9.4) any G(') is isomorphic to G, and there is a Frobenius endomorphism F = FG:G + G such that we get any F L as G G 3 G(') using a suitable isomorphism. We get especially for all r (1)

G, = ker(F').

We also have T = (TFJkand U, = (Ua,F,)kfor all E. So F stabilizes T and all U,. We get any x, from x , , ~ : 3 Ua,z and we have X ( T ) = X(Tz).

The Frobenius Kernels

215

Therefore the isomorphisms G, 2 Ua and T 2 (G,), (where r = r k T ) are compatible with the usual Frobenius endomorphisms on G, and G,. This implies (for all A ) (2)

F(t) = tP

for all t E T ( A )

and (3)

F(xa(a))= xa(aP)

for all a E R, a E A .

All the groups introduced in Chapter 1 (e.g., B,B+,P,,P:, U(R'), L,) are F-stable and F restricts to a Frobenius endomorphism on them. 3.2 Lemma:

n Ua,,x

n U-a3r

The multiplication induces isomorphisms of schemes (for any r) T, x

acR+

1:

U: x T, x U, N G,.

asR+

Proof: As 1 is contained in the open subscheme U +T U of G its inverse image G, = T r ( l )is contained in F-'(U+TU) = U'TU, cf. I.9.3(5).The multiplication induces isomorphisms of schemes

n UaxTx n U_,SU+xTxUrU+TU,

aeR+

aeR+

where 1 corresponds to the element having all components equal to 1. Now F stabilizes all factors and induces the Frobenius endomorphism on each of them. This implies the lemma. 3.3 Lemma:

The elements

with 0 In(a),m(i),n'(a) < p' form a basis of Dist(G,). Proof: We have by 3.2 (or even more directly using 1.7.2(5))an isomorphism of vector spaces

@ Dist(Ua,,)0 Dist(T,) 0 a@ Dist(U-,,,) sR+

aeR+

N

Dist(G,).

So the claim follows from the description of Dist(G,,,) = M(G,,,) and Dist(G,,,) = Dist(p,,,) = M(p(,)),where q = p' in 1.8.9. Remark: It is now also clear how to write down bases of Dist(U,), Dist(U:), Dist(T,), Dist(B,), Dist(B:), . . . .

216

Representations of Algebraic Groups

3.4 Recall that G (resp. B, B') operates on G, (resp. B,, B:) through conju-

gation leading to a representation on Dist(G,) (resp. Dist(B,), Dist(B:)), and that then Dist(G,)? (resp. Dist(Br)p, Dist(B:)y) is a one dimensional submodule, cf. 1.8.19,1.9.7.

Proposition: a) The operation of G on Dist(G,)? is trivial. b) The operation of B = TU on Dist(B,)p is trivial on U,and is given by -2(p' - 1)p on T. c) The operation of B+ = T U + on Dist(B:)y is trivial on U’, and is given by 2(p' - 1)p on T. Proofi a) The adjoint representation of G on Lie(G) factors through G / Z ( G ) ,which is semi-simple and admits no character. Hence det 0 Ad = 1 and the claim follows from 1.9.8. b) We have Lie B = Lie T @ CDaeR + (Lie G)-a, hence det(Ad(t)) = (rk(T)O

+

aeR+

(-a))(t) = ( - 2 p ) ( t )

for any t E T ( A ) ,so T acts by 1.9.8 on Dist(B,)p via (p' - 1)(-2p). On the other hand, the unipotent group U admits no character, hence U has to operate trivially. c) This is proved similarly. 3.5 Corollary: Let M be a B,-module and M' a B:-module. Then (1)

coindg;M

N

indg;(M @ 2 ( p r - 1)p)

and (2)

coind:;M'

N

indg;(M' @ (-2(p'

-

1)p)).

In case dim M < 00 resp. dim M' < co,we have

(3)

(indg;M)*

N

indg;(M* @ 2(p' - 1)p)

resp. (4)

(indgj M')*

N

indi;(M'* @ (-2(p'

- 1)p)).

Proof: This follows from I.8.17- 19 and from 3.4.

3.6 (Induced and Coinduced Modules) The isomorphism of schemes U: x B, 3 G, given by the multiplication is compatible with the action of U: by left multiplication on U: and G, and with the action of B, by right

217

The Frobenius Kernels

multiplication on B, and G,. It is also compatible with the action of T, by conjugation on U: and by left multiplication on B, and G,. Therefore, the isomorphisms k[G,] 3 k[U:] 8 k[B,] and Dist(U:) 6 Dist(B,) 3 Dist(G,) of vector spaces are also compatible with the representations of U: (resp. B,, T,) induced by these actions. We have for any B,-module M indi;M = (k[G,] 8 M)Br1: k[U:]

8 (k[B,] 8 M)Br,

hence indi;M

(1)

k [ U : ] 8 M.

N

This isomorphism is compatible with the representations of U: (acting via p, on k[U:] and trivially on M) and of T, (acting as given on M and via the conjugation action on k[ U:]). Similarly, coind;;M = Dist(G,) @Dist(B,) M

2:

Dist(U:) 8 Dist(B,) @Dist(B,) M,

hence (2)

coindi;M

N

Dist(U:) 8 M.

Again this isomorphism of vector spaces is compatible with the representations of U: and T, (defined similarly as above). By interchanging the rBles of U, and U: we also get for each B:-module M’ isomorphisms (3) and (4)

indi;(M’)

N

k[U,] 8 M’

coindi;(M’)

N

Dist(U,) 8 M‘

of U,-modules and T,-modules for the “obvious” representations on the right hand sides. = prlRflby 1.9.6(2)as U is reduced. This is also We have dim k[U,] = prdimU the dimension of k[U:], Dist(U,), and Dist(U:). So all the induced or coinduced modules considered above have dimension equal to (5)

dim(M)prlR’I.

3.7 Any 1 E X(T )defines by restriction a character of T,, which we usually also denote by A (if no confusion is possible). We get from the restriction an exact sequence (1)

0 + p ‘ X ( T ) + X ( T )--t X ( T , ) -P 0,

where the first map is the inclusion.

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Represeatatiomof Algebraic Groups

By extending its restriction to T, trivially to U,or U:, any I E X ( T )defines a one dimensional module (usually denoted by I ) for B, and B:. We can induce and coinduce these modules to G,. Let us introduce the following notations:

(2)

Z,(I) = coindg; I ,

(3)

Z : ( I ) = indg;I.

Now 3.5 implies: (4)

Z,(I) N indg;(I - 2(p' - l)p),

(5)

Z:(I) N coindg;(I - 2(p' - l)p),

(6)

zr(I)* N zr(2(P" - 1 ) -~ I),

(7)

ZL(I)* N ZL(2(p' - 1)p - I).

Note that we have by 3.6(5) (8)

dim Zr(I)= dim Z:(I) = prlRtI

for all I. As A + p'v and I have the same restriction to T,(for all A, v E X ( T ) )we get:

+ P'V)

(9)

zr(I

(10)

z;(I+ p'v)

= zr(L), = ZL(I).

It will become clear later on (9.7) that &(I) and Z:(I) define the same element in the Grothendieck group of all G,-modules. This is the reason for choosing the notation as above. 3.8 Proposition: Let I E X(T). a) Considered as a B,-module &(I) is the projective cover of I and the injective hull Of A - 2(p' - 1 ) ~ . b) Considered as a B:-module Z:(I) is the injective hull of I and the projective cover of I - 2(p' - 1)p. Proofi The injective hull of any simple module for groups like B: = T, pc U : and B, = T, pc U, has been determined in 1.3.11 (cf. 1.3.17). The injective hull, say, of I for B: is k [ U : ] 0 I , with U: operating only on the first factor (via p,) and with T, operating through the conjugation action on k [ U : ] tensored with the restriction of I.. On the other hand, ZL(A)= indg;I has exactly this form by 3.6(1). Thus we get the first part in b) and similarly the second part in a) using 3.7(4). The statements about projective covers follow now from 3.4 and 1.8.13.

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The Frobenius Kernels

3.9 (Simple G,-modules) It follows from 3.8 that all Zr(l)Urand Zi(lb)": have dimension one, as these spaces of fixed points are contained in the Br- resp. the B:-socle of the module. On the other hand, we have Mur # 0 # Mu$ for any Gr-module M # 0, as U,,U: are unipotent (by 1.2.14(8)). Arguing as in 2.3 we get (1) Any Zr(l) and any Zi(1) has a simple socle when considered as a Grmodule. Dualizing (using 3.7(6), (7)) we get (2) Any Zr(l)/radG,Zr(l)and any Z;(A)/radGvZ&4)is a simple G,-module. Arguing as in 2.4.a we get (3) For any simple Gr-module L there are I., ,L2,A,, A4 L

2

E

X(T ) with

socGrzi(n,) 2 socGrzr(A2) zi(n'j)/radG,z:(n,) zr(A4)/radG,zr(A4)*

Set for all A

E

X(T) Lr(A) = socG,z:(n).

(4)

Obviously, 3.7( 10) implies (for all 1,v E X(T)):

L'(1

(5)

+ p'v) = L'(1).

3.10 Proposition: W e have for all 3, E X(T): L,(A)"$ = A,

(1)

(2)

zA1)/radGrz,(n)

(3)

N

Lr(Jh

End&,(A) = k.

If A is a set of representatives in X(T )for X(T ) / p ' X ( T ) ,then each simple Grmodule is isomorphic to exactly one Lr(l) with rl E A. Proofi The first formula follows immediately from the definition of Lr(l)and from 3.8.b. This implies hence by I.8.14(3)

HornB: (1%Lr(n)) # 0,

0 # HOmGr(COind$ 1,L,(I))= HOm,(z,(A), Lr(A)).

As Lr(l) is simple, any non-zero homomorphism

&(A)

--*

Lr(l) has to be

220

Representatiom of Algebraic Groups

surjective and to factor through Zr(A)/radGrZr(A).Now 3.9(2) implies (2). In order to get (3) observe that any E EndG&,(A) stabilizes the line L,(A)'=, hence has an eigenvalue in k. Now Schur's lemma implies (3). It follows from (1) that Lr(A), Lr(p) are isomorphic if and only if A - p E prX(T). This together with 3.9(3) gives the last claim. Remark: As in 2.9, formula (3) implies that L,(A) is absolutely irreducible, i.e., that for each (perfect) field k' 3 k the (Gr)kt= (Gk,),-mOdUk &(A) @ k' is the unique simple (G,,),-module such that (&,)r operates via A on the (Ul,),-fixed points. Furthermore, we have (as in 2.9(1)) (4)

SOC(G,,,,(M@ k') = (socG,M) @ k'

for any Gr-module M .

3.11 As described in 1.2.15, we can construct for any g E G(k) and any Gr-module M a new Gr-module EM. Proposition:

For all g E G(k) and for all simple Gr-modulesL we have

N

L.

Proof: Obviously the "twist" M H EM commutes with field extension. Because of 3.10 and the remark to 3.10 we may assume that k is algebraically closed. Suppose, for the moment, L = L,(A) and choose u E Lu', u # 0. Any g E B + ( k ) normalizes U:, hence u is also in (EL)''. If t E T,(A) for some A and g E B+(k), then g - ' t g = t ( t - ' g - ' t g ) E B:(A) and t - ' g - ' t g E U + ( A ) , hence t-'g-'tg E U:(A). The operation of t on @ A maps u @ 1 to (g-'tg)(u @ 1) in L @ A, hence to t ( u @ 1) = A(t)(u @ 1). So (")" = A and N L . As this is true for any simple G,-module and any g E B+(k),and as G(k) is the union of its Bore1 subgroups, we get the proposition.

Proof: Recall that w, E W is the element with w,R+ = - R + . It satisfies w,’ = 1. Choose g E N,(T)(k) as a representative. Then g U + g - ' = U,hence g U : g - ' = U,.Identifying Lr(A)and as vector spaces we get Lr(A)": = (",(A)'.

As T, operates on the left hand side through A, it operates on the right hand

221

The Frobenills Kernels

side through w o k This implies using 3.1 1 0 # Hom~,(woA'Lr(A))N Homg7(woALr(A)),

hence (by 3.7(5))

+

0 # HOmG,(COind~y,~, Lr(A))= HOmG,(zL(W,~ 2(pr - l)p),L,(A)). As w,p = - p we get also

0 # HomGr(ZL(i),Lr(woA + 2(pr - l ) ~ ) ) ,

which yields the second isomorphism in the corollary (using 3.9(2)). We know by the above that L,(I) is the unique simple Gr-module (up to isomorphism) with L,(A)"r

1

wok

Now 3.9(1) and 3.8.a yield the first isomorphism in the corollary. 3.13 Corollary:

W e have for all 1 E X ( T ) : L,(1)* 2: L,( - won).

Proofi Obviously L,(A)* is again simple. We saw above HomB,(woil,Lr(l))# 0, and get by dualizing

0 # HOmBp(Lr(n)*,-won)

2:

HomGr(Lr(A)*,z:(- won)),

which implies the corollary. 3.14 Lemma: Let 1 E X ( T ) +and choose u E L(l)A,v # 0. Then Dist(Gr)uis a simple G,-module isomorphic to Lr(A). Proofi We may obviously assume that k is algebraically closed. As G is reduced, 1.6.16 implies that L(1) is a semi-simple Gr-module, hence so is Dist(G,)u. We have ku N A as a BT-module, so we get a homomorphism cp: Z,(A) = Dist(Gr) @Dis,(B:)

1 + Dist(Gr)o

mapping 1 @ 1 to v. Obviously cp is surjective. As its image is semi-simple, by 3.10(2)it has to be even simple and isomorphic to L,(A). 3.15 Set

(1)

X,(T)={IEX(T)IOI (1,a") < p r

forall U E S } .

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Representations of Algebraic Groups

Obviously, X , ( T ) c X , ( T ) c . * + c X'(T) c . . - X ( T ) + . Proposition: For each ;1E Xr(T ) the simple G-module L(2) is also simple as a

G,-module and is isomorphic to L,(I) for G,.

Proof: Choose u as in 3.14 and set L = Dist(G,)u. We have to show L = L(2). We can obviously assume that k is algebraically closed. For each g E G(k) the subspace gL = g L is another simple Gr-submoduleof L(I), so either gL = L or gL n L = 0. If g E B+(k), then g u E k X u , hence gL n L # 0 and gL = L . Consider now a simple root a and a representative of s, in N,(T)(k), e.g., let us take n,(l). Then n,(l)u E L(I)sEaand n,(l)u # 0. The proof of 2.ll.b shows @ L(&-,,, = Dist(U-,)u = kX-,,,u, n2O

n2O

hence L(IJa-,,== kX-,,,u for all n E N,and L(&-,, c Dist(G,)u for all n < p'. Now s,i = I - (IL,a") a and 0 I (A,a " ) < p' by assumption. This yields n,(l)u E Dist(Gr)u= L and, therefore, n,(l)L= L . As W is generated by the s, with a E S any w E W has a representative w E N,(T)(k) with GJL= L . Using the Bruhat decomposition G(k) = U , , , B + ( k ) w B + ( k ) we get gL = L for all g E G(k). So L is a non-zero G(k)-submodule of the simple G-module L(R).As G is reduced, this implies L = L(I), hence the simplicity of L ( I )as a Gr-module.

Remarks: 1) There is a possibly more elegant approach to this result in [Cline, Parshall, and Scott 71 using ideas from [Cartier]. One gets from 3.1 1 a projective representation of G on L , i.e., a homomorphism G + PGL(L). This can be lifted to a representation G' + G L ( L )of a certain covering group of G. The proof uses the (nontrivial) result due to Steinberg that for G semisimple and simply connected any projective representation lifts to a linear representation. The simple G'-module L can then be checked to be L ( i ) .There is still another proof in [Kempf 5],4.2. 2) For G semi-simple and simply connected X r ( T )is a system of representatives for X ( T ) / p ' X ( T ) . So in this case any simple G,-module can be lifted to G (cf. remark 1). In general one can replace G by a covering such that X r ( T ) contains a system of representatives for X ( T ) / p ' X ( T ) .One has, however, to observe for a covering G' + G that Gr is not necessarily a factor group of G:.

223

The Frobenius Kernels

3.16 If M is a G-module, then we can define another G-module structure on M by composing the given representation G + G L ( M ) with F': G + G. We

denote this G-module by Mrrl.If the given representation on M is defined over Fp, then MI'] N M"), cf. 1.9.10. Obviously G, acts trivially on MIr1. On the other hand, look at a G-module V such that G, acts trivially on V. Then the representation G -+ GL( V ) factors through G/G,. We know G / G , N G by 1.9.5, more precisely, we have a commutative diagram.

G

\,"a

can

Therefore V has the form MI'] for some G-module M usually denoted by I"’]. For example, we can apply this for any G-module V to V G r and, more generally, for any G-modules V, V’ with dim V' < co to HomGr(V',V ) . Suppose there is a system X : ( T ) of representatives for X ( T ) / p ' X ( T ) with X ; ( T ) c X,(T). Then we have for any G-module M an isomorphism of G-modules (by 1.6.15(2))

We can apply the remark above to all HornGr@@),M ) . Proposition:

W e have for all I E Xr(T ) and p E X ( T ) +: L(I

+ p'p)

2

L ( I )0 L(p)"'.

Proof: We may assume that there is X : ( T ) as above and that A E X:(T), hence that we can apply (1) to M = L(A + p'p). As this module is simple, there is only one summand fO, and the corresponding Horn(...)[-'] has to be simple. Therefore there are A' E X : ( T ) and p' E X ( T ) + with L(I + p'p) N L(A')0 L(p')[']. Comparing the highest weights yields I + p'p = I' p'p'. Now I - A' E p ' X ( T ) and A, I' E X i ( T ) imply I = I' and then p = p' and the proposition.

+

Remark: Suppose there is X : ( T )as above. The proposition implies that any simple G-module is semi-simple for G,, hence for any G-module M

(2)

SOCG M

c

SOCG,

M.

This also follows from 1.6.16 using 3.10(4) and 2.9(1).

224

Representations of Algebraic Groups

More precisely than (2), the proposition shows for all I E X r ( T )that each L(A)6 socGHOmG,.(L(A),M ) is a semi-simple G-module. On the other hand, each simple G-submodule of M is isomorphic to some L(A)6 L(p)['] with A E X : ( T ) and p E X ( T ) , . Under the identification in (1) it corresponds to some ,!,(Iu) @ I/ with I/ c HornG$@), M ) and V N L(p)['], hence V c SOCGHOmGr(L(A),M). This implies: (3)

3.17 We get from Proposition 3.16 by induction:

Corollary (Steinberg's Tensor Product Theorem): Let I , , 11,. . . ,A, E Xl(T) and set A = C;=, p i I i . Then

L(A) Cv L(I,) 0 L(A,)"l@ *..6L(A,)[,l. 3.18 (The Steinberg Module) We have for any I E X ( T )by 3.6(2):

&(I) = Dist(U,) 6A.

(1)

So all elements (for some fixed ordering of R’) with 0 I n(a) < p' form a basis of (3)

zr(A)"r

=k

&(A), cf. 3.3. Set q = p'. We have

n

aER+

X - u , ( q - 1)

6 1,

as the element on the right hand side is obviously annihilated by all X - u , n with a E R+ and 0 < n < p', hence invariant under U,, and as dim Zr(A)Ur= 1 by 3.8.a. Consider now I E X ( T ) +and choose u E L(&, # 0. There is by 3.14 a surjective homomorphism &(I) + Dist(Gr)u of G,-modules mapping 1 6 1 to u, hence a basis element as in (2) to X-u,n(u)u. This is a weight vector n(a) a. (for T ) of weight I - CuER+ Suppose (p' - 1)p E X ( T ) .(This is automatically satisfied for p # 2, but for p = 2 we may have to replace G by some covering.) Obviously ( p , a") = 1 for all a E S implies (p' - 1)p E Xr(T).So by 3.15 we have for 1 = (p' - 1)p that L(1) = Dist(G,)u, with u as above. The lowest weight of L ( I )is

HuGR+

w,A = w,(p'

- 1)p = -(p'

- 1)p = A - 2(p' - 1)p = I -

nuER+ nueR+

c (p' -

1) a

urR+

The only vector of the form X-a,n(u) u with all n(.) < p', which can have this weight, is X - u , q - lu (where q = p' as before). So (3) implies that

225

The Frobenius Kernels

the epimorphism Z,(A) L(A)is injective on the U,-socle of Z,(A). Therefore it is injective on the whole module. This implies --f

(4)

Lr((P' - 1 ) ~N) Zr((P' - 1 ) ~'Y) V(P'- 1 ) ~ )

and by dimension comparison also (5)

L,((p' - 1)p) = ZL((pr - 1 ) p ) = indi;(p' - 1)p.

We denote the G-module L((p' - 1)p) by St, and call it the rth Steinberg module (always supposing (p' - 1)p E X(T)). Recall (2.4) that St, is the socle of the induced module Ho((q - 1)p) = ind& - 1)p where q = p'. So we have 0 # Hom,(St,,indg(q - 1)p) N HOm,(St,,(q - 1)p) HOmGrB(St,, ind?,(q

- 1)p).

We know by I.6.13(1) that indCB(q - 1)p considered as a G,-module is isomorphic to indg;(q - l)p, hence to St, by (4), (5). It therefore has the same dimension. As St, is simple for G,, hence also for G,B, any non-zero homomorphism of G,B-modules St, + indfB(q - 1)p has to be injective, hence bijective because of the equality of dimensions. So we get (6)

St,

N

ind?'(p'

- 1)p

(as G,B-modules).

(We shall study modules of the form ind,G.,A more systematically later on in Chapter 9.) 3.19 Proposition: Suppose (p' - 1)p E X(T ) . W e have for each B-module M and each i E N isomorphisms of G-modules:

H'((p' - 1)p @ M"])N St, @ H'(M)"'.

