The Local Langlands Conjecture for GL

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics Series editors M. Berger...

Author: Colin J. Bushnell | Guy Henniart

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Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

Series editors M. Berger B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. Hörmander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates

S.R.S. Varadhan

335

Colin J. Bushnell . Guy Henniart

The Local Langlands Conjecture for GL(2)

ABC

Colin J. Bushnell King's College London Department of Mathematics