Proof: We can apply 1.6.11 to H = G,B and N = G,. We have G,B/G, N B/(G, n B ) = B/B, and can thus regard MIr1as a G,B/G,-module. So from 1.6.11 we get isomorphisms

The isomorphisms G/G, N G and G,B/G, N BIB, 1: B induced by the Frobenius morphism are compatible with the inclusions G,B/G, c G/G, and B c G. Furthermore, if we regard MIr1as a B-module via B N BIB,, then we get M. So the left hand side in (1) is

(2)

R'indg(M) = H ' ( M ) ,

when regarded as a (G N G/G,)-module. If, as in (l), we take the structure as

226

Representations of Algebraic Group

a G-module via G % G / G r , then we have to apply the Frobenius twist [ r ] again. So we get H'(M)"'

(3)

N

R'indZ,,( M"]).

We can now tensor this isomorphism with St,, use the generalized tensor identity (1.4.8) and the isomorphism from 3.18(6) to get H'(M)"' @ St,

N

RiindZrB(M['])@ St,

N

R'ind&(kf"] @ st,)

z

R'ind:,,

o

indFB(M[']@ (p' - 1)p).

As G r B / B N Gr/Br= U: is affine, the functor ind?' is exact (1.5.13). Therefore the spectral sequence I.4.5.c degenerates in this case to give isomorphisms R'indg,,

0

ind,GrBN R'indi.

If we apply this to the last formula, then we get the claim of the proposition.

Remark:

Take M = k. As H o ( k ) N k by 2.1(6), we get H 0 ( ( p r - 1 ) ~N) S t r ,

(4)

hence also S t ,

N

V((p' - 1)p) by duality.

4

Kempf’s Vanishing Theorem

Let k be a field throughout this chapter. If A E X(T)is a dominant character (equivalently, if Ho(A) # 0), then all higher cohomology groups (ie., with i > 0)

H i @ ) = H ’ ( G / B ,9 ( A ) ) = R’indgA vanish (are zero). This is Kempf’s vanishing theorem, which is the main result of this chapter. In the case of char&) = 0 it had been known for a long time, but in the case of char(k) # 0 it was only in 1976 that Kempf’s proof of this theorem appeared. The proof given here (4.5) is due to [Andersen 51 and [Haboush 41 independently. Before we can enter the proof of this theorem we discuss general properties of invertible sheaves, especially of ample sheaves, and describe the form Serre duality takes on G/B.This is done in 4.1-4.4. There is more known about line bundles on G/B than we discuss here. For more information see, e.g., [Kempf 31 or [Iversen]. After the proof of Kempf’s vanishing we discuss some first applications. For example, we prove formulas comparing B- and G-cohomology (4.6/7). We deviate to prove results on B-cohomology (4.8-4. lo), which are then applied to give some results on G-cohomology (4.11-4.13). For example, we describe the cohomology of the trivial module. Most of the results in 4.6-4.13

227

228

Representations of Algebraic Groups

were first proved in [Cline, Parshall, Scott, and van der Kallen]. The corollary 4.12, however, was first proved by W. van der Kallen who has communicated his proof to me. Many G-modules that occur in “nature” have an ascending filtration such that each successive quotient is isomorphic to some Elo@). We discuss such “good” filtrations in 4.15-4.21. There are some references mentioned in 4.19/20. One should add [Donkin 41 where the criterion in 4.16.b appeared for the first time. 4.1 For any parabolic subgroup P c G the canonical map n:G G / P is locally trivial by l.lO(5). This means that we can cover G / P by open subschemes Y for which there is a section 0,:Y-+ G, ie., a morphism with II 0, = id,. Consider now a P-module M and the associated sheaf Y ( M ) = Y G / p ( M ) as in 1.5.8. Note that the local triviality of G -P G / P implies by 1.5.16(2): -+

Q

(1) The OGip-moduleY ( M ) is locally free of rank dim M. We have for all P-modules M , M ’ and for each open Y c G / P a natural map Y ( M ) ( Y )@o,,p,~,Y(M’)(y)-+ Y ( M @ M’XY), mapping any f l @ f2 to the function g H fl(g) @ f2(g). Using the local triviality it is easy to show that we get thus an isomorphism (2)

Y ( M )@o,,, Y ( M ’ )

Y ( M @ M’),

Similarly one gets isomorphisms for each r E N: (3)

Y(S‘M)

N

S ‘ Y ( M ) and Y ( A ‘ M ) N A ‘ Y ( M ) ,

where the symmetric and exterior power on the right hand sides are taken within the category of OG/,-modules. Furthermore, one has in case dim M < co (4)

Y ( M * )N Y ( M ) ’ ,

where ( )” denotes the dual of a locally free sheaf, cf. [Ha], 11, exerc. 5.1. 4.2 Let I c S and set P = P,. As G/P is projective, I.5.12.c implies:

(1) If M is a finite dimensional P-module, then each R’indgM is a finite dimensional G-module.

229

Kempf's Vanishing Theorem

The dimension n(P) = dim G / P is equal to the number of roots a with U, q! Pr = P,hence: n ( P ) = IR+ - RrI.

(2) Recall from I.5.12.b that (3)

R'ind; = 0 for all i > n(P).

For any P-module M there is an action of G on the associated bundle G x p Ma (cf. 1.5.14/15) via left multiplication on the first factor. This operation is compatible with the canonical map nM:G x p Ma --* G / P and the obvious action on G / P . It is not difficult to show that any vector bundle over G/P with a G-action, such that the projection to G / P is G-equivariant, has the form G x p M a for some P-module M . (Take M as the fibre over l p ~ ( G / P ) ( k ) . ) For example, in the case of the tangent bundle we have M = Lie(G)/Lie(P), so we get using the comparison from 1.5.15(1): (4) The tangent sheaf on G/P is g(Lie(G)/Lie(P)).

(Of course, this generalizes to arbitrary G and P.) The canonical sheaf (cf. [Ha], p. 180) is oGlP = An(p)Y( Lie(G)/Lie(P))' N

Y(A"(')(Lie( G)/Lie(P))*).

The weights of T on Lie(G)/Lie(P)are the a E R+ - R,. Therefore A"(p)(Lie(G)/ Lie(P))*is the one dimensional P-module corresponding to the weight - 2pp, where

Hence, (6)

wG/P

= y(- 2 p P ) ,

especially (7)

~ G /= B

W-2~1,

The correspondence between vector bundles and locally free sheaves (on G / P ) maps bundles with G-actions as above to G-linearized sheaves as, say, in [MF], p. 30. On such sheaves Y the functorial property of sheaf coho, [MF], mology induces G-actions on each cohomology group H'(G/P, Y ) cf. p. 32 for i = 0. For any P-module M this operation on H ' ( G / P , Y ( M ) )

230

Representations of Algebraic Groups

coincides with that coming from H'(G/P, Y ( M ) )z R'indFM, ie., from 1.5.12.a.This can be checked directly for i = 0 and follows in general from the uniqueness of derived functors. This implies that Serre duality (cf. [Ha], 111. 7.7) is compatible with the G-action. As G/P is smooth, the canonical sheaf is equal to the dualizing sheaf (cf. [Ha], III,7.12). So for any finite dimensional Pmodule M : (8) The G-module R'indgM is dual to R"(P)-iindF(M*0 (-2pp)).

We get (taking P = B ) for all A E X(T)

Hi(/?) N If"-’( -(A

(9)

+ 2p))*,

where n = JR'J. We have, for example, (cf. 2.13(1)) (10)

V(A)N H"(w,l

-

2p) = H " ( w , . A)

for all 3, E X(T)+.

4.3 (Ample Invertible Sheaves) Let us apply 4.1 to G = GL(V) and P = StabGko for some finite dimensional vector space V (with dim V 2 2) and some u E V, u # 0. We have G/P N P(V) N P(dimV)-l, the projective space associated to V.

Let us denote the character of P, which describes the action of P on ko, by --w as in 2.16. The arguments there show for any Y c P(V) open, that we can identify 2'(rw)(Y)for each r E Z with the space of all regular functions on n-'(Y) u (0) c V that are homogeneous of degree r. This is the sheaf usually denoted by OP(")(r), cf. [Ha], p. 117. So we get (1)

YGL(Y)/P(ro) =

(I have to admit that the language above is not quite correct. We can argue like that only for k algebraically closed when working with varieties instead of schemes. The arguments carry over to arbitrary k and varieties with kstructures. As all our schemes are reduced, they correspond to such varieties.) Recall that an invertible sheaf 9 on some algebraic scheme X over k is called very ample if there is an immersion i: X + 9( V ) for some vector space V, 2 5 dim V c co such that 2’ N i*OP(")(l).It is called ample if there is m > 0 such that 2'"'is very ample. The following three conditions on 2' are equivalent:

(i) Y is ample, (ii) 2’"’ is ample for some m > 0, (iii) 2’"’ is ample for all m > 0. (For all of this one may compare [Ha], p. 153/4.)

23 1

Kempf's Vanishing Theorem

We shall have to use Serre's cohomological criterion for ampleness ([Ha], 111.5.3): (2) A n inuertible sheaf Y on an algebraic scheme X ouer k is ample if and only if there is, for each coherent sheaf % on X , an integer rn, such that = 0 for all i > 0 and m 2 m,. H i ( X , % @ 9"’) 4.4 Proposition: 0 for all a E S.

Let A E X ( T ) . Then YG/B(A) is ample if and only if ( A , a") >

Proof: Suppose at first ( A , a v ) > 0 for all a E S. Then there is for any p E X ( T ) some m E N,m > 0 with mA p E X ( T ) + . Denote by k some algebraic closure of k. We know (cf. [Bo], 5.1) that there is a representation G --t G L ( V ) in some finite dimensional vector space V and a line kv c V such that B(k) is the stabilizer of k(u @ 1) c V 0 ?I in G(C) and such that Lie@) is the stabilizer of ku in Lie(G). Let ~ E X ( Tbe) the weight through which B acts on kv. Choose m E N, m > 0 with mA p E X ( T ) + .We know from 2.6 (with B and B+ interchanged) that there is a G-module V’ and a line ku' c V' such that B(k) c Stab,&u' @ 1) and Lie(B) c StabLi,(G)kU'and such that B operates via -(mA p ) on ku'. Then an easy calculation shows that B(C) is the stabilizer in G(k) of k(v @ u' @ 1) c (V @I V') @ k and that Lie(B) is the stabilizer in Lie(G) of k(u @ 0’). Furthermore, B acts on k(u @ 0’) via -mA. Now the precise description of the stabilizers implies that g H g(v @ v') defines an immersion i: G/B P( V @ V'), cf. [Bo], 6.7. Now 4.3( 1) and 1.5.17(1) yield

+

+

+

--f

*OP(V Q V’)( l)

YG/B(mA).

So Y G , B ( d ) is very ample. As the functor 9 G / B commutes with tensor products, we have YG/B(mA) N Y,,,(A)"', hence YG,B(A) is ample. Let us now prove the converse. Take a E S . There is a homomorphism (p,:SL, + G as in 1.3(2) mapping each x-,(a), and each

(:

(::-')

(A

y)

to x,(a), each

to a'(b). Let us take B' c S L , as the Bore1 sub-

group of lower triangular matrices. We can identify SL,/B' with P'. I f p is the character

4.3(1). This implies

t+

b", then 9sL21Bp(p) is identified with O,,(n) by

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using 1.5.17(1). Now @a defines an isomorphism of P’ onto a closed subscheme of G/B, cf. 1.8(5). If Y G / B ( A ) is ample, then so is @ z Y G , B ( A ) , e.g., by [Ha], 111, exerc. 5.7, hence (A,av) > 0, e.g., by [Ha], 11, 7.6.1.

Remarks: 1) Let I c S and consider P = PI. We know from 1.18(4) that any A E X ( T ) with ( A , a') = 0 for all a E I defines a one dimensional repreon G/P. Using more or less sentation of P,hence an invertible sheaf YGIp(A) is ample if and only if the same arguments as before one shows that YGIp(A) (A,aV) > O f o r a l l a ~ S - l . 2) For another approach to this result see [Kempf 3],5.3. 3) If some YGIB(A) is ample, then it is even very ample, cf. 8.5(1). 4.5 Proposition: If

A E X ( T ) + ,then Hi@)= 0 for all i > 0.

Proof: Recall that Hi@)= R'indgA N H'(G/B,9'(A)). There is a finite central extension of G of the form G1 x G, with G, semi-simple and simply connected and with G, a torus. The inverse image of B resp. of T has the form B, x G, resp. T, x G, with B, a Bore1 subgroup in G, and Ti a split maximal torus in G,. Now A defines a character on T, x G,, hence A, E X(T,) and A, E X(G,). One has ( A , a v ) = ( A , , a v ) for all a E R, hence A, E X ( T , ) + . We can now apply either 1.6.11 and I.4.9(1) in order to show that it is enough to prove the proposition for G, and A,, or we can observe that G / B N G1/B,, and that under this isomorphism Y G / B ( A ) N YGl,Bl(Al). So we can assume that G is semi-simple and simply connected. We can therefore assume p E X ( T ) . As ( p , a " ) > 0 and (A, a v ) 2 0 for all a E S we get (A + p , a " ) > 0 for all a E S, hence Y(A+ p ) is ample by 4.4. Assume now at first char@) = p # 0. We apply 4.3(2) to Y = Y(A+ p ) and B = Y(- p ) . We get thus r E N with 0 = H ' ( G / B , Y ( - p ) @ Y ( A + p)"') = H'(p'(A

+p) -p)

for all i > 0. We have by 3.19 H'((p'

- 1)p + p'A)

N

St, @ H i @ ) [ ' ] ,

so the vanishing above implies H i @ ) = 0 for all i > 0, as claimed. As G and B arise from GZ and BZ via base change and as also A E X ( T ) 1: X ( Tz) comes from a B,-module, we can apply the universal coefficient theorem 1.4.18.b. The vanishing established for all Fp so far proves the vanishing of the R'indgA for all i > 0, from which the result then follows for all fields of characteristic 0.

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Kempf's Vanishing Theorem

Remarks: 1) In the case of characteristic 0 we can also apply Kodaira's vanishing theorem (cf. [Ha], 111, 7.15) in order to prove the proposition. (One has to use that 9(- 2p) is the canonical sheaf on G/B,cf. 4.2(7).)There is another proof (in case char(k) = 0) in [Donkin 91, 12.3. 2) In the case of characteristic p # 0 the proposition was first proved in [Kempf 13. We shall call it Kempf's vanishing theorem even in those cases where characteristic 0 is still included. 4.6 Proposition:

Let I c S and set P = PI

a) W e have R'indik =

I

for i = 0, for i > 0.

k

0

b) We have for any P-module M and any i 2 0 R'indg M

N

R'ind; M .

c) We have for all A E X ( T ) +with (,?,av) = 0 for all a E I R'indgA = 0

for all

i > 0.

Proof: a) We apply 1.6.11 to the normal subgroup U, and we apply Kempf's vanishing theorem to the reductive group L , N P/U, and its Bore1 subgroup B n L , N B/U,. Then we get a) immediately from 2.1(6) for i = 0 and from 4.5 for i > 0. b) We get from the generalized tensor identity (1.4.8) and from a) R'indgM

N

M Q R'indgk

I

M

N

0

for i = 0, for i > 0.

Now the spectral sequence from I.4.5.c immediately implies b). c) This follows from b) and from Kempf's vanishing theorem. 4.1 Corollary: Let I t S and set P = PI. Let V, V' be G-modules and i E N. a) For any I E X ( T ) + with ( A , a") = 0 for all a E I there is an isomorphism Extd(V', V @ Ho(A)) N Extb(V', V @ A). b) There is an isomorphism Extb(V', V ) 'Y ExtL(V', V ) .

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Represeatatiom of Algebraic Groups

c) There is an isomorphism H'(G, V ) N H'(P, V ) .

Proof: a) The generalized tensor identity (1.4.8) and 4.6.b/c show R'ind;( V @ A)

N

V @ R'indgA

N

{;@H'(I.)

for i = 0, for i > 0.

We apply now the spectral sequence I.4.5.a to N = V' and M = V @ A and get a). b) This is the special case A = 0 of a). c) This is the special case V' = k of b). Remark: Let M, M' be P-modules and let A E X ( T ) with (A,a")2 0 for all a E I. Then the same arguments as above yield isomorphisms (1)

Extb(M', M 8 indiA) N ExtL(M', M 0 A)

and (2)

Extb(M', M ) N ExtB(M', M )

and (3)

H'(P, M) N H'(B, M).

4.8 We want to apply 4.7 to P = B and to get information about G-cohomology using information about B-cohomology. We have to know more about the latter in order to be able to do this. Recall from 1.3.11 (cf. 1.3.17) that for each R E X ( T ) the injective hull Y, of ;i = k , as a B-module is (1)

Y, = "1

8 A,

with T acting on k [ U ] via the conjugation action and with U acting on k [ U ] via pr or p , . The isomorphism of varieties

given by the multiplication is compatible with the conjugation action of T. So k [ U ] considered as a T-module can be identified with the symmetric algebra S(Lie(U)*). It is a polynomial ring in IR'I-generators y,(a E R'), each y, being a weight vector of weight a. This shows:

(2) All weights of k [ U ] belong to NR' = NS.

235

Kempf 's Vanishing Theorem

(3) Each weight space of k [ U ] has j n i t e dimension. (4) The zero weight space of k [ U ] is equal to k l and has dimension 1. We define the height ht(3J of any A E ZS by ht(A) =

zacS n, if A

=

EaeSn,a.

Combining ( 1 ) and (2)-(4) we get for any iE X ( T ) :

(5) The weight spaces of Y, are Jinite dimensional. Each weight p of Y, belongs to 3. + NR’ and satisfies ht(p) 2 ht(A). I f ht(p) = ht(A), then p = A. The A-weight space of Y, has dimension one.

4.9 Lemma:

There is an injective resolution

O+k+I,-rIl+12-+I,-t~** of k as a B-module such that I , is the injective hull of k, such that all weight spaces of all 4 are j n i t e dimensional, and such that all weights p of Ij belong to NR’ and satisfy ht(p) 2 j .

Proof: We construct the (I,), as the obvious minimal resolution. If we already have Io,...,Ij-l,then we set I, equal to the injective hull of socB(Ij-,/imIj-2). So 4 = @ r o J ( j , YLlj for some index J ( j ) set where N OIEJ(,)Alj. The embedding of soc,(lj-,/im into socB(Ij_,/im I, can be extended to an embedding of 4- l/im 4- and leads thus to an exact sequence I, - -+ I, - + I,. Any A,, is a weight of Ij- 1, so by induction A I j E NR’. So any weight p of I, belongs to NR’ by 4.8(5). For any ,u E NR’ there are only finitely many A E NR’ with p E A + NR’. As the iweight space of Ij- is finite dimensional there can be only finitely many 1 E J ( j ) with A = AIj. So again by 4.8(5)the p-weight space of Ij has finite dimension. Any A,, is a weight of Zj- so there has to be a direct summand Y, of I,such that Alj is a weight of 6 . By construction SOC,(I~_ 1) is contained in the image of 4 - 2 (resp. of k for j = l), so A # A,,, hence ht(A,,) > ht(A). Now induction yields ht(p) 2 j for any weight p of Ij.

,

4.10 Proposition:

Let P

3

B be a parabolic subgroup of G and let M be a P-

module. a) If dim M co,then dim H j ( P , M ) < co for all j E N. b) I f H j ( P , M ) # 0 for some j E N , then there is a weight - A E NR’ and ht( - A) 2 j .

-=

A of M with

Proof: Because of 4.7(3)we can assume P = B. Tensoring the resolution ( I . ) from 4.9 with M we get a resolution ( M @ I,) of M . So H'(B, M ) is the

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Representations of Algebraic Groups

cohomology of the complex ( M 0 ZJB. Therefore Hj(B, M ) is a subquotient of

Now dim(M €3 Zj)' c co if dim M c co by 4.9 and if ( M €9 Zj)' # 0, then there has to be a weight 1 of M such that - A is a weight of Zj. Again 4.9 yields the claim. 4.11 Corollary:

W e have for all i E N:

H'(G, k ) N H'(P, k ) N H'(B, k ) N

k 0

for i = 0, for i > 0.

Proof: The result for i = 0 is obvious. For i > 0 we can apply 4.10 observing ht(0) = 0. Let M be a finite dimensional P-module. Then H'(P, M ) r lim Hi(Pr,M ) for all i E N.

4.12 Corollary: c

Proof: Combine 4.10.a and 4.1 1 with 1.9.9. Remark: In the special case P = B this result was proved in [Cline, Parshall, and Scott 61, in the special case P = G and p > h in [Friedlander and Parshall 31. This general result is due to W. van der Kallen, cf. 1.9.9. The result for P = B also yields immediately that Hi(&M ) = l&H'(BrT, M ) . One can show more precisely, cf. [Cline, Parshall, and Scott 61, that there is some ro(M,i ) such that Hi@,M ) N H'(BrT,M ) for all r > ro(M,i). The Hi@,, M ) , however, will not stabilize for large r (in general). 4.13 Recall the notation V(A)= Ho(- woA)* from 2.13.

Proposition:

Let A, p E X(T ) , . Then

H'(G, Ho(A)@ H o ( p ) ) =

k 0

if i = O and A = -wop, otherwise,

Ex&( V(A),H o ( p ) ) =

k 0

i f i = O and otherwise.

and 1=p,

237

Kempf ’s Vanishing Theorem

Proofi We have by 4.7 H'(G, Ho(A)@ Ho(p))N H i ( & Ho(A)@ p ) N H'(B, Ho(p)@ A).

If this cohomology group is non-zero, then there have to be weights v of Ho(A) and v' of Ho(p)with p + v , A + v' E -NR+, and ht(p + v), ht(A + v ' ) I-i. We have seen in 2.2 that w,A resp. wop is the smallest weight of Ho(A) resp. Ho(p). So p + w,A, A + w,p E -NR+. Now wz = 1 implies A w,p = w,(p won),hence

+

+

A + w,p E (-NR’)

This proves A =

- w,p.

n w,( -NR+) = (-NIX+) n (NR’) = 0.

Furthermore,

0 = ht(A

+ w,p) Iht(A + v’) I- i

yields i = 0. On the other hand, HO(A)"

N

wol,

hence (HO(A)@ (-won))" = k = (Ho(A)@ ( -

This implies Ho(G,Ho(l)@ Ho(- won))= k and proves the first part of the proposition. The second one follows from the first one using 1.4.4. Remarks: (1)

If

1) We have H o ( 0 ) = V(0)= k. Therefore we get as a special case

p # 0, then H'(G, Ho(p))= 0.

2) We have for all A E X(T)+and all i > 0 (2)

Extb(L(A),Ho(A)) = 0.

Indeed, the corollary 4.7.a implies Extb(L(A),Ho(A))N Extb(L(A), A)

N

H'(B, L(A)* @ A).

All weights p of L(I)* 03 1 satisfy p 2 0. Therefore the claim follows from 4.10. 4.14 Proposition:

L e t A, p E X(T ) , with A

+ p. Then

Extb(L(A),L ( p ) )N Extb-'(L(rZ), Ho(p)/socGHo(p)) for all i E N,i > 0.

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Representations of Algebraic Groups

Proof: We apply the functor Hom,(L(i), ?) to the short exact sequence 0 + L ( p ) + H o ( p ) + Ho(p)/socGHo(p)+ 0.

The long exact sequence for the Ext-groups will yield the claim if we can show 0 = Extb(L(l),Ho(p))

for all i > 0.

(Note that always HOmG(L(I),L ( p ) ) N HOmG(L(A),H o ( p ) ) as SOC,Ho(p) = L(p).)This vanishing is equivalent to 0 = H'(G, L(1)* 0 H o ( p ) ) N Hi(&L(1)* 0 p)

(using 4.7 and 1.4.4). The smallest weight of L(1)* 0 p is p - 1. By our assumption p - 1 -# 0. This implies p - 1 $ - N R + in case p # 1, hence v q! - N R + for each weight v of L(I)* 0 p. This implies by 4.10 the vanishing of Hi(&L(A)* 0 p) in this case. If p = 1,then 0 is the only weight of L(A)* 0 p in -NR+ and ht(0) = 0. So again 4.10 yields the desired vanishing. Remarks: 1) From the proposition we get by dualizing (for 1,p as above (with 1 p) and all i > 0)

+

ExtL(L(p),L(1)) N Ex&- '(rad V ( p ) ,L(il)). 2) Note that these results generalize 2.14 (the proposition and the remark). 4.15 Lemma: Let V be a G-module and let ilE X ( T ) +with Hom,(L(A), V ) # 0. If Horn,(L(p), V ) = 0 = Exti(V(p), V ) = 0 for all p E X ( T ) + with p < 2, then V contains a submodule isomorphic to Ho(il).

Proof: The assumption implies Ext;(L(p), V ) = 0 for all p E X ( T ) +with p < 1 and hence Ext;(Ho(l)/L(il), V ) = 0. Therefore the given homomorphism L(1) + I/’ can be extended to Ifo(),). As SOC,H~(I)= L(1) this extended map has to be injective. 4.16 An ascending chain

0 = V, c

v, c V, c ...

of submodules of a G-module V is called a good filtration if V = if each is isomorphic to some Ho(Ai) with 1’ E X ( T ) + .

v/&-,

u < and iL

Proposition: Let V be a G-module. a) If V admits a good filtration, then for each 1 E X ( T ) +the number of factors isomorphic to H o ( n ) is equal to dim Horn,( V(n),V ) .

239

Kempf’s Vanishing Theorem

b) Suppose dim Hom,(V(A), V ) c cc for all I properties are equivalent:

E

X ( T ) + . The following three

(i) V admits a good filtration. (ii) Ex&( V(1), V ) = 0 for all A E X ( T ) + and i > 0. (iii) Extk(V(A), V ) = 0 for all 1E X ( T ) + .

Proof: a) This follows immediately from 4.13. b) The implication (i) * (ii) is another consequence of 4.13, and the implication (ii) s-(iii) is trivial. Let us suppose that (iii) is satisfied and prove (i). Choose a total ordering L O , l 1 , 1 ,... , of X ( T ) + such that Li < L j implies i c j . We may assume V # 0, hence that there is some 3, E X ( T ) + with Hom,(V(A), V ) # 0. Let i be minimal with Hom,(V(i,), V ) # 0. Each weight p E X ( T ) + with p c Ai has the form l j with j c i. This implies HomG(V(p),V ) = 0, hence also Hom,(L(p), V ) = 0. Therefore 4.15 implies that V contains a submodule V, isomorphic to HO(1,). As V, satisfies (ii), we also see that V/V, satisfies (iii). Furthermore, HomG(V(pL),V/Vl) ‘v HomG(V(pu),V , for all p # li whereas dim Horn,( V(Ai),V/V,) = dim Horn,( V(di),V ) - 1. Itc V of suberating we get a chain V, = 0 c V, c V, c * * * c V’ = modules with dim Horn,( V ( l ) ,V ’ )= dim HornG(V(n),V )for all 1 E X ( T ) + ,such that each F/F- is isomorphic to some H o ( p i ) .Then Horn,( V(1), V / V ’ )= 0 for all 1 E X ( T ) + ,hence V/V’ = 0 and V = V’ has a good filtration.

uiz0

4.17 Corollary: Let 0 + V + V-, V” + 0 be an exact sequence of G-modules with dim Hom,(V(l), V’’) c cc for all l E X ( T ) + . If V’ and V have a good filtration, so has V”.

This is obvious. Remark: One can generalize 4.16/17 and introduce the notion of a “good filtration dimension”, cf. [Friedlander and Parshall 2],3.3. 4.18 Proposition: Let 1 E X ( T ) + . The injective hull of L ( l ) has a good filtration. Any H o ( p ) with p E X ( T ) + occurs as a factor in such a filtration exactly [ V ( p ) :L ( l ) ]times.

Proof: Obviously the injective hull Q satisfies (ii) in 4.16.b. Furthermore, dim Hom,(V(p), Q) = [ V(p):L(A)]by standard properties of injective hulls.

240

Representatiow of Algebraic Groups

Remark: We can embed any G-module into an injective G-module using an embedding of its socle. Therefore the proposition implies: Any finite dimensional G-module can be embedded into a finite dimensional G-module admitting a good filtration. 4.19 Proposition: Let V, V' be G-modules admitting a good filtration. Then so does V 6 V , except possibly in the case where char(k) = 2 and where R has a

component of type E , or E , .

We shall not give a proof here and refer instead to the original proofs in [Wang 11 for char&) large, and in [Donkin 91 for the general case. The exception made in the proposition is expected to be unnecessary. 4.20 Proposition: Consider k[G] as a (G x G)-module via pI x p,. Then k[G] admits a good filtration. The factors are the Ho(A)6 Ho( - won) with 1 E X(T)+each occurring with multiplicity 1.

Proof: Observe that T x T is a maximal torus in G x G and that B x B is a Bore1 subgroup. We can identify X(T x T ) with X(T)6 X(T)and get Ho(rl,p) = H0@)6 Ho(p)with ( g l , g z ) E G x G acting as g 1 6 g 2 . As 1 x G is normal in G x G and as we can identify G x 1 with (G x G)/ (1 x G),we get from 1.6.6(3) a spectral sequence H'(G x l , H j ( l x G , H o ( A , p ) 6 k[G]))*H'+j(G

x G,Ho(A,p)6k[G]).

As k [ G ] is injective for G, this sequence degenerates and yields isomorphisms. H'(G x l,Ho(A) 6 (Ho(p)6 k[G])')

N

H'(G x G , H O ( l , p )6 k[G]).

We can identify ( H o ( p )6 k[G])' N indgHo(p) N Ho(p). So the left hand side above is Hi(G,Ho(A)6 Ho(p)),which we have computed in 4.13. Now the claim follows from 4.13 and 4.16.b. Remarks: 1) If we embed G diagonally into G x G,then the representation above restricts to the conjugation representation of G on k[G]. Now 4.19 implies (where we can apply it) that k[G] has a good filtration as a G-module under the conjugation action. 2) Proposition 4.20 is due to Donkin (unpublished) and [Koppinen 31. A special case appeared before in [De Concini 21. 4.21 Set g = Lie(G) and let Z = {f E R[G] If(1) = O}. Then g*

1/12 as a G-module if we take the adjoint action on k[G]. More generally Sjg* N Z j / Z j + l for all j , as G is smooth. N

241

Kempf's Vanishing Theorem

Suppose there is a G-submodule M of k [ G ] with I = 1 ' 0 M . Then I' = I J + ' + M J and M j is a homomorphic image of S J M N Sjg* (for all j E N). Therefore I' = I J + l 0 M' 3: I ) + ' 0 SJg* and k[G] N ( @ { = o S i g * ) 0 Ij". As k[G] has a good filtration (remark 1 to 4.20), so has each S'g* (modulo some problem for p = 2 and type E , , Ea). One can show that one has such a submodule M most of the time. (One embeds V @ V* N End(V) for some G-module V into k[G] using matrix coefficients and one embeds g into End(V) via its action on V,) There is a different type of argument for G = SL,. In both cases let me refer you to the proofs in [Andersen and Jantzen], 4.3/4. The final result there is: If G is semi-simple and simply connected and if p is good, then Sg* has a good jiltration.

(1)

Here a prime p is called good if p > nap for all

ct

E

R + , fl E S where

a = &Esnbpfl. (So we exclude p = 2 resp. 3, 5 if R has a component not of

type A resp. of exceptional type resp. of type Ea.) If p is not good, then Sg* has no good filtration. This has been shown by S. Donkin and C. Musili.

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5

The Borel-Bott-Weil Theorem and Weyl's Character Formula

Throughout this chapter let k be a field. In characteristic 0 the Borel-Bott-Weil theorem gives complete information about all H i @ ) : If I E X(T)+and w E W, then H i ( w A) = 0 for i # I(w) and H1(")(w.A) N Ho(A). Furthermore, ch Ho(A) is given by Weyl's character for, H ' ( p ) = 0 for all i. mula. If p E X(T)with p # W . X ( T ) + then This result does not generalize to characteristic p , not even in the weak version where we replace the isomorphism H'(")(w.A) N Ho(A)from above by an identity of characters ch H'(")(w A) = ch Ho(I). This weak version holds if and only if the root system has only irreducible components of rank 1 (assuming char&) # 0). We start this chapter with an explicit description of the Hi@)in the case where G has semi-simple rank 1 (for short: rk,,G = 1).More precisely, we take for arbitrary G a simple root a E S and describe in 5.1-5.3 all R'ind;(*)(p) where P(a) = P{.,.We have G = P(a) for rk,,G = 1, so in this case we get the H i ( p ) . On the other hand, the formula 1.4.9 for R'ind in the case of a semidirect product reduces the computation of R'indi(")M in general to the case where rk,,G = 1 and then to the well known computation of cohomology of line bundles on the projective line P'. (We use the computations for SL, in 2.16 instead.) In this case (rk,,G = 1) we see that the weak version of the BorelBott-Weil theorem still holds, but a computation of the socle of H ' ( p ) in 5.12 shows that we have, in general, no isomorphism Ho(A) N H'(s, A) for I E X(T ) + (and rk,,G = 1).

-

-

243

244

Representations of Algebraic Groups

We then use the complete information about the R'ind;(")(p) to deduce results about the H i @ ) . For example, we prove the Borel-Bott-Weil theorem (5.5)in characteristic 0,while in characteristic p we exhibit some 1. E X ( T ) +for which the Hi(w A) behave as in characteristic 0. This approach to the BorelBott-Weil theorem is due to [Demazure 51 whereas the part about characteristic p was first proved in [Andersen 11. Another application (of the description of all R'ind;(")(p)) is a proof of Weyl's character formula for the ch HO(A)with A E X ( T ) + or, more generally, for the Ci,, (- 1)'ch Hi(A)for all A E X ( T ) .The approach used here is due to [Donkin 91. These results about R'ind;'") are applied to give a determination of all p with H 1 ( p ) # 0, of the socles of these H ' ( p ) and of their largest weights (5.13-5.18). All this is due to [Andersen 21. As a corollary we get a complete description of H'(B, p) as in [Andersen lo].

-

5.1 We have already used ( p , f i v ) = 1 for all fi E S in 3.18. This implies ssp - p E ZR for all p E S, hence wp - p E ZR c X ( T ) for all w E W = (ss I /?E S ) . This shows: If we define w

(1)

*

A = w(A + p ) - p

-

for all w E W and A E X ( T )6R, then A E X ( T ) implies w A E X ( T ) for all wE

w.

We shall use this "dot convention" more systematically later on. For the moment fix some u E S.Then (2)

s,

.A = s,l - a = A - ((A,

a")

+ 1)a

for all A.

Set P(u) = P(a,. 5.2 Proposition: Let u E S and A E X ( T ) . a) The unipotent radical of P(u) acts trivially on each R'ind;(")A. b) If (A, u v ) = - 1, then R'ind;(")A = 0. c) If (A, u " ) = r 2 0, then R'indE(")A = 0 for all i # 0 and ind;(")A has a basis u o , v l , .. .,v, such that for all i (0 I i I r) and A : (1)

toi = (A - iu)(t)vi

for all

t E T(A),

for all a E A , (3)

245

Borel-Bott-Weil and Weyl

If (A, av) I-2, then R'indi(")A = 0 for all i # 1 and R1ind;@)Ahas a basis ub, u ; , . , . ,v: where r = - (A, a') - 2 such that for all i ( 0 I i I I ) and A : d)

.A - ia)(t)u:

(1')

tu: = (s,

(3')

x-,(a)u: =

c)aj-iuj

for all t E T ( A ) ,

for all

aE

A.

j=i

Proofi As the unipotent radical of P(a) is contained in U and as U operates trivially on kA = A, the statement in a) is an immediate consequence of 1.6.11. Furthermore, we may form the quotients by this unipotent radical. So we may assume that G = P(a) has semi-simple rank 1. Then G is a factor group of some group of the form SL, x T; with T i a torus. This can be done in such a way that T' is the image of {diagonal matrices in SL,} x T ; and such that x, and x-, come from the "standard" root homomorphisms in SL,, Using 1.6.1 1 again we may assume that G = SL, x Ti. Now 1.4.9(1)implies that R'ind;A is isomorphic to R*indEL,2sL,(A as an SL,-module. On the other hand, T i acts on each R'indiA through the restriction of A to T; by 2.10(1). This is compatible with our results as a vanishes on T; and as s, I - I E Za. We may therefore assume G = SL,. Now ind;A is described in 2.16. The character denoted by o there maps (for

IBnSL2)

a 0

SL,)any (o

a-l)

to a, hence ( w , a V ) = 1 and X ( T ) = Zo.So any A E X ( T )

is equal to (A,crv)w. For (A,av) = r 2 0 we get as indiA the rth symmetric power of the dual of the natural representation. Taking the basis consisting of all monomials (in two variables) and changing some signs, we get the action as described in (1)-(3). If (A,av) < 0, then Ho(A) = indgA = 0 by 2.6. Because of 4.2(3) we have R'ind; = 0 for i # 0,l. So we have to look only at R'ind; and can now use Serre duality (4.2(8))

H'(A)

N

HO(-(A

+ 2p))*.

If ( -(A + 2p), a V )< 0, i.e., if (A, a") > -2 this is zero, whereas for (A, a') I -2 we get (1')-(3') using the dual basis up to sign changes.

5.3 Corollary: Let a E S and A E X(T)with (A, u ' ) 1 0. a) I f char(k) = 0, then ind;(")A N R'ind;(,)(s, A). b) If char@) = p # 0 and i f there are S,M E N with 0 < s < p and (A,av) = spm - 1, then ind;@)A N Rlindg(")(s, A).

246

Representations of Algebraic Groups

Proof.- Let us use the notations from 5.2. The map

mapping each 0: to

0

-

R ind;(")(s, A)

--*

indi(")A,

ui is a homomorphism of P(a)-modules. (This follows

from 5.2 using elementary calculations.) It is an isomorphism if each

(9

(9

with

0 Ii Ir = ( A , a v ) is non-zero in k. This is automatically satisfied if char k =O. If char(k) = p # 0, then we have to know for which r no

with 0 Ii 2 r

is divisible by p. This is done using the standard formula for binomial coefficients modulo p (cf. [Haboush 31, 5.1), and it immediately implies the claim.

-

Remark: Suppose that G = P(a). Then the map R 'indg(s, A) --* indg(A)has image equal to L(1), as 2.16(7) shows. 5.4 Proposition: Let a E S and A E X ( T ) . a) If ( & a v ) = - 1, then H'(1) = 0. b) If (A, a') 2 0, then ( f o r all i )

H i @ ) = R'ind&,,(indi(")A) N H'(ind;(")A). c) If (A,av) I-2, then ( f o r all i )

H'(A) N R i - 'ind~(,)(R'indi(")A)N H i - l(R1indi(a)A). d) Suppose (A,&) 2 0. If char(k) = 0 or i f char&) = p # 0 and ( & a v ) = sp" - I for suitable S,M E N with 0 < s < p , then ( f o r all i)

H'(A) 1: H i

+

1(s, '

A).

Proof;. We use the spectral sequence R'ind~(,,(Rjind~(")A) * R'+jindiA = H"j(1) from I.4.5.c. By 5.2 at most one Rjindg(")Ais non-zero. So the spectral sequence degenerates and yields isomorphisms R 'ind&,(Rjindi(")l) N R 'jindgi, where j is the unique integer with Rjindi(")A # 0 if (A, a') # - 1, whereas for (A, a') = - 1 we get H'(1) = 0. This proves a) and the first isomorphisms in b) and c). The second isomorphisms follow from 4.6.b. Finally d) is an immediate consequence of b), c) and 5.3.

247

Borel-Bott-Weil and Weyl

5.5 Set

if char k = 0, and

Cz = {A E X ( T )I O I (A + p, pV) I p

for all

p E R+},

if char(k) = p > 0. Recall the definition of l(w) and its elementary properties from 1.5. Coroflury

a) If A E Cz with 1 4 X ( T ) + ,then

H'(w

A) = 0

for all

w E W.

b) If A E czn X ( T ) + ,then we have for all w E W and i E N

Proof: We want to use induction on l(w).If I(w) = 0, then w = 1. In this case b) is a consequence of Kempf's vanishing theorem 4.5. If we are in the situation of a), then there has to be some a E S with ( A , a v ) < 0. As 0 I (A + p , a v ) = ( A , a v ) + 1, we get ( A , a v ) = - 1 and can apply 5.4.a. Suppose now I(w) > 0. Then there is some a E S with w-'(a) E - R + , hence with l(s,w) = l(w) - 1. Set fl = - w-l(a) E R'. We have (saw * A , a v )= (s,w(A =

-(w(A

+ p) - p,av) + p ) , a v ) - 1 = (A + p , p ' )

- 1 2-1

and (saw A,av) < p . Now 5.4.d (and 5.4.a) imply

H'(w ' A) N H'-'(s,w

(Observe that s, (w

' A).

- A) = (saw) A.) We can now use induction. .

Remarks: 1) If char(k) = 0, then each element of X ( T )has the form w A for some A E Cz and some w E W ,So in this case the corollary yields each Hi(p).It is known in this case as the Borel-Bott- Weil theorem. 2) We can avoid using Kempf's vanishing theorem by working with descending induction for all H ' ( w . A) with i > l(w) using H ' ( p ) = 0 for i > IR+I.

248

Representntiom of Algebraic Croup

5.6 Corollary: If 1 E X ( T ) +n c,, then L(1) = HO(1). Proof;. We have w,(p) = - p , hence -w,(C,) (1)

=

c,. Furthermore, we have

w, (-won) = w,(-ww,l + p ) - p = -1 - 2p

for any 1E X ( T ) ,hence by Serre duality (4.2(8)) (2)

H"(w, (-won))

N

HO(A)*,

where n = IR+I = l(w,). So 5.5.b implies for each 1E (3)

c, nX ( T ) + :

HO(i)* N HO(- won).

We get from L( - won) = socGHo(- won) by dualizing (cf. 2.6) (4)

soCGHo(d)= L(1)N Ho(n)/radGHo(n).

As L(1) occurs with multiplicity 1 as a composition factor in Ho(l), this = Ho(l),hence Ho(l)= L(1). implies radGHO(l)= 0 and SOC~HO(A) Remark: If char(k) = 0, then L(1) = HO(1) for all 1E X ( T ) + , hence Extk(L(i),L(p))= 0 for all 1,p E X ( T ) + by 2.12 and 2.14. This implies Exth(M,MI) = 0 for all finite dimensional G-modules M, M', hence that any G-module is semi-simple (using the local finiteness). 5.7 In characteristic 0 we have X(T)+c so in this case we can apply 5.5.b to all 1 E X ( T ) + .The corresponding statement is false in characteristic p as we shall see below (5.18). There is, however, an Euler characteristic analogue that

c,

always holds. Let us define for any finite dimensional B-module M the Euler characteristic x(M) =

C (-

1)' ch H'(M).

120

This is a well defined element in Z [ X (T)] as each H'(M) is finite dimensional (4,2(1)), and as H'(M) = 0 for i > IR+I by 4.2(3). The long exact sequence for

the derived functors of indi yields (2)

X(M) = X W ' )

+XW")

for any exact sequence 0 + M' + M + M" + 0 of finite dimensional Bmodules. The generalized tensor identity (1.4.8) implies (3)

X ( V 0 M) = Ch(V)X(M)

for each G-module V and any B-module M with dim V < 00 and dim M < 00.

249

Borel-Bott-Weil and Weyl

We use especially the notation x(A) = ~(k,).Note that Kempf's vanishing theorem implies

x(A) = ch Ho(A)

(4)

for all A E X(T)+.

a) The ch L(A)with A E X(T)+form a basis of Z[X(T)IW. b) One has for all A E X ( T )and all c e a ( p )E Z[X(T)Iw: 5.8 Lemma:

c

x(4 W e ( d= c

c Q(P)x(A + P I c

Proof;. a) Set

the sym(p) As each v E X(T)is conjugate under W to exactly one p E X(T)+, with p E X(T)+form a basis of Z[X(T)]"’. If A E X(T)+, then ch L(A)E Z[X(T)IWand dim L(A), = 1 by 1.19(2) and 2.4, hence

where we sum over p E X ( T ) + with p # A. Again by 2.4 only p < A have to be considered. Therefore the transition matrix from the (sym(iJ)Ato the (ch L(A)),is unipotent triangular, and also the ch L(1) form a basis. b) Because of a) we may assume xea(p)e(p) = ch Y for some finite dimensional G-module V. Now the desired formula follows from 5.7(3), (2) as the B-module V @ A has a composition series with dim(VN)factors k,+, for each p, Remark: The same proof as for a) shows that also the ch Ho(A) = x(A) with A E X(T)+form a basis of Z[X(T)]’". 5.9 For any o! E S and A E X(T)with (A

+ p, ctv)

2 0, Proposition 5.2 implies

ch ind;(")A = ch R'ind;(")s,

- A,

Therefore 5.5 implies x(A) = - ~ ( s a

which holds for all A E X(T),even if (A (s, a E S ) we get: (1)

I

x(w.A) = det(w)x(A)

A),

+ p,ctv) = - 1 by

5.5.a. As W =

for all w E W and 3, E X(T).

250

Representations of Algebraic Groups

We have by 2.1(6) and by 5.7(4) ~ ( 0= ) e(0) = 1.

(2)

Therefore 5.8.b implies for all C,a(p)e(p) E Z[X(T)IW (3) Set for any A

E

X(T)@ Q

1det(w)e(wl) E Z[X(T)8 Q],

A(A) =

(4)

W € W

Then obviously wA(1) = det(w)A(L) for all w E W, hence A(A)A(p)E Z[X(T)@ Q ] for all 1,p E X(T )8 Q. 5.10 Proposition (Weyl’s Character Formula): We have for all A E X(T):

x(4 =

+ p)/A(p).

Proofi To start with, A(A + p ) / A ( p ) is just an element in the quotient field of the integral domain Z[X(T)+ Zp],which is a localized polynois a basis of the free abelian group X(T)+ Zp,then mial ring over Z.(If (wJi Z[X(T) Zp] = z[e(oi),e(oi)-’ I i ] . ) We get for any A E X(T) using 5.9(3),(1)

+

A(1

c

+ + w‘p) 1 det(w’)e(w(1+ p + w’p)) c W det(w’)X(w(L+ p + w’p))

+ p)A(p)=

det(ww’)e(w(L p )

W.W’EW

=

W.W’€W

=

W,W’€

=

+ w’p + w-’p)) det(ww’)X(A + w’p + wp). det(w’)X(w (A

W.W’EW

= W,W’EW

For A = 0 we get A ( P )= ~

c

det(ww’)X(wp

+ w’p)

E

Z[X(T)IW,

W,W’EW

hence by 5.8.b A@

+ P)A(P) = A ( p ) 2 x ( 4 .

As p has trivial stabilizer in W we get A ( p ) # 0, hence we can cancel A ( p )in the

25 1

Borel-Bott-Weil and Weyl

integral domain Z [ X ( T ) + Zp] and get

hence the proposition. Let A E X(T)+. Then

5.11 Corollary:

ch Ho(A) = ch V(A)= x(A) = A(A + p ) / A ( p ) . Prooji The equality ch Ho(A) = x(A) is an immediate consequence of Kempf's vanishing theorem 4.5. As -w,A E X(T)+ also ch V(A)= ch Ho(- w,A)* is independent of the field k. So we can assume k = Q, where

V(A)N L( - w,A)*

N

L(A) N HO(A),

especially ch V(A)= ch Ho(A). Remark: The modules V(A)are nowadays called Weyl modules. This name was introduced in [Carter and Lusztig 11 where the case G = GL,, is considered and where the V(A)are constructed in the "same" way as in Weyl's book on classical groups. Observe that by Serre duality

V(A)N HO(- w,A)*

N

H"(w,A - 2 p ) = H"(wo * I).

5.12 Let us assume in this section that G has semi-simple rank 1. Denote by a the only simple root. Take I E X ( T )with (A, a v ) 2 0 and consider the basis (0:) of H ' ( s , * A) as in 5.2.d. Recall from 5.3:

( 1 ) If char(k) = 0 or if char k = p and ( A , a V ) = up" - 1 f o r some a, n E N, 0 < a < p , then H ' ( s , . A) N L(A).

We want to determine soc,H'(s,. A) in the remaining cases using explicit computations. There is also a more conceptual proof, cf. 11.13. Lemma:

Suppose char(k) = p # 0. Write n

r = (&av)=

1ajpj, j=O

with 0 Iaj < p for all j and a,, # 0. Suppose that there is some j < n with aj < p - 1 andset m = min{jlaj c p - I } .

-

a) Then H ' ( s , A)"' = ZkT-!mkv',,,,, + kvb, where i(m') = x y l o a j p j b) soc,H'(s, A) 2: L(A ajp' + 1)a) = L(s,. A + a,,p"a).

(xj":;

+ 1.

252

Representations of Algebraic Groups

Proofi a) Obviously H1(s,. A)" is spanned by its weight vectors, hence by the u: contained in it. Because of 5.2.d(2') these are exactly all i such that p

divides all

(:Ii)

with 0 I j c i. Obviously ub will occur.

Suppose now some i > 0 satisfies this condition. Consider the p-adic bjp' with 0 I bj c p for all j . As r - i < r we development r - i = have bj c aj for the largest j with bj # aj. Therefore there are j with bj c p - 1. Let m' be minimal with b,,,, c p - 1. Using the standard formula for binomial coefficients mod p (cf. [Haboush 31, 5.1), we see that p does not r-i+p" . hence that pm' > i and divide r--1

(

),

r-i

+ pm' = C

m'-1

( p - 1)pj + (b,,,, + 1)p""

n

+ 11 bjp'

> r > - i.

j=m'+

j=O

Comparing the p-adic developments of r - i + p"", r, and r - i we get b, = aj for j > rn' and bm, = a,,,, - 1 and aj < p - 1 for at least one j c m', hence m < m'. Furthermore, 1

c ( p - 1)pj + (am,

m'-

r -i =

-

I)+

j=O

+

j=m'+l

ajpj,

hence

-

So at most the u$,,,) with m I m' < n occur in H'(s, A)’". On the other hand, for all these m' any r - i(m') + j with r - i(m') < r - i(m') + j < r has a p-adic development of the form CI=oc,p', where c, c p - 1 for some 1 I m', or c, c a, for some 1 < m' + 1, or c,,,.+~< a,,,,,, - 1 as otherwise r - i(m') + j r - i(m') + j > r. Therefore p has to divide all standard formula for binomial coefficients. This yields the converse inclusion. b) The socle of H1(s, A) is spanned by those weight vectors in H1(s,. A)" (i.e., by those u: as in a)) that generate a simple submodule. Now 5.2.d(3') shows

-

1 kui, r

Dist(G)ub =

j=o

so ub cannot occur. If m I m' c m" c n, then p does not divide

(g,8;) by

253

Borel-Bott-Weil and Weyl

the standard formula and by the definition of i(m'),i(m"). Therefore u:(,..)

E

Dist(G)u~,,r,

by 5.2.d(3'), hence Dist(G)vi,,,, is not simple. So only ui("- remains. Its weight is given by 5.2.d(1') and is the one mentioned. 5.13 We want to apply 5.12 to get for arbitrary G all A with H ' ( 1 ) # 0 and then the structure of SOC,H'(A). If char(k) = 0, then we can apply the Borel-Bott-Weil theorem (5.5) and get complete information. (So H '(A) # 0 if and only if there is a E S with s, A E X ( T ) + and then H'(A) N L(s,. A).) Let us therefore suppose from now on that char(k) = p # 0. If ( p + p , a v ) 2 0 for all a E S, then Hi@)= 0 for all i > 0 by Kempf's vanishing theorem and by 5.4.a. So, if H ' ( p ) # 0, then ( p p,av) < 0 for some a E S and then H ' ( p ) = ind&,(R indi(")p)by 5.4.c. In other words, we get all non-zero H'(p)'s by looking at all a E S and A E X ( T ) with (A p, a') > 0, and by taking

-

+

'

+

-

H'(s, A)

N

ind&,(R'indg(")(s, A)).

-

The unipotent radical of P(a) acts trivially on Rlindi(")s, A and the Levi factor L,,) acts as on "the H ' ( s , A) for L,,,", cf. the proof of 4.6.a. So 5.12 describes the socle of R indi(%, A as an L{,,-module and the U,-invariant elements of this module. From this we get information about H'(s, A) using a general result to be proved first.

'

-

.

5.14 Lemma: Let I t S and set P = PI. Let M be a P-module. a) Suppose the socle of M as an L,-module is simple with highest weight A E X ( T ) .If indFM # 0, then soc,indgM 1: L(A) and A E X ( T ) + . b) Let A be a maximal weight of indFM. Then A E X ( T ) + and (Mu+nLr)A # 0. c) Let A E X ( T ) +with (Mu' "Lr)A # 0. Suppose that U,acts trivially on M. Then A is a weight of indgM.

Proofi a) The evaluation map e M : indgM M is a homomorphism of P-modules. According to the remark to 2.2 it is injective on (indgM)U(P) where U ( F )= U:. If L ( p )is a simple submodule of indgM, then L(p)'@) is the simple L,-module with highest weight p (by 2.1 l), hence so is ~ ~ ( L ( p ) ~ ( p ) ) . Our assumption yields A = p E X ( T ) + and soc,(indgM) N L(A). b) If A is a maximal weight of indgM, then --f

0 # (indgM), c (indgM)" c (indgM)U(P)

254

Representatiom of Algebraic Groups

by 1.19(7), hence 0 # &,(indgM), c MAn M("+nLr).

c) Let u E MA,u # 0 be (U’ n L,)-invariant. Using the left exactness of indg we may assume M = Dist(P)u = Dist(U)Dist(T)Dist(U' n L,)v = Dist(U)v = Dist(U n L,)u.

Therefore M' = @,,
+ M + A + 0,

hence an exact sequence of G-modules *

*

a

+ Ho(M)+ Ho(A)+

H'(M')+*.-.

We know by 2.2 that Ho(A), # 0. If we can show H 1 ( M ' ) A= 0, then necessarily H 0 ( M ) , N (indgM), # 0 (cf. 4.6.b) as claimed. Of course, it is enough to show H ' ( P ) ~= 0 for each weight p of M'. Suppose H 1 ( p ) A# 0 for such a p. By 5.13 there is a E S with ( p p, a') < 0 and H ' ( p ) 1: ind&,(R ind;(")p). Furthermore, s, p is the largest weight of R'ind;(")p by 5.2. As H ' ( P ) ~# 0 there is a maximal weight A' of H1(p)with A' 2 A. Now b) implies that A' is a weight of R'ind;(")p, hence that A I A’ Is, p. On the other hand, p is a weight of M' c Dist(L,)u. This implies p < A and A - ~ E Z I .Now p < A ~ s , . p = p - ( ( ( p , a " ) + l ) a = s , p - a implies A - p E Za, hence a E I. As M is an L,-module, sap is also a weight of M, hence sap I A and s, p < A contradicting 3, I s, p. So H ' ( p ) , = 0 as we had to show.

+

'

-

.

5.15 Proposition: (A, a') 2 0.

.

Suppose char(k) = p # 0. Let a E S and A E X(T) with

a) Suppose (A, a") = up" - 1 for some a, n E N with 0 < a < p. Then

H'(s, A) # 0 0 1 E X(T)+, and i f so, then L(A) N soc,H'(s,

*

A).

b) Let (A,a’) = x3,0ajp' with 0 I aj < p for all j and a, # 0. Suppose

255

Borel-Bott-Weil and Weyl

there is some j < n with aj < p

-

1. Let m be minimal with a, < p - 1. Then

H ' ( s , . A) # 0 o s, A

+ anpnaE X ( T ) + ,

and if so, then SOCGH'(S,* L) N L(s, * A

+ a,p"a).

I f A E X ( T ) + ,then A is the largest weight of H'(s, A). I f not let m' 2 m be minimal for p = s, * A + ajp'a E X ( T ) + . Then p is the largest weight of H '(s, A),

cy=,, .

Proofi a) In this case H'(s, A) N Ho(A)by 5.4. Therefore the claim follows from 2.6 and 2.4(1). b) We apply 5.14 to M = R'ind;(,)(s,. A) and P = P(a). If H ' ( s , . A) 'Y indg(,,M # 0 (cf. 5.13), then 5.14.a and 5.12 imply s, A + a,pna E X ( T ) + and SOC~H'(S, A) N L(s, .A + a,pna). O n the other hand, 5.12 and 5.14.c show that any p E X ( T ) +n {A, A - i(m')a I m I m' < n } is a weight of ind&,M. This yields especially: If s, A + anpncl= A - i(n - l ) a E X ( T ) + , then 0 # ind&,M N H'(s, .A). If p is a maximal weight of H ' ( s , A), then p is by 5.14.b also a weight of R'ind;(')(s, A)"=, hence

-

-

8

p ~ x ( T ) n+{ A , A - i ( m ' ) a ~ m I m ' < n } .

As A > A - i(m)a > A - i(m + l ) a > * * > A - i(n - l)a, this implies the formula for the maximal weight of H'(s, A).

-

.

Remark: If H'(s, 3,) # 0, then (s, the situation of b) we get 0 I (s,

*

A, p') 2 0 for all p E S, /3 # a. Indeed, in

- A + a,pna, P') 5 (s,

A, P'),

as ( a , p') I 0. The argument is similar in a). 5.16 Corollary: Let p

E X ( T ) .I f

H ' ( p ) # 0, then socGH'(p) is simple.

Remark: This does not generalize to all Hi.For example, there is in [Jantzen 51 an example of a Weyl module V(A)1: H o ( -woA)* 1: H"(wo A) where n = JR+I (using 4.2(9)) with a socle which is not simple. In [Andersen 41, p. 58 there is even an example where some H'(,u) is decomposable.

-

5.17 The description of all H 1 ( p )# 0 gets nicer if we look at it from the

opposite point of view. Because of 5.16 it is enough to describe for any A E X ( T ) +all p E X ( T ) with L(A) = socGH'(p).

256

Representations of Algebraic Groups

Proposition: Suppose char(k) = p # 0. Let 1 E X ( T ) + .Define for each a E S and n E N an integer c,(n, a) through (c,(n, a) - 1)p" < (A

+ p , a')

5 ca(n,a)p".

Then

I

{ p E X ( T ) L(A)

socGH1(p)}= {A - ca(n,a)p"a I a E S , n E N,cA(n,a) c p } .

Proof;. Consider at first some p E X ( T ) with L(1) = soc,H'(p). There has to be a unique a E S with ( p + p , a v ) < 0. Write (s, p,uv) = xy=oajp' with 0 I aj < p for all j and a, > 0. If al = p - 1 for 0 I j c n, then we get from 5.15.a that A = s, p, hence

-

p = s,

*

A

= A - (A

+ p,aV)a = A - 1 ajp' + 1 L:o

)

a =A

- (a, + 1)p"a.

On the other hand, (A + p , a v ) = (a, + l)p", hence c,(n, a) = a, + 1 5 p and p = 1 - c,(n,a)p"a. If a, = p - 1, then c,(n + 1,a) = 1 and p = 1 c,(n + 1, a ) p n +'a. Suppose now aj p - 1 for some j < n. Then 1= p + anpnaby 5.15.b. Furthermore,

-=

(1+p,aV)=2a,pn+(p+p,av)=2a,p"-(s,(p+p),aV)

=2a,p"-

1 (p-l-aj)pj.

n

n- 1

j=O

j=O

1ajp'-l=(a,-l)p"+

As p - 1 - aj > 0 for some j < n we get c,(n,a) = a, < p , hence p = A - c,(n, a)p'a. Thus we have proved one inclusion so far. The converse requires more or less the same calculations and is left to the reader. 5.18 Corollary: Let p E X ( T ) . Then the trivial G-module L(0)= k is the socle of H1( p ) if and only if there is a simple root a E S and a natural number n E N with p = -p"a.

Pro08 We have ( p , av) = 1 for all a E S, hence co(n,a) = 1 for all n E N.So the corollary follows immediately from 5.17. Remark: Let a,/? E S with (a,Bv) < 0. If n > 0, then s, (-p"a) = (p" - l)a 4 X ( T ) + as ( ( p " - l)a,/?') c 0. So p = -p"a is a weight with H 1 ( p )# 0, but p 4 w ( X ( T ) + ) .This shows that the Borel-BottWeil theorem will not generalize to characteristic p-not even in its weak

ul(w)=l

Borel-Bott-Weil and Weyl

257

version, which says if J. E X ( T ) +and H'(w A) # 0, then i = I(w)-as soon as R has a component of rank 2 2. This was first observed by Mumford for SL, and p = 2. Then Griffith determined for SL, and any p all p and i with H ' ( p ) # 0. The results can be found in [Andersen 21. (Note that we can restrict ourselves to i = 0 , l using Serre duality and IR+1 = 3. So the result follows from 2.6 and 5.15 and some combinatorial considerations.) Using additional ideas, all non-vanishing H i ( p ) for rank-2-groups are determined in [Andersen 91 though there are minor errors in the figure for type G, and in the statement about the walls, cf. [Humphreys 201. 5.19 We have by I.4.5.a for each A E X ( T ) +and p E X ( T ) a spectral sequence

with (1)

Ei' = Extb(L(J.),H'(p))

Extp'(L(L),p).

For p E X(T)+we have used this sequence in 4.7 (in a more general situation) to get isomorphisms (2)

Extb(L(A),HO(p))Y ExtB(L(J.),p)

for p E X ( T ) + .

For p # X ( T ) + (and char(k) # 0) the situation is more complicated. But we know at least H o ( p ) = 0 for such p, hence E i o = 0. Therefore, the five term exact sequence corresponding to (l), cf. 1.4.1(4), yields an isomorphism (3)

Ext&(J.),

= Hom,(L(J.), H1(p))

for p $ X ( T ) + .

The right hand side in (3) is completely known by 5.16/17. So we get using the notations from 5.17: (4) Let J. E X(T)+and p E X(T),p # X(T)+.If there are a E S and n E N with p = A - cA(n,u)p"aand c,(n,a) < p , then Exti(L(A),p) = k. r f not, then ExtA(L(A),p ) = 0.

5.20 Proposition: Suppose char(k) = p # 0. a) W e haoe H1(B,-p"a) = k for all u E S and n E N. b) For all p E X(T) with p # -p"a for all a E S and n E N we have H'(B, p) = 0. Proofi Part a) and part b) for p 4. X(T)+are the special case L = 0 of 5.19(4), But H ' ( B , p ) # 0 cf. 5.18. It remains to show H ' ( B , p ) = 0 for all p E X(T)+. implies by 4.10 that p E -NR+ and ht(-p) 2 1. If we take a W-invariant scalar product (,) on X ( T )&R then p E -NR+ and ht( - p ) 2 1 implies ( p , p ) < 0, whereas p E X ( T ) +implies ( p , p ) 2 0. This yields H 1 ( B , p )= 0 for p EX(T)+.

258

Representations of Algebraic Groups

5.21 Let Z c S and set P = PI. Write Hf(p) = R'ind;(p) for all i E N and p E X(T).Regarded as an L,-module, H : ( p ) is isomorphic to R'ind;&,,(p), and V, acts trivially on Hi@), cf. 1.6.11. Set L,(p) = soc,H,O(p). So either L,(p) is 0 or a simple module with highest weight p. Let 1E X(T)+. We have L,(A) N @vEzrL(A)a-v by 2.11. Applying this in characteristic zero (where L(A) = HO(1))we get dim Ho(l)a-v= dim H,O(lJa-v for all v E ZZ using the independence of ch Ho(A) = ~ ( 1of ) k. We have Ho(A) N indgH,O(A) by the transitivity of induction. The evaluation map e:H0(1)-H:(1) is (by 2.11(1) and 2.2(1)) injective on @ v E Z , H O ( A )The l-v. equality of dimensions yields:

If we intersect a composition series of Ho(l)with @ v E Z r H o ( A ) l - vthen , we get (by (1) and 2.11) a composition series of Ho(,i) with some factors equal to 0. This implies for all A , p E X ( T ) + : (2)

[HO(l):L(p)l= [H,O(A):Lr(/i)] i f

A

- /L E

ZI.

There is also a result similar to (1) for the V(A),cf. [Jantzen 11, p. 15. That it leads to results like (2) is clear and was mentioned in a similar situation in [J], 1.18. It was applied in our situation in [Schaper], p. 65, and also appears in [Donkin 81.

6

The Linkage Principle

Throughout this chapter, let p be a prime and k a field of characteristic p . Whereas in characteristic zero each Ho(A) with A E X(T)+is simple, this is no longer so over k (except in the trivial case G = B = T ) . We want to know more about the composition factors and their multiplicities. Let us denote the multiplicity of L ( p ) as a composition factor of V by [ V : L ( p ) ]for any p E X(T)+and any finite dimensional G-module V. So we ask for inforWe know mation about the matrix of all [Ho(A):L ( p ) ] with A,p E X(T)+. = 1 and get from 2.2 that [HO(A):L(p)] # 0 implies by 2.4 that [HO(A):L(l)] p IA.

We want to show that we can replace “ p I A in this statement by “ p I.”, where 7 is an order relation refining 1.To be more precise, let us consider (affine) reflections of the form x H x - ((x + p , a v ) - rp)a with a E R and I E Z. Then by definition p 7 1 holds (for arbitrary A, p E X(T)) if and only if either p = A, or there are reflections sl, s2,. . . ,s, of this type with

1 2 s,.A 2 s2s1.12 . . * 2 (s;.*s,).A

= p.

So p t A implies p I1,but the converse does not hold in general. That [HO(l):L(p)]# 0 should imply pLf A was first conjectured in [Verma], then proved for type A , and p > n in [Jantzen 21, for p 2 h in [Jantzen 31, and for arbitrary p in [Andersen 41. There the statement is generalized to

259

260

Representations of Algebraic Groups

all H’(w.2) with i E N and w E W. We follow here Andersen’s approach (6.12-6.16). It also yields complete information about [H’(w.A):L(il)] for w E W, and i E N. all ilE X(T)+, A more careful analysis of Andersen’s method also yields more detailed results about [H‘(w.A):L(p)] with ,u not “too far away” from il (in a sense made precise in 6.22). This was first observed in [Koppinen 13 where some special cases treated in [Jantzen 31 were generalized. Since writing the first version of this text, I have received copies of [Koppinen 61 and [Wong 3,4] containing other generalizations of these results. As a corollary to the above results we get the “linkage principle” (6.17):If Ext;(L(A),L(p)) # 0 for some il,p E X ( T ) + ,then 3, E Wp.p,where Wp is the group generated by all reflections as above. We call W, the ufJine Weyl group of R. It is isomorphic to the affine Weyl group W, as in [B3], ch. VI, 52. This chapter begins with a discussion of this group, looking at the system of alcoves and facets defined by it. Then we describe some properties of the order relation 1, most of which are quite straightforward. (Proposition 6.8 and 6.11(5) were first proved in [Jantzen 31.) 6.1 Let us denote by s ~ for, all ~ fl E R and n E N the affine reflection on X ( T )

or X ( T )BZR with

, B~ E R~ and n E Z . for all A. Set W, equal to the group generated by all s ~with We call Wpthe ufJine Weyl group (with respect to p). It is obviously isomor, ~ phic to the affine Weyl group, as in [B3], ch. VI, 52, generated by all s ~with /?E R, n E N. One easily shows (as in [B3], ch. VI, 52, prop. 1) that W, is the semi-direct product of W, and the group pZR acting by translations on X ( T )OZR using s,+~,~,(~J= il - npP for all 1:

(2)

WpN W w p Z R .

We shall always consider the dot action w . 1 = w(il + p ) - p of Wp on X ( T ) and X ( T ) @ , R . So we regard sg,,, as a reflection with respect to the hyperplane (3) We can apply the general theory of reflection groups, as in [B3], ch. V, to the group W,.

26 1

The Linkage Principle

6.2 The reflection group W, acting on X ( T )OZR defines a system of facets. A facet (for W,) is a non-empty set of the form

(1)

I + p , a V ) = n,p

F = {A E X(T)OzR (A

(nu- 1 ) p < (A + p , t l v ) c n,p

for all a E R i ( F ) , for all a E R : ( F ) }

for suitable integers n, E Z and for a disjoint decomposition R' = R : ( F ) u R:(F). Then the closure P of F is equal to

(2)

F

=

{A E X ( T )OZR I (A + p, a v ) = n,p (nu- l ) p I (A

+ p,a")

for all

tl E

Ri(F),

I nap

for all z E R : ( F ) } .

= n,p

for all a E R:(F),

We call (3)

F = {A E X ( T )O ~ I R (A + p,av) (n, - l ) p < (A

+ p , a v ) I n,p

for all

tl E

R:(F)}

the upper closure of F. Obviously f c F and both f and P are unions of facets. Any facet F is an open subset in an affine subspace of X(T)OZR, more precisely, in {A 1 (A + p, a V ) = nap for all a E R i ( F ) } using the notations from above. The codimension of this subspace is equal to dim~,,RJ(,,RE. A facet F is called an alcove if RA ( F ) = 0 (or, equivalently, if F is an open subset of X(T)OZR). The alcoves (for W,) are precisely the connected components of the complement in X ( T )OzR of the union of all reflection hyperplanes, i.e., of X(T)OzR -

u u {Al(A + p , a V )

= np}.

aERneZ

The union of the closures of the alcoves is X ( T )OZR.Any A E X ( T )OzR and any facet belongs to the upper closure of exactly one alcove. Let me quote from [B3], ch. V, 53, th. 2:

(4) If F is an alcove for W,, then its closure F is a fundamental domain for Wp operating on X ( T ) OZR. The group W, permutes the alcoves simply transitively.

As Wpstabilizes X ( T ) ,we get immediately:

( 5 ) If F is an alcove for W,, then P n X ( T ) is a fundamental domain for W, operating on X(T ) .

262

Representations of Algebraic Groups

There is one alcove (our "standard alcove") to which we shall usually apply (5). Set (6)

C = {A E X ( T ) @ ~ R I O < (A

+ p,av) c p

for all a E R').

As {a' I a E S} is linearly independent and as p' E z a E s N a Vfor all /?E R', it is elementary to show that C # 0, hence C is an alcove. Notice that

C n X ( T )= Cz,

(7)

in the notations of 5.5. Suppose that A E C n X ( T ) .Then for all a E S 0 < (A

hence (%,a') 2 0 and A hence

E

+ p, a')

=

(A,a')

+ 1,

X(T)+.This implies (%,p') 2 0 for all p E R + ,

(P,P'>

(2

+ P , B V ) < P.

This shows (8)

C n X(T)#

0o ( p , p v ) < p

for all /?E R +

o0E

C.

Set

+ 1 Ifi E R'}.

h = max{(p,p')

(9)

In case R is indecomposable this is the Coxeter number of R, cf. [B3], ch. VI, $1, prop. 31 and use that R and R'have the same Weyl group, hence the same Coxeter number. In general, it is the maximum of all Coxeter numbers of the irreducible components of R . We can reformulate (8) as: C nX ( T )#

(10)

0o p

2 h.

Note that we can replace C by any alcove, as these are all conjugate to C under W,.

6.3 A facet F is called a wall if lR;(F)I = 1, i.e., if there is a unique p E R + with (A + p , p ' ) E Z p for all A E F. If so, then there is a unique reflection sF = sP,,, with n p = (A + p , p ' ) for all A E F in W, (called the reflection with respect to F ) which acts as the identity on F. Let C' be an alcove for W,. Denote by X(C') the set of all reflections sF, where F is a wall (for W,) with F c Then Z ( C ' )generates Wpas a group, more precisely, ( W,, C(C'))is a Coxeter system (cf. [B3], ch. V, $3, th. 1). For any A E the stabilizer

c.

c

W,O(A)={wEW,lw.A=A)

263

The Linkage Principle

is generated by CO(1, C') = {s E C ( C ' ) s. A = I ) ,

I

and (W;(A),C o ( I ,C'))is a Coxeter system, cf. [B3], ch. V, $3, prop. 1 and 2. In the case of the standard alcove C from 6.2(6) we simply write C = C ( C ) and C o ( I ) = Co(A,C). The walls contained in and hence also X can be described explicitly, cf. [B3], ch. VI, $2, prop. 5. We get that C consists of all s, with a E S and of all S~,,, where p is the longest short root of an irreducible component of R. Suppose for the moment that there are wh E X ( T ) for all a E S with (oh, p') = liuOls (the Kronecker delta) for all a, p E S . In the semi-simple case the oh are just the fundamental weights (cf. 1.6), so the assumption amounts to G being simply connected in this case. In general, the restrictions of the wb to T n 9 G are the fundamental weights of 9 G , hence they exist if and only if 9 G is simple connected. We claim:

c

(1) If 9 G is simply connected and if p 2 h, then F n X ( T ) # 0 for each wall F for W,. As all alcoves are conjugate under W, we may assume that F c

c. Let us

look at first at a wall given by (A + p , a v ) = 0 for some a E S. In this situation -a: E F as ( p - w:,a') = 0, a n d 0 < ( p - oh,P') I( p , / 3 ' ) c h I p for any p E R + , p # a. (Note that ( p - c y p s o ~ , p =v 0 )for all p E R+ as ( p , y ' ) = 1 for all y E S. Note also that we can write 8' = ~ y s S m y with yV my E N,and that p # a implies m, > 0 for some y # a,) Now let p be the longest short root of some component of R. Write /I'= Cues maav. If p' E R+ belongs to the same component as /3 and if p" = ~ , E S m ~ then a v , mi Ima for all a. Suppose at first that there is some a E S with m, = 1, i.e., with ( w ~ , p ' )= 1. (This is always satisfied if the component is not of type E , , F4, or G , , cf. the tables in [B3], ch. VI.) Then I = ( p - ( p , P ' ) ) o : , will work as (A p , p ' ) = p , as (A + p,y') 2 ( p , y ' ) > 0 for all y E R+, as (A + p , y v ) = ( p , y ' ) c p for y E R+ not in the component ofS, and ( ~ + p , y ' ) = ( ~ , ~ ' ) + ( p , y ' ) ~ ( I , Y ' > + ( P , B ' > I ( I + P , P " ) = P for y E R+ in the component of /3. If p belongs to a component of type E , , F4,or G , , then there is some a E S with ( o ; , p ' ) = 2. We can then take A = $ ( p - (p,p'))wh, as in this case both p 2 h and ( p , p ' ) are odd. (Look at the tables in [B3], ch. VI, to get ( p , p') = 29, 11, 5 in these three cases.)

+

t on X ( T ) . We want I T p to hold if and only if there are pl,.. .,p r E X ( T ) and reflections sl,. . . , s r + E W, with

6.4 Let us introduce an order relation

(1)

I Is1 * A = p1

Is 2 . p 1 = p,

I *.. Is r . p r - l = pr Is , + l . p r = p,

264

Representations of Algebraic Group

or if p = A. We have, obviously,

(4

Atp=.AIp

and A E W , . ~ .

If we write si = sB,,,,, with pi E R + and n, E Z, then the condition p i I s i + l . p i in (1) amounts to ( p i , & + 1 ) I n i + , p for 0 I i I r (where p o = 2). If 1 E X(T)and a E R + , then there are unique n,, d , E Z with (A + p, a') = nap + d, and 0 < d , I p . Then S , , ~ = ~ . A= A - d,af?,. If d, c p , then s,,(,,=- l)p.(A - d,a) = 1 - p a t A - d,a. Therefore, in any case:

1 -pa t 1

(3)

for all a E R + 'and all A E X(T).

Another elementary property of f is the following: We have for all A,p, v

E

X(T)

Afp*1+pvfp+pv.

(4)

(If 1 t p and if si = sBi,,ip are as in (l),then the sBi,(n,+(v,B,>)p will do for 1 and p p v . ) We have, furthermore,

+

( 5 ) If A E X ( T ) with w E w.

(A + p , a v ) 2 0 for all

aE

+pv

R + , then w . 1 7 2 for all

This is proved using induction on l(w). If l(w) = 0, then w = 1 and the result is obvious. If l(w) > 0, then there is a E S with l(s,w) = I(w) - 1, i.e., with w - l a c 0. Then ( w ( 1 + p),a') = (1 + p , w - ' ( a ) ' ) I 0, hence w.?,t(s,w).A. Now we can apply induction to saw to get (s,w).A t 1.

t on the set of alcoves for W,. Let C,, C, be two alcoves. For any Al E X(T)n C, there is a unique A, E C, n Wp.A,as C2 is a fundamental domain for W,. Then we want the following to hold:

6.5 We can also define an order relation

(1)

~lt22-ClfCz.

It is elementary to show that the left hand side in (1) does not depend on the choice of 1, in C1 n X(T).So we would take (1) as the definition if we were certain that C1 n X(T)# 0. As this is not true for small p by 6.2(10), we have to give another definition. If a E R + and n E Z, then either (A + p , a v ) < np for all 1 E C,, or (A + p , a v ) > np for all A E C1. In the first case set C1 t S,,,~.C~, and in the second one set s,,,~. C1 t C1. This is a definition of s. C, t C1 for any reflection s in Wpand any alcove C1.Now we set C, t C, if and only if there are

265

The Linkage Principle

reflections s,,.

. . ,s,

,in W, with

c1 ts,.C,

t(S2Sl)'Cl T..*T(S,+,*.*S,).C, = c2,

or if C, = C2. It is then obvious that we get an order relation and that (1) holds. For A1 E C1 n X ( T ) we only have a weaker statement: If A2 is the unique element in Gn Wp.A,, then

c1 t c2

(2)

A1

t A,.

The same proofs as in 6.4 show for all alcoves C,, C2 (3)

c1 - P a t C 1

(4)

C, C2 0 C,

for all a E R + ,

+ p v t C2 + p v

Let us call an alcove C1 dominant if (A I E C,. Then:

(for all v E X ( T ) ) .

+ p , a v ) > 0 for all

a E R'

and

(5) If C , is dominant, then w.C1 t C1 for all w E W. 6.6 For any alcove C 1 and any a E R+ there is a unique nu E Z with n,p < (A p,av) < (n, 1)p for all A E C,. If so,then set

+

+

If C, is dominant (i.e.,if nu 2 0 for all a E R'), then d ( C , ) is exactly the number of reflection hyperplanes for W, such that C, and the standard alcove C lie on different sides of this hyperplane. (The equations of these hyperplanes are (A + p , a v ) = i p with a E Rf and 1 I i I nu.) Lemma: L e t C1 be an alcove for W, and s a rejection in W,. Then d(s. C,) < d(C1). S . C1 t C1 Pvoofi Let nu (a E R + ) be as above and let mu(a E R') be the corresponding integers for s. C,. Choose A E C,. Then there are d , E R for each a E R+ with 0 < d, < p and (A + p, a') = n,p + d,. Suppose s = s,,.,~ for some b E R + and r E Z. Let us assume s.C, t C1, i.e., r In,,. We have s.A = A - ((n,, - r)p d,,)P, hence ( s , A p,j?') = (2r - ns)p - d,, < nsp, hence m,, < n,,. It is obvious for each a E R + that we have mu 5 nu if ( & a v ) 2 0, and mu 2 nuif ( B , a') < 0. Consider a E R + with (/a') I,< 0. Then also (a, b') < 0, hence y = s,,(a) = a - (a,pv)/3E R + with y # a and ( P , y ' ) = -(P,a").

+

+

266

Representations of Algebraic Groups

Furthermore, we have y' (I

= a'

- ( p , a')/?', hence

+ p , ~ ' )= n,p + d, = (nu- (P,a'>ns)p + d , - (S,a'>ds.

This implies (s.?.

+ p, a')

(s.2

+ p , ~ ' )= ( n u - r(P,a'>)p + d,,

= (n,

+ r(P,a'))p + d ,

and hence m, + my = npr+ n y . So the contribution to d(C,) - d(s.C,) from all a E R+ with ( p , a ' ) < 0 cancels with the contribution from all y E R+ with (p, y ' ) > 0 and ss(y) > 0. As any other y E R+ with (p, y ' ) 2 0 contributes something non-negative, and as y = p contributes something positive, we get d(C1) > d(s. Ci). If s.C, t C , does not hold, then C 1 f s . C , , hence d(C,) c d(s.C,) by the part already proved, and the claim follows. Remark: Let A E X ( T ) . Define nu, d, E Z with (A + p,a') = nap + d , and 0 < d, Ip for all a E R+. Set d(A) = C,,on,. The same proof as above shows for any reflection s E Wpwith s . I # I :

s . I t I o d(s.A) < d(A).

el.

(2)

There is a unique alcove C , with I E Then d ( I ) = d(C,). One can also (2) from the lemma by observing: If s.Af I , then d(s.A) Id(s.C,) as deduce s.I E s.c,. Let C , be an alcove for W, and F a wall with F c c1 and Let sl,sz,. . . ,s, be re$ections in w, and set wi = s i s i - , * * * s for l 1 5 i 5 r. Suppose w,. C 1 w,sF. C 1 and

6.7 Lemma:

c1t SFC,.

(1)

WI.C,fW,_,.C1~.~.~W,.Cl~C1.

a) W e have: (2)

W r S F . c1

Wp- 1 S F .

c, 7 7 w1 s p . c1 t S F . c,, * * *

or there is some i(l Ii Ir) with (3)

WrsP. c1 t w,-

1S F . c 1

t " ' 7 wisp. c1 = wi- 1

a

b) Suppose d(w,.C,) = d(C,) - j for all j , 1 Ij Ir. Then d(wjsF.Cl)= d(s,.C,) - j for all j in the case of (2), resp. for all j 2 i in the case of (3). If

267

The Linkage Principle

wj. C , is dominant for some j in the case of (2)resp. for some j 2 i in the case of (3), then wjsF.C1 is dominant. Proof: We have wjsF.C1 = x w ; ' . ( w j . C , ) = s,,.(w~.C,), where F' = wj.F is a wall contained in wj.C1. Therefore the two alcoves wj.C1 and wjsF.Cl are separated only through one reflection hyperplane (with reflection s,.). For any reflection s # s,, in W, the alcoves wj.C1 and wjsF.c1 lie on the same side of the reflection hyperplane, hence wj.C, ts.(wj.C,) holds if and only if wjs,.C1 T s.(wjsF.Cl) holds. a) If (2) does not hold, then choose i 2 1 maximal such that wisF.c1 t wi- 1sF. C, does not hold (where wo = 1 for the moment). As wi- 1 s F . C , = siwiSFac1 = si.(wisF.c1), as wi.cl Tsi.(wi.cl) holds and wisF.c1 t s i . (W&.c1) does not, the remarks above imply si = wisFwf hence wiSF.c, = siwi.C1 = wi-,.C1. In this way we get (3). b) Because of 6.6 we have in the case of (2):

',

d(C,) - r = d(w,. C , ) c d(w,s,. C,)

< d(Wr-1S,.Ci) < * * * < d(S,.C,)

= d(C1)

+ 1.

(For the last equality use that C 1 and s,.C, are separated only by one reflection hyperplane.) This chain of inequalities can only be satisfied if d(WjSF.Cl) = d(C,) + 1 - ( j + 1 ) = d(S,.C,) - j for all j . The arguments in the case of (3) are similar. We have d(wj.C,) = d(wjsF.Cl)- 1 for all j resp. for all j 2 i, hence

Suppose wj. C , to be dominant. The hyperplane containing wj. F cannot have an equation (A + p, a') = 0 with a E R as otherwise w~s,w;' = s, E W and wjs,. C, t wj. C1 by 6.6(5). As this was the only reflection hyperplane separating wj.C1 and wj~,.C1, these two alcoves lie on the same sides of all hyperplanes (A + p,av) = 0 with a E R, hence also wjsFSc1 is dominant. 6.8 Proposition: Let C , be a dominant alcove for W,. Let tl E R + and n E N with 0 c np c ( A + p, a V ) for all A E C,. There is a unique w E W with WS,,,~.C~dominant, and there are dominant alcoves Ci (0 I i I r for some r) with d(C:) = d ( C l ) - i for all i, and

c;= WS,,,,. c, t c; - 1 t t c; t c; = c,. ''*

Proof: The existence and uniqueness of w is obvious as there is for each A E C, a unique w E W with ( W S , , ~ ~ . , ~ p, Pv) > 0 for all /3 E S.

+

268

Representations of Algebraic Group

Let us use induction on d(C,). We have d(C,) 2 n > 0. There has to be a wall F c C1 such that the supporting hyperplane of F separates C1 and the standard alcove C. As C, is dominant, this implies s F .C1t C1and d(s,.C,) = d(C,) - 1. Furthermore, s F .C1 is dominant. If sF = s,,,~, then W S , , , ~ . C, = s F .C1and (1) is satisfied with r = 1. SO let US suppose sF # s , , , ~ . Then C, and sF.C, lie on the same side of the hyperplane (A + p , a v ) = np. We can now apply induction to S F . c , and s , , , ~ : If W ' S , , , ~ . ( S ~C,) . is dominant for some w' E W,then there is a chain as in (1) from w's,,,psF. c1 to SF. c1. Suppose at first w' = w. If WS,,,~.C~ 7 W S , , , ~ S ~C,, . then we can prolong the chain as in (1) from W ' S , , , ~ ~C, ~ . to sF.C, by adding W S , , , ~ . C1and C, at its ~ . T W S , , , ~ . C,, then we can two ends and get a chain as in (1). If w ' s , , , ~ s C, apply Lemma 6.7 to sF.C, and W S , , , ~ , instead of C1 and w,. Then we immediately get (1). Consider now the case w' # w. Then W ' S , , , ~ . C , is not dominant, hence there is y E S with (w's,,,,(A p), y') < 0 for all A E C,. As W ' S , , , ~ . C, and w's,,,psF. C, lie on different sides of the hyperplane (A + p , y v ) = 0, we get S ~ W ' S , = ,~~ W ' S ~ , , ~ Shence ~, w = syw' and W S , , , ~ . C1 = w ' s , , , ~ s C,. ~ . Therefore we get a chain from WS,,,~.C~ to C, by taking the one from WS,,,~.C~ to sF.C1and adding C , .

+

Remark: Note that WS,,,~.C~ # C, as s , , , ~ $ W. E X ( T ) with ( A + p , B v ) 2 0 for all P E S. Let a E R+ and n E N with 0 < np < ( A p,a'). Let w E W with ( W S , , n p ( A + p ) , P ' ) 2 0 for all j E S . Then W S , , , ~ . I 7 A and W S , , , ~ . A < A.

6.9 Corollary: Let I

+

c.

Proof: There is a dominant alcove C1 with A E Then np < ( p + p , a') for all p E C1, hence we can apply 6.8. If W'S,,,~.C, is dominant for some w' E W, then (w's,,,,(A p ) , P ' ) 2 0 for all /? E S, hence W'S,,,~.A = WS,,,~.A. Now W ' S , , , ~ . C1 T C, by 6.8, hence W'S,,,~.A t A by 6.5(2). If W S , , , ~ . A = A, then

+

(1+ P, 1, + P ) = ( s a , n p ( A + PI,

Sa,np(A

+ PI)

= (A

+ p - ( ( A + p , a v ) - np)a,A + p - ((A + p , a v ) - np)a) = (A + p , A + p ) - 2((A + p,a') - np)(A + p , ~ ) + (0+ p , a'> - nP)2(a,4, hence 0 = ((A contradicting 0 < np < (3,

+ p , a'>

- np)(a,a)( - np),

+ p , ~ ' ) .Hence, WS,,,~.A < 1.

269

The Linkage Principle

6.10 Corollary: Let C', C" be alcoves for W, with C" t C'. Then there are alcoves Ci(OIi II = d(C') - d(C"))with d(Ci) = d ( C ' ) - i for0 Ii II , and

C" = c, 7 c,- 1 7 '

t c1 co = C'.

Proof: We may assume C" = s. C' for some reflection s E W,. If we translate our alcoves by pv for some v E Z R , then d(C") - d ( C ' )and the relation C" t C' do not change. We can therefore assume that C', C" are dominant. Then we get the chain from 6.8(1). 6.11 Let F be a facet. Let us use the notations from 6.2(2). Then Ro(F)=

R i ( F )u ( - R i ( F ) ) is a root system in its own right, and its Weyl group can be identified with the group W:(F) generated by all s,,., with a E R i ( F ) . By 6.3 we have W:(F) = W:(A)for any 3, E F. There is a unique alcove C- = C - ( F ) with F c C - . It is given by (1)

C - = { A E X ( T ) OzR [(nu- 1)p < (3-

+ p , a v ) < nap for all a E R'}.

The group W : ( F )permutes simply transitively all alcoves containing F in its closure, i.e.,

{C'I F

(2)

c

F } = {w.C- 1 w E W:(F)).

At the other extreme there is an alcove C + = C + ( F )with F c C f and (A + p , a v ) > n,p for all a E R i ( F )and all ,lE C + .(It is the alcove containing F in its upper closure, if we work with - R f instead of R + as the positive system and if we forget the shift by p.) One proves by an induction argument similar to that in 6.4(5) using (2):

(3) If C' is an alcove with F c

c,then C - 7 C' 7 C'.

This yields:

(4) Let 3. E F n X ( T ) and w E W,. Then I t w.A

o C-

t w.C-.

We already know one direction, cf. 6.5(2). Suppose now I t w . IThen . there are reflections siE W, with

w.A > SlW.3, > S,S,W.E.

>

* "

>.S,."S1W.3.

= 1.

As w ' w . 1 ~w'w.C- for all W ' E W,, and as sw'w.2 < w'w.1 implies sw'w.C- 7 w'w.C- for any reflection s E W,, we get s,

'* *

s1w

.c- 7 s, -

1

* * *

s1 w

.c- f

* *

. t s1 w

.c- 7 w .c-.

270

Representations of Algebraic Groups

Furthermore, 1 E s, .s1 w. C - , hence C - f s, * * * s1w. C - by (3), hence C - f w. C - as desired. We still want to prove the converse to (3), i.e., {C'lF c

(5)

F } = { C ' I C - f C'f C+}.

Indeed, suppose C- t C' t C+.Then there are w1 = 1, w 2 . . .,w, E W,, and reflections si E W, with wi+ = siwi such that (with a suitable j )

c- = w1.c-

w2.c-

t...?w j . c -

w,.c- = c+.

= C'f...?

Then for any 3, E F

I

= w1.I Iw,.A I* . * IWj.I

c,

I ..' Iw,.I

= I,

(We have to modify the hence wj.I = I and 3, E wj.C- = hence F c definition of Ifor a moment in order to have the inequalities as above. We have to replace N by the non-negative real numbers.) Let a E S and I E X ( T ) with ( I + p , a v ) 2 0. Then there are finite dimensional B-modules &(A), N : ( I ) , N",],) with

6.12 Lemma:

+ npa), where the sum is over all n E N with 0 < np < (3. + p,av) such that there are ch N ; ( I ) = ch N;(A) = xe(s,.A

(1)

n

long exact sequences of G-modules

(2)

+Hi(sm,%)+Hi-l

(A)+ H'(N",A))

+H i +

&.I)

+ *..

and (3)

* * *

+ H'(N"jA)) + H'(N$(A))+ Hi-l(N;(A)) + H'+'(N!(A)) + .*..

Proof: Let us assume at first that p E X ( T ) . We shall write H : ( M ) = indE(")M for any B-module M (where P(a) = P{,) as in 5.1). Set r = (A + p, uV). We have described H:(A + p ) explicitly in 5.2.c: There is a basis ( u i ) O s i s r such that T acts on ui through I p - ia,we know how U,,U-, operate and that each U-,with p E R’, p # a acts trivially. It is then clear that H:(A + p)- = Z{=,kui and that ku, N s,(A + p ) are B-submodules of H z ( I + p). Set H z ( A + p)" = H:(A p)-/ku,. This yields exact sequences of B-modules that we tensor with - p , and thus get

+

+

(4)

O+H,O(A + p ) - o ( - p ) + H , O ( I + p ) O ( - p ) + I + O

and (5)

O+s,.A-+H,O(I+p)-

o(-p)+H:(A+p)"O(-p)-'O.

271

The Linkage Principle

Suppose for the moment that r 2 2. We can then also form H t ( A + p - a). Let us denote the corresponding basis (as in 5.2) by Fi (0 I i I r - 2). The formulas in 5.2 imply that the map ui H (r - i)Fi- induces a homomorphism of B-modules H:(A + p)" + H:(A + p - a), hence also H:(A + p)" 0 (-p) + + p - a) 0 ( - p ) . Let us denote the kernel, cokernel, and image of this homomorphism by N",& N:(,?), N:(jl). We have, therefore, short exact sequences of B-modules

(6)

0 + N"I(I)+ H:(A

+ p)" 0 (- p )

+

N!(A) + 0

and (7)

0 + N;(A)

+ H,O(l,

+ p - a) 0( - p )

+N;((I)

-+

0.

Furthermore, the explicit description of the map shows that ch &(A) ch N",A) are as given in (1). We have ( - p , a V ) = - 1, hence R'indi(")(- p ) = 0 by 5.2.b, hence R'indi(")(M0 ( - p ) )

1M

=

0 R'indi(")( - p ) = 0

for any P(a)-module M (using the generalized tensor identity I.4.8), hence finally R'ind,G(M 0 ( - p ) ) = 0 (using the spectral sequence 1.4.5.c). We can apply this to M = H t ( A + p ) and to M = H t ( A + p - a). Therefore (4) and (7) yield isomorphisms for each i (8)

Hi(,?) N H ' + ' ( H t ( A

+ p)-

0( - p ) )

and (9)

H'(N:(A)) 'y H'+'(N",A)).

We can apply indg to ( 5 ) and ( 6 ) and get long exact sequences. They contain the right hand sides of (8) resp. (9) and we use (8), (9) to replace these terms by the left hand sides. We then get (2) and (3) with N",(3,) = Ht(A + PI" 0(-PI. For r I1 we set IVY(,?)= 0 for i = 0, 1,2. This is certainly compatible with (l)and(3).Ifr =0,then(3.,av) = -1ands,.A = A,henceH'(A)= H'(s,.%) = 0 by 5.2 and (2) holds. If r = 1, then H f ( A + p)" = 0, hence (2) follows from ( 5 ) and (8) as above. This proves the lemma in case p E X ( T ) .In general there is a central extension G' + G with a split maximal torus T' + T such that p E X(T')3 X ( T ) . We can then carry out the constructions as above for T'. Let B' c G' be the inverse image of B. The B'-modules Np(A) have all their weights in X ( T ) , hence the kernel Z of B' + B (contained in T') acts trivially and the IVY(,?) are 8-modules in a natural way. (We have T N T / Z , hence B -N B'lZ and

212

Representations of Algebraic Groups

G N G ' / Z . ) Using R'indgN N R'indg,'N for any B-module N (cf. I.6.11), we see that the sequences (2), (3) for G' yield the desired sequences for G. 6.13 Proposition (The Strong Linkage Principle): Let A E X ( T ) with ( A p , a v ) 2 0 for all a E R+ and p E X(T)+. I f L ( p ) is a composition factor of some Hi(w.%)with w E W and i E N, then p t A.

+

We shall prove this result via induction on A for I in the next sections, always supposing A, p to be as above. The first step is the point where the induction hypothesis enters. We use the notations N@') as in 6.12. 6.14 Lemma: Let a E S and w E W with ( w ( 1 + p),a') 2 0. If L ( p ) is a composition factor of some H'(N",w.A)), then p t A and p c A.

Proof: Because of 6.12(3) it is enough to prove the same result for Nt(w.A) and N;(w.A), hence for each composition factor of these B-modules. Because of 6.12(1)we have to look at all Hi@') with Al of the form A, = s,w.A + npa for some n E N with 0 < np < (w(A + p),av) = ( A + p , w-'(a)'). By our assumption on A we must have p = w-'(a) E R + , and we can write A1 = s,w.(A - npp). There are A, E X(T)and w' E W with A1 = w'.A2 and (A, + p , y v ) 2 0 for all y E R'. As A, E W.(A - npp) we get from 6.9 that A, t A and A, < A. We can, therefore, apply the induction hypothesis to A, and the composition factor L ( p ) of some Hi(w'.A,). We get p L f , , hence p t A and p I A2 c A.

Let i E N and w E W with l(w) # i. I f L ( p ) is a composition factor of Hi(w.A), then p A and p c 1.

6.15 Proposition:

Proof: Let us suppose at first i c l(w) and use induction on i. If i = 0, then Ho(w,A) # 0, hence w.A E X ( T ) + by 2.6. Therefore w.A = A, as {A' E X ( T )BZR (A’ + p, a v ) 2 0 for all a E R + ) is a fundamental domain for the "dot" action of W on X ( T )&R. Furthermore A E X ( T ) + ,hence A has a trivial stabilizer in W (under the dot action), hence w.1 = 1 implies w = 1 and l(w) = 0 = i. This is a contradiction and settles the case i = 0. Suppose now i > 0. If (w(A + p ) , a " ) 2 0 for all a E S, then H'(w.2) = 0 by 4.5 and 5.4.a. Therefore we can find some a E S with (w(A p ) , a " ) C 0. We can now apply Lemma 6.12 to s,w.A instead of A. From 6.12(2) we get that L(p)iseither acomposition factor of H'-'(N",s,w.A)), or of H'-'(s,w.A). In the first case we apply Lemma 6.14, in the second one we use induction over i as l(s,w) = l(w) - 1 > i - 1. (Note that (A + p, w-'(a)") < 0, hence w-'(a) < 0, hence l(s,w) = l(w) - 1 by 1.5(3).)

I

+

273

The Linkage Principle

This settles the case i c l(w). The case i > I(w) follows either using Serre duality, or by descending induction. (Then we have to use that Hi(A') = 0 for i > n = IR+I, and that H"(I') # 0 if and only if (A' + p , a v ) c 0 for all a E S by a special case of Serre duality.) Remarks: 1) Observe that this settles the strong linkage principle for all A 4 X ( T ) + .For these we can find a E S with s,.A = I , hence W S , . ~ = w.1. for all w. If Hi(w.A) # 0, then either i # ~(ws,),or i # I(w), so we can apply the proposition above. 2) This proposition contains more information than the strong linkage principle, which does not imply p < %.(For X ( T ) + this inequality is obvious, of course.) 6.16 Proposition: Suppose I E X(T)+.Then L(A) is a composition factor with multiplicity one of each H""'(w.A) with w E W. A n y composition factor ~ ( pof) H""'(w. A) satisfies p t 1.

Proof: Suppose that either p t 1 does not hold or that p = I . For any w E W and any a E S with I(s,w) = I(w) + 1 we have by 6.12(2) an exact sequence H""'(N",w.A))

+ Hi(")+' (S,W.A) + H""'(w.A) + H""'+'(Na,(W.A)).

Lemma 6.14 implies that L ( p )is not a composition factor of any H'(N$(w.I)). It is therefore not a composition factor of the kernel or of the cokernel of the homomorphism HI(")+ (s, w. A) -+ H'(")(w.A). We can choose a sequence wb, w;, . ..,w; E W where n = IR+I, such that l(w:) = i for all i (hence wb = 1, wh = wo), such that there are simple roots ai E S with w: = s,,wI-l for 1 I i I n, and such that w;(") = w. (By definition ~ ~ Sw: , ~= . s,, "'s,, for i I I(w). of l(w) there are ai E S with w = S , , ( ~ ) ~ ~ ~ S Set Then I(w:) = i, because l(w:) c i implies I(w) = I(sui(w)*** sai w ; ) c I(w). We have l(w0w-') = l(wo) - Z(w-l) = n - l(w), cf. [B3], ch. IV, $1, exerc. 22a. Therefore there are ai E S with wow-' = sun s,i(w, L. Then take w: = s a i . . . suit,)+w for i > I(w). We have I(w:) = i as I(w;) c i implies l(wo) < n.) This sequence wb, w;, . . . ,w; leads, by applying the argument as above to each (w:,ai+') instead of (w,a), to a sequence of homomorphisms of Gmodules +

+

H"(wh.3,)+H"-’(wh-’.A)+’..+H’(W;.A)-,HO(wb.I). By the argument from above L(p) does not occur in the kernel or cokernel of any of the maps H i + l ( w ; + l . l ) + Hi(w;.I). Therefore it occurs in each

214

Representations of Algebraic Groups

H'(w',.A) with the same multiplicity as in the image M of the composed map H"W0.A) = H"w:,.A)

-+

HO(wb.A) = HO(A).

As this applies especially to p = A and as L(A) is a composition factor of H o ( i ) , it is also one of M , hence M # 0. We have by Serre duality

H"(w0 * A) = H"w,A - 2p)

N

HO(- W,A)*

= V(A),

cf. 2.13, hence for the homomorphic image M by 2.14( 1): MlradGM N L(L). On the other hand, as M c H o ( l )we get from 2.3 SOCGM1: L(3,). As L(3.) occurs with multiplicity one in in 5.6)

H0(%), hence in M, this implies (as

M = L(A). So the only L ( p ) as above, which is a composition factor of some H'(w)(w.A) is L(A),and it occurs exactly once. Remark: The proof also shows that the composed map H"(wo.A) -+ Ho(A) is non-zero and has image equal to L ( l ) . In fact, any non-zero homomorphism from H"(w,.A) to H o ( l ) has image L(A). 6.17 Corollary (The Linkage Principle): Let A, p E X ( T ) + . If Ext&!,(A), L(p))# 0, then 1 E W,. p.

Proof: Because of 2.12(4) we may assume p

+ A.

Therefore 2.14 implies

[ V ( l ) : L ( p ) I= CHO(A):L(p)l# 0,

hence p

E

W,.A by the strong linkage principle.

6.18 Suppose some A E X ( T ) with (A + p, /3') 2 0 for all /3 E S has the property that there is no p E X ( T ) + with p ? A and p # A. Then 6.16 and 6.15 imply for each w E W that H'(w.2) = 0 for all i # E(w) and that H'(")(w.A) N L(1)if A E X ( T ) + ,whereas H'(")(w.A) = 0 if A # X ( T ) + .

This argument can be applied especially in case A I 0 as ( - C a e s N a ) n X ( T ) + = (0). We get thus:

H'(w.0)

=

k 0

if i = E(w), otherwise.

The Linkage Principle

275

Furthermore, (2) H'(w.I)=O

for all A
with ( L + p , P V ) 2 0

for all P E S .

Proposition: Let n E N. Then H'(A"(Lie G/Lie B)*) = 0 f o r all i # n, and H"(A"(Lie G/Lie B)*) is the trivial G-module of dimension I{w E W I l(w) = .}I. Proof: Let us abbreviate M = (Lie G/Lie B)*. The weights of Lie G/Lie B are just the positive roots, hence those of M the negative roots (each with multiplicity 1). Set

for all subsets J c R + . Then the weights of A"M are just the - p ( J ) with J c R + and IJI = n. We have

where J' = R + - J , hence each w ( p - p ( J ) ) has the form p - p(Jl) for some J1 c R". This implies especially w.(-p(J)) = -p(Jl) 5 0 . We can choose w E W with ( w ( p - p ( J ) ) , p v ) 2 0 for all P E S. If w . ( - p ( J ) ) # 0, then the argument above and (2) show H ' ( - p ( J ) ) = 0, so this weight will not contribute to H'(A"M). O n the other hand, given w E W set v, = w-l .O = w-'(p) - p. It is by now clear that only weights of A"M of the form v, can contribute to H'(A"M). According to [Kostant 13, 5.10.2 there is for each w E W exactly one subset J, c R + with w - ' ( p ) - p = -p(J,) and one has IJ,I = l(w). So we get from (1) that weights of this form contribute only to H"(A"M).So H * ( A " M )= 0 for i # n, and H"(A"M) has a composition series with trivial factors and has the right dimension. As G is reductive, the trivial module has only trivial extensions with itself, so H"(A"M)is a trivial G-module. Remarks: 1) This proof works equally well in characteristic zero, so the result is true there also. 2) Note that we compute above the cohomology of the exterior powers of the cotangent bundle on G/B. 6.19 (Central Characters)The centre of Dist(G) acts by scalars on each simple G-module, i.e., via some central character Z(Dist(G)) + k (by 2.8).Similarly, the subalgebra of all G-invariant elements in the enveloping algebra U(Lie G) acts

276

Representations of Algebraic Groups

by some central character U(Lie G)“ -, k. If there is a non-split extension of L(I) and L ( p ) for I , p E X ( T ) + ,then the centre has to act on both simple modules by the same scalar, as otherwise the generalized eigenspaces of Z(Dist(G)) or of U(Lie G)“ would make the extension split. Using the Casimir operator in U(Lie G), the first results in the direction of 6.17 were proved in [Springer]. In [Humphreys 13 the linkage principle was proved (using U(Lie G)“) for all p > h (the Coxeter number). Then U(Lie G)“ was determined for all p in [Kac and Weisfeiler]. Their result almost implied 6.17: One had to replace “IE Wp.p by “IE W.p + pX(T)”. As obviously I - p E Z R this makes a difference only for those p dividing the order of ( X ( T )A Q R ) / Z R . This exception was already known to be unnecessary for G of type A , due to [Carter and Lusztig 13, where Z(Dist(G)) was looked at. There are more results about this centre in [Haboush 31, but it is still not completely understood. 6.20 Recall the notation d(A) from the remark to 6.6.

Proposition: Let I,p E X ( T ) + . If Ext’,(L(I), H o ( p ) ) # 0, then p t I and i Id(I) - d ( p ) .

Proofi

We use induction on i. For i = 0 we have

Suppose now i > 0. If 3, = p, then we have to prove Extb(L(I),HO(p))= 0, which we proved already in 4.13. Let us suppose now A # p. Then the exact sequence 0 -,rad V(A)+ V(A)-, L(I) --t 0 together with 4.13 yields an isomorphism (1)

Extb(L(I), H o ( p ) )N Extb-’(rad V(Q H o ( p ) ) .

If this group is non-zero, then there is a composition factor L(A’)of rad V(I) with Extb-’(L(A’), H o ( p ) )# 0. Now the strong linkage principle implies A‘ I and I‘ # I , hence d(I’) c d(I) by the remark to 6.6. On the other hand, induction yields p t A‘ and i - 1 5 d(A’) - d(p), hence p 7 A and i 5 d(A) d(P). 6.21 The same induction as in 6.20 also proves

dim Extb(L(A),H o ( p ) )c co

277

The Linkage Principle

for all I , p E X ( T ) + ,hence dim Extd(V,Ho(p)) < CL)

(1)

for each finite dimensional G-module V. Furthermore, 6.20 implies that there is for each such V some n ( V ) with Extb(V,Ho(p)) = 0 for all i > n ( V ) and all p. We can therefore form the alternating sum (2)

cp(V ) =

c (- 1)’dim Ex&( V,Ho(p)).

iL0

Like any Euler characteristic the function c,, is additive: If dimensional G-modules and a j E Z with

5

are finite

then

There is a unique expression (3) We have ~ ( p=) ch H o ( p ) = ch V ( p ) ,and 4.12 implies for all I , p E X ( T ) +

(4)

c,(V(4) = a*,,

(the Kronecker delta). Therefore (3) yields (5)

b, = c,(V).

In other words, we have for all finite dimensional G-modules V :

6.22 We now return to the technique used in 6.14-6.16 to get information about composition factors L ( p ) of H i ( w . I ) in cases where p is not “too far” away from I . More precisely, consider I , p E X(T ) , with p t I . We say that p is

close to A if and only if

(1) There is no a E R+ with p t I - pa. (2) If p I ’ 7 I for some I ‘ E X(T ) ,then I ‘ E X ( T ) + . Note that p t I’ 7 I and p close to I imply that I’ is close to I (obviously),and that p is close to A‘ (as I‘ - pa I - pa for all a E R ).

278

Representations of Algebraic Groups

Let 1 E X ( T ) + . Let F , be the facet with A E Fl and let F be a facet with F c Fl. Use the notation W:(F) as in 6.11, and consider p = w.1 with w E W:(F) and ,u T 1. Then 6.1 l(4) and (5) imply that any R' with p T A’ t 1 has the form 1’ = w'.1 with w' E W:(F).This shows especially that 1' # 1 - pa for all a E R+. Furthermore: If (x + p, a") > 0 for all a E R+ and x E F, then 1' E X ( T ) + . So in this case p is close to 1. 6.23 Proposition: Let 1,p E X ( T ) +with p t 1 and p close to 1.Let w E W and i E N. a) If i # l(w), then L ( p ) is not a composition factor of Hi(w.A). b) The multiplicity of L ( p ) as a composition factor in H'(")(w.1)is positive and independent of w.

Proofi We want to use induction on 1 - p. The case A = p has been dealt with in 6.15/16. So let us suppose p < A. There are n,, d, E N with (A + p , a v ) = nap + d, and 0 < d , I p for all a E R + . Set 1, = 1 - d,a = S,,~.~.A. The proof will show more precisely max C H o ( 2 a ) : ~ ( ~I ) CHO(JJ:L(p)I I I C CHO(Aa):L(p)I,

(1)

u>o

u>o

using the notation [?: L ( p ) ] for the multiplicity of L ( p )as a composition factor as in 2.7. Consider some w E W and a E S with /? = w-la > 0. Let us look at the long exact sequences in 6.12 for w.A instead of 1. If L ( p ) is a composition factor of some Hi(N;(w.A)) or H'(N",w.A)), then there has to be a weight I 1 of Nt(w.1) or N%(w.A) with [Hi(A1):L ( p ) ] # 0. The possible A1 have the form s,w.1 npcl with 0 < np < ( w ( 1 + p),a') = (1 + p,/?'). Furthermore, ,I1 = wss.A + npwp = wsa,,,.A. There is w 1 E W such that A2 = w1.Al = w1wss,,,.A satisfies (1, + p,y') 2 0 for all y E S. We have p T I., T 1 by 6.9 and 6.13 as [Hi(w;1.A2):L(p)] # 0. If wl(a) < 0, then

+

,u

T 1, = wls,w.l

- np(- wla) T wlsuw.,l 1

(cf. 6.4(3),(5)), contradicting that p is close to 1.Therefore wl(a) > 0. As

+ n ~ w , ( a =) swl(a),nps,I(u)w1saw.1 - swl(u),npw1w*A

1, = w1saw.J

and as

279

The Linkage Principle

we get 1, t w l w . A t A. The closeness of p to ;I implies w1 = w-' and 1, = S , , ~ ~ . AFurthermore, . necessarily n = n, and A2 = A,. So the only possible A1 is Al = w . 1 , . As w.A, is a weight of N",w.A) and N",w.A), this implies [H'(NY(w.A)):L ( p ) ] = [H'(w.A,):L(p)]

for j = 1,2 and all i . If this term is non-zero, then A, E X ( T ) +(as p p is close to A,, By induction we get (for j = 1,2) [H'(N;(w.A)):L(p)]=

(p)] > 0

t A, 7 A) and

for i = I(w), for i # I(w).

Therefore 6.12(3) yields [H'(N",w.

A)):

(p)] > 0

L(p)] =

for i = I(w), l(w) + 1, otherwise.

We get from 6.12(2) that L ( p ) is not a composition factor of the kernel of one of the maps H'+'(s,w.A) H'(w.1) for i l(w), or of the cokernel of one of these maps for i > I(w).This implies as in 6.15 that [H'(w.A):L ( p ) ] = 0 for i # I(w), i.e., that a) holds. Furthermore, we get from 6.12(2) that the multiplicity of L ( p ) both in the kernel and in the cokernel of --f

H'(W)+

-=

1 (s,w.A) + H""'(w.A)

is equal to [HO(A,):L(p)].(Note that I(s,w) = l(w) + 1.) This implies [H""'(w.L):L(p)]= [H""'+'(s,w.A):L(p)] 2 [HO(A,):L(p)] > 0.

Working with a sequence w b , ...,wk as in the proof of 6.16, this yields b) and also (1) as L ( p ) is not a composition factor of the image of H"(w,.A) HO(A). --f

6.24 Corollary: Let A E X ( T ) + . Suppose p E X ( T ) is maximal for p a n d p # A. If p E X ( T ) +and v p # A - p a for all a E R+,then

(1)

tA

[HO(A):L ( p ) ] = 1

and

(2)

HomG(Ho(A),Ho(p))=k.

Proof; The assumption implies that p is close to A. Furthermore, p has to be of the form iLI for some (unique) 01 E R+ (using the notations of the last proof). Therefore (1) follows from 6.23(1).

280

Representations of Algebraic Groups

Let Q, be the injective hull of the G-module L(p). As dim HomG(V,Q,) = [ V :L(p)] for each finite dimensional G-module V, we get

from (1). Let cp: H'(A) + Q, be a non-zero homomorphism. We can regard H o ( p ) as a submodule of Q, because of s~c,H'(p) = L(p). We want to prove cp(Ho(I))c H'(p). Then (2) follows immediately. If cp(Ho(I))$ H o ( p ) , then cp(Ho(A))+ H o ( p ) contains H o ( p ) properly, so there is a submodule V c cp(H'(I)) + H o ( p ) with V 3 H o ( p ) and V / H o ( p ) simple. Let p' E X ( T ) + be the highest weight of V/H'(p). As L(A) has to be contained in the kernel of cp, we have p' 7 A and p' # A. The extension O + H o ( p ) + V + L ( p ' ) + O cannot split, as V c Q, has a simple socle. Therefore Exth(L(p'),H o ( p ) ) # 0. Now 6.20 implies p 1p' 7 I and we get p # p' from 4.13(2), hence a contradiction to the maximality of p. Therefore there cannot be any I/ as above and cp(Ho(I))c H o ( p ) as desired.

Remark: Suppose I E C1 for some alcove C1.Then p E W,.I satisfies the assumption of the corollary if and only if the alcove C2 with p E C2 satisfies C27 C1and d(C2)= d(Cl) - 1. This follows easily from 6.10. 6.25 The proof of 6.24(2)given here is taken from [Koppinen 61. There, more # 0 is shown for I , p E X ( T ) +with p 7 A and p generally, HomG(Ho(A),Ho(p))

close to A. (In fact, the situation there is still more general.) In special cases (type A , B), 6.24(2) had been proved in [Andersen 61 where 6.24(1) is also proved (without restriction). Before the work of Andersen and Koppinen, homomorphisms had been constructed in a different way. Notice that one has (by dualizing) HOmG(H'(I), H o ( p ) )N Horn,( V(p), V(A))N V(A):'. Take u E V(I),, u # 0, so V(A)= Dist(G)u. In order to get a non-zero homomorphism V ( p )+ V ( I ) we have to find u E Dist(G) with uu # 0 and uu E V(1,)"' such that u is a weight vector of weight p - I for the adjoint action. Take, for example, a E S and p = I a = I - daa (using the notations from 6.23). If , I? > 0, then u = X-.,d= will do it. For a 4 S one has to replace X-.,d, by more complicated elements in Dist(G). For R of type A , this was done in [Carter and Lusztig 13 and (for more p ) in [Carter and Payne], and for arbitrary R in [Franklin].

7

The Translation Functors

Let p and k be as in Chapter 6. We can decompose the category of all G-modules into a direct product of subcategories as follows. Consider for each I E X ( T ) the category ALof all G-modules having only composition factors of the form L ( p ) with p E Wp.,l. The linkage principle easily implies (7.2/3) that { G-modules} is the direct product of all different MA, i.e., of all AAwith I in a system of representatives for the W,-orbits in X ( T ) .We usually take c, as our system of representatives. We want to compare different A’ and introduce “translation functors” Tr: + Apfor all I , p E c,, cf. 7.6. (For the sake of convenience we regard them as functors from { G-modules} to itself. In fact, we do not use the notation Ahexcept in this introduction.) It turns out that T $ is an equivalence of categories if I and p belong to the same facet (7.9). This is sometimes called the translation principle. Even in more general situations the T’;are useful. For example, if p belongs to the closure of the facet of I , then T’;behaves nicely on cohomology groups H’(w.3.) and simple modules L(w.,l) with w E W,, cf. 7.11 and 7.15. This reduces the computation of composition factors of the H i ( w . p ) , or of character formulas for the L(w.p)to the same problem for ,l.

281

282

Representationsof Algebraic Groups

For example, if p 2 h, then we get all chL(L) with A E X ( T ) +if we know all ch L ( w . 0 ) with w E W, and w . 0 E X(T)+. For the coefficients aw,w,in chL(w.0) = C U ~ , ~ , ~ ( W ‘ . O ) ~

W’

(sum over all w’ E W, with w’.O E X ( T ) +and w ’ . O t w . 0 ) there is a conjecture in terms of Kazhdan-Lusztig polynomials as long as w . 0 is “not too large”. (See 7.20(3)for a precise version of “not too large”.) This conjecture is related to properties of the functors T i with p E cz and A E Cz,cf. 7.20. The main sources for this chapter are [Andersen 91, [Jantzen 2,3,4,5,7], and [Koppinen 21. 7.1 (Blocks of an Algebraic Group) Let H be an algebraic group scheme over a field. Consider on the set of simple H-modules (or rather on the set of isomorphism classes of such modules) the smallest equivalence relation, such that two simple H-modules L , L‘ are equivalent whenever Exth(L, L‘) # 0. The equivalence classes are called the blocks of H.Let us denote the set of all blocks by g ( H ) . let Mbbe the sum of all submodules For any H-module M and any b E B(H) M‘ of M such that all composition factors of M’ belong to b. Then Mb is the largest submodule of this type. Lemma:

We have for any H-modules M , M’

and

Proofi It is obvious that the sum in (1) is direct and that Hom,(Mb, Mb.) = 0 for two different blocks b, b’. It is therefore enough to show for any M that the submodule N = CbEI(H) Mb is all of M. We may assume dim M c 00. Suppose N # M. Then there is a submodule N ’ c M with N c N ‘ and N ‘ J N = L simple. There is a block b with L E b. The exact sequence 0 + N + N ‘ + L + 0 cannot split, as otherwise N ’ N N 0 L and L c M, c N . Hence,

If Ext&(L,Mb,)# 0, then there is a composition factor L of Mbj with Exth(L, L‘) # 0. Then L’ belongs to the same block as L, hence b = b’.

283

The Translation Functors

Set N" = O,., Mb, c N c N ' . By the definition of blocks the exact sequence 0 + N" + N' + N'/N'' + 0 splits, hence N' N N" 0 (N'IN"). Obviously ("IN") = ( N ' / N " ) , c M b . This implies N ' c Mb, = N , a contradiction. Therefore N = M.

a,,

Remarks: 1) The lemma implies that M more, (2) generalizes to

H

Mb is an exact functor. Further-

2) Suppose M # 0 is an indecomposable H-module. Then there is a unique block b with M = Mb. This b is called the block of M . 3) If H is a finite algebraic group (1,8), then the blocks of H correspond to the indecomposable two-sided ideals in M(H). This is a general fact about representations of finite dimensional algebras. 7.2 (Blocks of C ) In the case H = G we can regard blocks also as subsets of X ( T ) +via A H L(A). Denote for any A E X ( T ) +the block of is (or of L(A))by b(A) t X ( T ) + .Obviously, 6.17 implies

(1)

b(A) c W , . A n X ( T ) + .

We shall not need more precise information than that. Let me, however, mention without proof the precise result. Most of the time we have equality in (1): (2) Let A E X ( T ) +and suppose there is some u E R + with Then b ( i ) = X ( T ) +n W,.i.

(A + p , u v ) 4 Zp.

This is proved in [Donkin 51. The case where (A + p , u v ) $ Zp for all u E R+ is especially easy and was already done in [Humphreys and Jantzen], using 6.24. For 1 not satisfying the assumption of (2) we no longer have equality in (1). This was first observed in the case of S L , in [Winter], and then proved in all cases in [Haboush 31 for the (p' - 1)p with r E N,using central characters. In order to formulate the precise description of b(A)in this case, let WF)denote for any r the group generated by W and all translations by p'v for all v E Z R . So W t )is a subgroup of W,, isomorphic to W,. Now:

( 3 ) Let AEX(T)+.Let r be the largest integer such that p' divides all ( A + p , u V ) w i t h u E R . Thenb(A)= Wt).AnX(T)+. In the case where G is semi-simple and simply connected we can write A = p'(L' + p ) - p with I.' E X ( T ) + .Then Ho(A) N H0(L')[']0 St, by 3.19. So the composition factors of Ho(A) are the L(p'(p' + p ) - p ) N L ( p ' ) I r 1 0S t ,

284

Representations of Algebraic Groups

(cf. 3.17) with L(p’)a composition factor of Ho(A’). Then at least b(1) c WF). A follows easily, but the precise statement also requires not too much work, cf. [Humphreys and Jantzen], 2.4. or 10.5 below. One can now easily extend this result to G of the form G, x G2 with G2 a torus and G, semi-simple and simply connected. In general, G has a central extension G’ + G with G’ of this form (G’ N G, x G2). If T’ + T is the corresponding extension of the splitting maximal tori, then the blocks of G are (obviously) exactly the blocks of G’ contained in X ( T ) .

7.3 For any G-module V and any 1E X ( T ) set pr, V equal to the sum of all submodules of V such that all its composition factors have a highest weight in W p . l .Then pr,V is the largest submodule with this property. It is clear from 7.2(1) that pr,V is a direct sum of some V, with b E a(G). If we choose a system 2 of representatives for the Wp-orbits in X(T),then 7.1(1)-(3) yield

and Horn,( V, V ’ ) N

(2)

or (more generally) for all i E N Ext!JV, V ’ ) ‘Y

(3)

n

asz

Horn,( pr,V, pr,V’),

n Extb(pr,V,pr,V’).

ZEZ

Each pr, is an exact functor. Note that we have, by definition, for all p E X(T)+ L(p)

(4)

if p E Wp.l, otherwise.

The strong linkage principle implies for all p E X(T ) and i E N praHi(p)=

(5)

{ti(’)

if p~ Wp.L, otherwise.

x;=

7.4 Let V be a finite dimensional G-module. The exactness of p r , implies: If ch V = aich for some ai E Z and some finite dimensional G-modules 6 , then ch p r , V = I aich(pr, F).For example, we can write

with almost all a, = 0 and all a,

E

2.(This expression is unique if we take

The Translation Functors

285

only p E X ( T ) +.) Then (2)

by 7.3(5) and 5.9(1). If M is another finite dimensional G-module, then

(3) by 5.8.b. (Regard M Q ? as an exact functor!) Let us take the composition of the two functors p r , and M @ ?. Then (2) and (3) yield ch(pr,(M Q V ) )=

(4)

c a,dim(MM,u + 4,

P*V

where we sum over all pairs ( p , v) E X(T)x X ( T ) with p

+ v E Wp.%.

7.5 Lemma: Let %, ,u E X ( T ) and let M be a finite dimensional G-module. a) The functors pr, 0 ( M @ ?) 0 pr, and pr, 0 ( M * Q ?) 0 pr, are adjoint to each other. b) Let V be a finite dimensional G-module with pr, V = V. Write

1w, a,x(w.A)

chV=

WE

with a, (1)

E

Z, almost all a, = 0. Then ch(pr,(M Q V ) )=

c a , ~ d i m ( l l / l , ) ~ ( w .+( ~v)), v

WCW,

where we sum over all v E X ( T ) with % + v

E

Wp.p.

Proof: a) We have by 7.3(2)for all G-modules V, V canonical isomorphisms (using I.4.4( 1)) H O ~ G ( P ~ , ( P AQ ( VMI,

v)= Homdpr,(pr,(V

C3 MI, pr,Vr)

= HomG(prd(V)6 M , pr,,Vr) = HomG(praV,pr,(Vr)6 M * ) = HomG(praV,pr,(pr,(V’) 6 M * ) ) = HomG(V,pr,(pr,(V’) 6 M*N. This proves that pr, 0 (A4@ ?) 0 p r , is left adjoint to p r , o ( M * @ ?) 0 pr,. As the situation is symmetric in % and ,u and as (M *)* E M, we also get that it is right adjoint. b) We have (cf. 7.4(3)) ch(M 0 V ) =

1

a, w~w,

1

vex(r)

dim(M,)x((w.%)+ v).

286

Representations of Algebraic Groups

As (w.A) + v = w.(A we also get

+ w l v )for a suitable w 1 E W and as ch(M) E Z [ X ( T ) I w ,

ch(M C3 V ) =

1

wow,

a,

1

V€X(T)

dim(M,)X(w.(A + v)).

From this (1) follows immediately (as in 7.4).

7.6 Consider 1 , p E C, (cf. 5.5, 6.2(7)). There is a unique v1 E X ( T ) +n W ( p - 1).We define the translation functor Tf;from A to p via (1)

Tf;V= pr,(L(v,) C3 p r m

for any G-module V. It is a functor from { G-modules} to itself. Lemma: Let ~ , E pC,. a) T h e functor Tf;is exact. b) The functors TS and T t are adjoint to each other. Proofi The first claim is obvious, as TS is a composition of exact functors. The second claim follows from 7.5.b as - wovl E W ( l - p) n X ( T ) + and as L(v1)* N L( - wov1) by 2.5. Remarks: 1) It will follow from 7.7 below that we can replace L(vl) by any finite dimensional G-module M with dim(M,,) = 1 and such that all weights v of M satisfy v I vl. For example, Ho(vl)or V(v,) would do. 2) The adjointness of Tf;and T i yields for any G-module V an isomorphism of functors Horn,(?, V ) 0 T i N Horn,(?, T S V ) , hence also isomorphisms of derived functors (cf. 1.4.1(3)) Extd(?, V ) 0 T i N Ex&(?, T S V ) . We have therefore for each G-module V' isomorphisms Ext';(TiV', V ) N Extb(V', T E V ) for all i.

7.7 Lemma: Let A , p E cz,and let v1 E X ( T ) + n W ( p - A). a) W e have A + wv q! Wp.p for any w E W a n d v E X ( T ) + with v < vl. b) If w E W with A + wvl E Wp.p, then there is some w 1 E W, with wl.A = A and w l . p = 1 + wvl.

287

The Translation Functors

Pro08 The set {wv I w E W, v E X(T)+, v 5 v,} is exactly the set of all weights v of Ho(v,) or of V(v,), cf. [Hu 11, 21.3. So we can express the claim of the lemma as follows: If v is a weight of Ho(v,) with I + v E Wp.I, then v E Wv, and there is w1 E Wp with w , . I = 3, and w1.p = I + v. Let us work with this last formulation. Suppose A + v E Wp.p. There is some Let d be the number of reflection hyperplanes alcove C’ with I + v E separating C and C’. Let us use induction on d. If d = 0, then C = C‘. Both p and A + v belong to and are conjugate under Wp, hence equal. So we can take w , = 1 and have v = p - A E Wv,. Suppose now d > 0. There is a wall F of C’ such that C and C’ are on different sides of the reflection hyperplane containing F. There are a E R+ and r E Z such that this hyperplane is given by (x p , a v ) = rp. Let us suppose (x + p , a’) < rp for x E C‘ and (x + p , a’) > rp for x E C . (The other case can be treated similarly.) Set C” = s,,,~. C’. Then C” and C are separated by one reflection hyperplane less than C’ and C , so we can apply induction in case A + v E c .Suppose therefore A + v I$ p,i.e., (A + v + p , u v ) < rp. As I E we have (A p , a’) 2 rp, hence (v, a’) < 0. We can write S,,,~.(A v) E Wp.p in the form A v’, where

c.

e

+

e

+

+

+

+

v’ = s,(v) - ((3, p , a v ) - rp)a = v + (rp - (3- + v + p,a’))a. Therefore v c v’ Is,(v) and v’, too, has to be a weight of Ho(v,), cf. [Hu 13, 21.3. As A v’ E Wp.p n C”, we can apply induction and get v’ E Wv, and the existence of w2 E Wp with w,.A = I and w 2 . p = A + v’. As v’ E W(v,) is an extremal weight of Ho(v,), it is impossible that both v‘ + a and v’ - a are weights of Ho(v,). So the inequalities v < v’ Is,(v) with weights v,s,(v) of Ho(v,) yield v’ = s,(v), hence v E Wv, and (A + p , a v ) = rp. This implies, W ~= S,,,~.(A + v’) = I + v. So finally, I = s,.,~. I = S , , , ~ W ,. I , whereas S , ~ , ~ .p w1 = S , , , ~ W works. ~

+

Remark: Let M be a finite dimensional G-module. a) If all composition factors of M have the form L(v) with v v,, then the lemma together with 7.5(1) implies pr,,(M 0 V ) = 0 for any G-module V with pr, V = V. (One has to apply 7.5(1) to all finite dimensional submodules and then to take direct limits.) b) If [M:L(v,)] = 1 and if all other composition factors L(v) of M satisfy v < v,, then pr,(M 0 V) N pr,,(L(v,) 0 V) for all V as above. This is an immediate consequence of a) and of the exactness of pr,,. It proves the first remark to 7.6.

-=

7.8 Proposition: Let I , p E cz, and let V be a finite dimensional G-module with pr,V = V. We can write ch V = a,X(w.A) with a , E Z and almost

cw,wp

288

Representations of Algebraic Groups

all a, = 0. Then

ch T$V =

(1)

c a,Cx(ww,.p),

wrw,

w1

where w , runs through a system of representatives for Stabwp(A)/ (Stabwp(4 Stabw.(p)).

”

Proof: Take v, as in 7.7 and apply 7.5(1) to M = L(v,).By 7.7.a all weights v of L(v,) with I + v E W,.p belong to Wv,,hence satisfy dim L(v,),= dim L(v,),, = 1. Furthermore, by 7.7.b all such A + v have the w , . p with w1 E Stabw,(p). On the other hand, each w1. p with w, E Stabw,,@)has the form 3, + v with v E Wv,as p - I E Wv,.Therefore the A + v occurring in 7.5(1) are exactly the w,.p with w1 E Stab,(p). Each w l . p occurs only once in 7.5(1), so we have to take a system of representatives as claimed.

7.9 Proposition: Suppose A, p E cz belong to the samefacet. Then T $ induces an equivalence of categories from { G-modules V with pr,V = V } to { G-modules V with prpV = V } . The functor T i T’; is isomorphic to pr,. 0

Proof: The adjointness of T’;and T i (cf. 7.6.b) yields for each G-module V a canonical isomorphism (1)

Hom,(V, T ; T $ V ) ‘Y HomG(T’;V,T’;V).

Let rpv: V + TtT’;V correspond to the identity on T$V. Then V I + rp, is a natural transformation from the functor V H V to T i T $ . Suppose for the moment dim V < co and pr,V = V. Then 7.8 implies ch TiT’;V = ch V, hence T$V # 0 if V # 0, and rp, # 0 in this case. If V is simple, then rpv has to be an isomorphism. The same follows for any V using induction on the length of V. Let V be arbitrary again. If pr,V = V, then the local finiteness implies that rpv is an isomorphism. In general T’;(pr,,V)= 0 for all I‘ E cz,A‘ # A, hence also rpv(pr,.V) = 0 for all these A’. So rp, induces an isomorphism pr,V r TiT’;V. Therefore pr, and T i T’; are isomorphic. This shows that the restriction of T i 0 T’;to {G-modules V with pr,V = V } is isomorphic to the identity, As the assumptions are symmetric in I and p, we get that T $ induces an equivalence of categories as claimed. 0

0

cz and let v1 E X ( T ) +n W(p- A). We want to describe the effect of T’;on G-modules of the form H’(w.A) = R‘ind;(w. A) with w E W,and i E N.

7.10 Let A , p E

289

The Translation Functors

The generalized tensor identity (1.4.8) yields T';H'(w.A) = pr,(L(v,) 0 R'indg(w.1)) (1) N

.

pr,,(R'indg(L(v,) 0 w A)).

Let us take a composition series of L(vl) considered as a B-module and tensor it with w.1. We get a composition series of L(vl) @ w.1, which we denote by 0 = MO

c

MI c M,

c

'*.

c M, = L(V1) Q w.A.

The factors Mj/Mj- have dimension one. There is 1, E X(T )with Mj/Mj- N l j + w.1. The A j are the weights of T on L(vl), counted with their multiplicities. We may and shall choose the composition series of L(vl) in such a way that A j c A,, implies j < j ' . The short exact sequences 0 -,Mj- + Mj -+Mj/Mj- + 0 with 1 5 j 5 r yield long exact sequences (recall 2.1(5))

... +H’(Mj-,)-+H’(Mj)-,H’(llj + w.L)+H'+'(M. J - 1

(2)

) + . a .

hence, using the exactness of pr,,, (3) + pr,,H'(Mj-l) + pr,H'(Mj)

-+

pr,,H'(Lj + w.%) + pr,,H'+'(Mj-,)

-+

* * a

Here

(4)

pr,H'(Aj

+ w.l) =

HZ(dj+ w.2)

if Aj + w.1 otherwise.

E

Wp.p,

For j = r we get in (3) the term pr,H'(M,), which is by (1) just T;H'(w.I). 7.11 There are a few cases where these long exact sequences collapse.

Proposition: Let 1,p E cz,and let F be the facet with 1 E F. If p E F, chen for all w E W, and i E N: TyH'(w.1) N H'(w.p)

Proof: By 7.718 there is exactly one j ' (in the notations of 7.10) with 1, + w . 2 ~Wp.p and it satisfies Aj, + w.A = w.p. Therefore 7.10(3) yields pr,H*(Mj) = 0 for j c j' and pr,H'(Mj) = H'(w.p) for j 2 j ' (and for all I ) , especially for j = r. This implies our claim. 7.12 Proposition: Let 1,p E Cz. Suppose p E C and suppose that there is s E 'c with Stab,@) = { 1, s}. L e t w E W, with ws.p < w.p. Then there is a long exact

290

Representations of Algebraic Groups

sequence of G-modules: 0 -b HO(ws.p)

+ T$HO(W.il) -b

HO(w.p)

H'(ws.p) -+..'

' . . -+Hi(ws.p)-i TrH'(w.3,)-rH'(w .p)-+H'+'(ws.p)-b....

+

Proof: By 7.718 there are exactly two j ' with Ijr w.1 E Wp.p. They satisfy AY + w.A = w.p resp. = W S . ~ .As ws.p c w.p, the j'with 1, w.A= ws.p is smaller than that where we get w.p. Now everything follows from 7.10.

+

7.13 Proposition: L e t 2, and W E W, with W . A E X ( T ) + . Then T $ H o ( w . l ) has a filtration such that the factors are the Ho(wwl.p) with w1 E Stabwp@) and wwl .p E X ( T ) + . Each dinerent wwl.p occurs exactly once. Proof: We know by 7.718 that the A j + w . l in W,.p are exactly the wwl.p with w1 ~ S t a b ~ , ( I )As . w . A E X ( T ) + and as w . 1 ~wwl.C we have ( x + p , a " ) > O f o r all x E w w l . C a n d a E R + . Now w w l . p E w w l . C , hence ( w w l . p + p , a v ) 2 0 for all a E R+ and Hi(wwl.p) = 0 for all i > 0 by 4.5 and 5.4.a. Therefore 7.10(3) implies inductively pr,H'(Mj) = 0 for all I > 0 and all j. Furthermore, we get for Ij + w.A 4 W,.I an isomorphism p r , H 0 ( M j - , ) N pr,Ho(Mj) and for ilj + w.1 E W,.A a short exact sequence 0 -+ p r , H o ( M j - , )

-b

pr,Ho(Mj) -+ HO(1,

+ w.A) -+ 0.

This yields the claim. 7.14 Lemma: Let I, p E Cz such that p belongs to the closure of the facet containing1. Let w E W, with w.1 E X ( T ) + .If w.p $ X ( T ) + ,then TrL(w.I) = 0. I f w.p E X ( T ) + ,then either T r L ( w . 1 ) il! L(w.p) or T$L(w.A) = 0.

Proof: There is a homomorphism (where n = IR'I) H"(wow.1)

N

V(w.A) + HO(w.1)

with image equal to L(w.I),cf. 2.14(1), 4.2(10). The exactness of T $ together with 7.1 1 implies that TSL(w.1) is the image of some homomorphism (1)

H"(w0w.p) -+ HO(w.p).

If w . p # X ( T ) + , then H o ( w . p ) = 0, hence also T $ L ( w . p ) = 0. If W . ~ E X ( T ) + ,then any non-zero homomorphism as in (1) has image L(w.p), cf. the remark to 6.16. This implies what we claim.

The Translation Functors

291

cz

7.15 Proposition: L e t A, p E such that p belongs to the closure of the facet containing A. L e t w E W, with w.2 E X ( T ) + ,and denote b y F the facet with w.2 E F. Then

{i(w*p) fi, w.p E otherwise. if

T’;L(w.A)

N

Proofi By the definition (6.2(3)) of the upper closure w.p $ X ( T ) + implies w.p 4 @.So in this case the claim follows from 7.14. We may assume from now on w.p E X ( T ) + . We know T’;Ho(w.2) N Ho(w.p) by 7.11, so there has to be a composition factor L(w’w.A) with w‘ E Wpof Ho(w.A) such that L(w.p) c Ho(w.p) is a composition factor of T$L(w’w.A). By 7.14 we get more precisely T:L(w’w.A) N L(w’w.p) = L(w.p), hence w’ E W;(w.p) = Stab,,(w.p). Furthermore, L(w’w. A) is the only composition factor L of HO(w.1,) with T’; -L(w.p), as L ( w . p ) occurs with multiplicity one in Ho(w.p). Suppose at first w . p E F. Then each reflection s E W ; ( w . p ) satisfies sw. A 2 w.%.Now [B3],ch.VI,$l,prop. 18implies w’w.A 2 w.Aforal1 W ’ E W;(w.p). Therefore L ( w A) is the only composition factor of H o ( w . A) having the form L(w’w.A) with w‘ E W;(w.p). Therefore the argument from above implies TSL(w.2) N L(w.p). Suppose now w.p 4 fi. Then there is a reflection s‘ E W;(w.p) with s’w.A < w.2. We can find w 1 E W ; ( w . p ) with w1w.2f w.A and sw,w.2 2 w,w.A for all reflections s E W,O(w.p). (If s ‘ w . ~does not do it, then iterate!) We know by the first case that T’;L(w,w.A) N L(w,w.p) = L(w.p). If we can show that L(w,w.A) is a composition factor of Ho(w.A), then the uniqueness statement from above implies T’;L(w.A) = 0 as claimed. Because of 6.23.b it is enough to show that w,w.2 is close to w.A. This, however, follows from 6.22.

.

7.16 Let us denote by QA (for any I module L(A).

E

X ( T ) + ) the injective hull of the G-

Corollary: Let A, p E cz and w E W, with w.p E X ( T ) + . Suppose w.p is in the upper closure of the facet containing w.A. Then TiQ,., z Qw.A. Proofi Being a direct summand of Q , , , @ E for some G-module E, TiQ,., is also an injective G-module (cf. 1.3.10.c), hence a direct sum of G-modules Qw..A with W ’ E W, and w ’ . A e X ( T ) + . Each such Q,..,. occurs exactly dim Hom,(L(w’.A), TiQ,.,) times. By the adjointness of T $and T i : Homc(L(w’.A), TiQ,.,)

N

H o m c ( T $ L ( w ’ .A), Q,.,).

292

Representations of Algebraic Groups

Therefore the multiplicity of Qw..lis 1 for Tf;L(w'.A)=L(w.p), i.e., for w'.A= w. A, and it is 0 otherwise. This implies the claim.

c,

7.17 Corollary: Let A, p E and w E W, with w.3. E X ( T ) + . Suppose that w.p is in the upper closure of the facet containing w.A. a) W e have for all w1 E W, and i E N:

.

.

.

[Hi(w 1 I*):L(w A)] = [Hi(w 1 p):L(w .I ) ] .

b) If ch L(w.A) = x w , E W p a w , w , ~ ( w 'with . 3 , ) almost all aw,w,= 0, then chL(w.p) =

C

U~,~,X(W'.~).

W'EWp

These are immediate consequences of 7.1 1 and 7.15 using the exactness of T $. 7.18 Proposition: Let A E C n X ( T ) and w E W, with w.A E X ( T ) + .Let s E X with w.A < ws.2. Then we have for all w1 E W, and i E N:

.

.

[H'(wl A) :L(w A)] = [H'(wl s. A) : L(w .%)].

Proofi We may assume that 9 G is simply connected, as going to a central extension does not change the multiplicities that we are considering. The existence of a A as above implies p 2 h by 6.2(10), hence there is by 6.3(1) some pE with X o ( A ) = {s},using a notation from 6.3. Now w.A < ws.A implies that w.p is in the upper closure of w.C. As w l . p = wls.p both sides in the proposition are (by 7.17.a) equal to [Hi(wl.p):L(w.p)].

c,

7.19 Lemma: Let A E X ( T )n C and p E Suppose there is s E C with X o ( p ) = (s}. Let w E W, with w.A E X ( T ) + and w.1 < ws.A. a) There is a short exact sequence of G-modules

c,.

0 + Ho(w.A) + T:Ho(w.p) + Ho(ws.A) + 0. b) W e have

L(w.A) N soc T i H o ( w . p ) N soc T i L ( w . p ) N

T:L(w.p)/rad TiL(w.p).

c) W e have [TiL(w.p):L(w.A)] = 2.

Proofi a) This is a special case of 7.13. (We have w.p E X ( T ) + ,as w.p is in the upper closure of w. C.)

293

The Translation Functors

b) Let w‘ E Wpwith w’.A E X ( T ) + .By the adjointness of T $ and T i (7.6.b) and by 7.15 we get

.

Horn,( L (W ’ A), T iH (W ./A)) N

Hom,(T$L(w’.A),H’(w.p))

N

k 0

for w’ = w, otherwise.

.

This implies L(w .A) N soc TjHo(w.p). If we replace Ho(w.p) by L(w p), then the same argument as above implies L(w.2) N soc TiL(w.p). If we look at Hom,(TtL(w.p), L(w’.A)) and argue as before, then we get the last part of the claim. c) We have a unique expression chL(w.1) = ~ u ~ , ~ , ~ ( w ’ . A ) , W‘

summing over all w’ E Wp with w’.A E X ( T ) + and w’.At w.A. We get from 7.17.b that

hence from 7.8(1): ch T;L(w.p)

+

= C U ~ , ~ ~ ~ ( ~W U ~ ’, ~. , ~X ()W ’ S . A ) w‘

W‘

(3) We now have to express each x(w’s.1) in terms of the chL(w”.A) and to determine the coefficient of L(w.1). We have uw,w= 1 and [HO(ws.A):L(w.A)] = 1 by 6.24 and the remark to 6.24. This together with chL(w.A) yields a contribution of 2 to [TiL(w.p):L(w.A)]. Wenowhavetoshowthatallw‘withw‘.Af w.Aandw‘.A # w.Acontribute zero to this multiplicity. We have d(wl.1,) = d(w’.C) < d(w.2) = d(w.C) and d(w’s.1) = d(w’s.C) = d(w‘.C) k 1, as the two alcoves w’ C and w‘s.C are separated only by a common wall. Therefore w. A t w’s iimplies

.

.

-

d(w C) I d(w’s ’ C) I d(w’. C) + 1 I d(w * C), hence d(w.C) = d(w’s C) and w = w’s. But w’ = ws does not occur as w. A t ws .A. This proves (using the strong linkage principle), for all summands aw,w.x(w’s.A)in (3) with w’ # w and w’s .A E X(T)+,that [Ho(w’s.A): L(w.A)] = 0. If w‘s.A 4 X ( T ) + ,then the adjacent alcoves w’.C and w’s.C are separated by a unique hyperplane, the equation of which has to have the form

294

Representations of Algebraic Groups

+ p,/?’) = 0

for some /? E S. Then spw’s.C = w‘ C and x(w’s.I) = -x(w’.I). As w‘.A c w.I thecoefficient of chL(w.A)in x(w’s.A)isagain zero. This implies c).

(x

7.20 The proof of 7.19.c shows (for 2, p, s as there) that we have an equation of the form

ch TkTtL(w.I) = chL(w.I)

+ x(ws.I) + zb,,X(w’.A), W‘

where the last sum is over all w’ E W, with w’.l E X(T)+and d(w’.E.) Id(w.2). The b,. can be computed from the aw,w,in 7.19(1). Suppose we want to get the [Ho(wl.l):L(w2.A)] using induction over d(w, .A), and suppose we have them for all w, with d(w, .I) 5 d(w.I). Then we also know the aw.w,from 7.19(1),hence the bw,in (1).Therefore ( 1 ) tells us that the determination of all [Ho(ws.l):L(w2.1)] is equivalent to that of all [T;1T$L(w.I):L(w2.1)].We may assume w2 # w by 7.19c, hence we have by 7.19.b to determine all [rad TiL(w.p)/soc T i L ( w . p ) : L ( w ,.I)]. Inspired by work of Vogan, and Beilinson and Bernstein on Verma modules for complex semi-simple Lie algebras, one may ask: (2) I s rad TiL(w.p)/soc T;L(w.p) a semi-simple G-module as long as ( w ( I p),a’) Ip ( p - h + 2) for all a E R+?

+

If this is so (for all w and s), then one can compute the multiplicities as above. This is shown in [Andersen 111, 2.12. In fact, [Andersen 113, 2.16 implies that a positive answer to (2) is equivalent to the following conjecture from [Lusztig 31: (3) Let I E X ( T ) n C and w E W, with w.2 E X ( T ) + and ( w ( n + p),a’) I p ( p - h + 2) for all a E R+. Then chL(w.I) =

z(-

l)d(w*a)-d(w‘ ~a’

~w~wo,wwo~~~x~~’.~~

W’

where we sum over all w’ E W, with w’. I E X ( T ) , . Here PX,+ denotes the Kazhdan-Lusztig polynomial for W, as introduced in [Kazhdan and Lusztig 11. The boundary (w(A + p), a’) I p ( p - h + 2) is introduced for the following reason. If w’ E W, with w’.I f w.I and w’.I E X(T)+, then write w’.l = P I , + I2 with &,I2 E X ( T ) + and (A2,/? ) c p for all /? E S.Then L(w’.I) N L ( l , ) [ ’ ]0 L(I,) by Steinberg’s tensor product theorem 3.17. We want to

295

The Translation Functors

have J!,(%~) = Ho(A,), as otherwise the formulas cannot have the amount of independence of p as they have in (3). Now the bound in (3) insures that we can apply 5.6 to I,, hence that indeed Ho(A,) = L(Al). Let me mention some other evidence for the truth of (3). In [Kato 21 it is proved that (3) is compatible with Steinberg's tensor product theorem. If the filtrations of the H o ( p )to be introduced in Chapter 8 are compatible (in some sense) with certain homomorphisms, then (3) holds according to [Andersen 123, where some ideas of Gabber and Joseph are carried over to our situation. 7.21 Proposition: Let A E C n X ( T )and w E W,, s E C with w. A E X(T)+and w.A < ws.A. a) One has for all w' E Wpwith w'.A E X(T)+and wl.1 < w's.1 and for all

iEN Ext b(L(w's. A), Ho(w. A))

.

'

Ext b- (L(w's A), Ho(ws. A)).

N

b) We have for all i E N

.

.

Ext b(L(ws A), H O (w I))Y c) W e have for all i E N Ext;(HO(ws.A), Ho(w. A))

1:

I

I

k 0

k 0

if if

i=l, i#1.

for i = O,l, otherwise.

Proof: a) We may suppose that there is some p E cz with C o ( p ) = {s}, cf. the proof of 7.18. Apply HOmG(L(W'S.A),?) and HOmG(Ho(W'S.A),?) to the short exact sequence in 7.19.a. By the adjointness of T $and T i and by 7.15 we get

.

.

.

Ex t b(L(w Is. A), T i H O( w p)) N Ext (; T $ L (w 's A), H O( w p)) = Ext~(O,HO(w.p))= 0.

This implies a). b) This follows from a) and from 4.13. c) We have

.

.

Ex t b(H (ws A), T H (w p)) Y

Extb(TXHo(ws.I), Ho(w.p))

Y

Extb(Ho(w.p), Ho(w.p)) =

k 0

for i = 0, for i > 0,

296

Representations of Algebraic Group

by the adjointness as in a), by 7.1 1, by 2.8, and by 6.20. We get for i > 1, using 7.19.a and 6.20, Extk(Ho(ws.A),Ho(~A))

N

Extk-l(Ho(~s.A),Ho(~s.A)) = 0.

Furthermore, we have an exact sequence 0 -+ HOmG(Ho(WS.A),Ho(W-A))+ HOmG(H0(WS.il),TtHo(W.p)) (1)

+ HOmG(Ho(WS.A),Ho(WS*

A)) + Ext6(Ho(ws.i),Ho(W.A))

-+

0.

The two terms in the middle are isomorphic to k, by the computation above and by 2.8. Any homomorphism cp: Ho(ws.A) -+ TtHo(w.p) annihilates soc Ho(ws.A) = L(ws.2) as SOC T ; H o ( w . p )N L(w A) by 7.19. Therefore L(ws.A) is no composition factor of im($ 0 cp), where $: TiHo(w.p) -+ Ho(ws.A) is the map in 7.19.a. But L(ws.A) = soc Ho(ws.A), hence $ 0 cp = 0. Therefore in (1) the middle map is zero, hence all four terms are isomorphic to k. Remarks: 1) The fact that HomG(Ho(ws.A),Ho(w.A)) N k is also a special case of 6.24(2). 2) One problem related to 7.20(2) is the question of whether Ext;(L(ws.l), L(w A)) # 0 for w.A as in 7.20(2), cf. [Andersen 11],2.11.

-

8

Filtrations of Weyl Modules

Assume that k is a field of characteristic p # 0. In this chapter we construct for any 3, E X ( T ) + a filtration V(A)= v(3,)O2 v(3,)'2 V(J.), 2 ... of the Weyl module V(3,),with V(A)' = 0 for large i and with V(A)/V(3,)'1: L(A). Furthermore, we have an explicit formula for x i , o c h V(3,)i,cf. 8.19. This sum formula together with results in earlier chapters is, so far, the most efficient tool for computing the ch L ( p ) with p E X(T)+. One gets them all for low rank groups (of type A , , A , , A , , B , , G,). If these filtrations are compatible with certain homomorphisms between Weyl modules, then one can compute all ch L(A) for p large enough. (This is proved in [Andersen 123 and I do not go into this here.) The filtration arises from a filtration of V(3,), and the sum formula follows from a result over Z. So we are concerned in this chapter most of the time with representations of GZ or, more generally, of G, for some Dedekind ring A of characteristic 0. (The assumption char(A) = 0 can be dropped at the beginning.) Denote by K the field of fractions of A . We start (8.2) by a criterion for an A-lattice in a G,-module to be G,-stable. This is then applied to the V(A),. We determine in 8.4 the annihilator of a highest weight vector in V(A). This can be used to prove that certain line bundles are very ample. We consider the induction functor from B, to G, and its derived functors (denoted by H L , i 2 0). We discuss for which 3, and i the G,-modules Hi(3,)

29 7

298

Representations of Algebraic Groups

are torsion modules over A and what the torsion free quotient looks like (8.7). We generalize Kempf's vanishing theorem to A (8.8) and identify V(A), in 8.9 with H;(w,,.A) where n = IR'I. The filtrations are constructed using a homomorphism cp: V(A), + Hi@) by taking V(A& = ( u E V(A), cp(u) E p ' H i ( A ) } and V(A)' as the image of V(A& 0, k in V(A)= V(n), OZk. The sum formula follows from formulas for the determinant of this map on each weight space. In the case where G has semi-simple rank 1, one can write down the map explicitly and then read off the determinant (8.13/ 14). The general case is reduced to this special case (8.15/16). The filtration can also be (and was, in fact, at first) constructed using a symmetric bilinear form (the "contravariant" form) on V(A),. This other method is discussed in 8.19. This bilinear form was first mentioned in [St 11 and independently in [Wong 11. Both authors prove: If we reduce the bilinear form modulo p , then its radical is the radical of the G-module V(A),so the quotient by this radical is ,!,(A). For G of type A , the determinants of this bilinear form were computed in [Jantzen 11. There was also a conjecture (in that paper) for the general case. In [Jantzen 31 it was proved that the p-adic valuation of the determinant is as conjectured if p 2 h. That one can use the bilinear form to construct filtrations is one of the main ideas behind [Jantzen 31, but is made explicit only in [Jantzen 51. The proof for all p together with the new construction via homomorphisms and reduction to the case of semi-simple rank 1 is due to [Andersen 123, where one can also find some results on arbitrary H'(")(w.A). For all other non-trivial results one can find references within the text. At the end of this chapter (8.23) we discuss some special aspects of the representation theory of the general linear groups, especially some relationship with the representation theory of the symmetric groups.

I

8.1 Let A be an integral domain and K its field of fractions. We can identify A[G,] resp. Dist(G,) with subalgebras of K[GK] N A[GA] 0, K resp. Dist(G,) N Dist(G,) @ , K , as these algebras are flat over A , cf. 1.1, 1.12. Similar statements hold with G replaced by the big cell Uf;B,, cf. 1.10. Recall that A[GA] is a subalgebra of A[U;B,] and that Dist(G,) = Dist(U:B,, l), as U i B , is open in G,. (We can also replace A by K in these statements.) Furthermore, (cf. 1.9(5)) (1)

A[G,] = K[GK] n A[UIB,].

Using this we can show (2) A[GA] = {f E K[G,]

1 p(f) E A

for all

p E Dist(G,)}.

299

Filtrations of Weyl Modules

By definition A [G,] is contained in the right hand side of (2). In order to get the other inclusion it is by (1) enough to show that the right hand side of (2) is contained A [U: B,]. So we want to prove that the scheme X = U: BA satisfies (3)

A [ x ] = {f E K [ & ] ( p ( f ) E A

for all p E Dist(X, I)}.

Now X is a direct product of schemes isomorphic to Go,, or Gm,A.Using the explicit description of Dist(G,) and Dist(G,) in 1.7.3/8, an elementary computation (left to the reader) yields (3). 8.2 According to our general conventions from Chapter 1 the group G satisfies the assumptions of Proposition 1.10.12and we might have deduced 8.1(2)from that. I have, however, preferred the above approach, as the smoothness as G is

really proved by arguments like those leading to 1.9(5) and 8.1(2). As in 1.10.13 we get from 8.1(2): (1) Let V be a Jinite dimensi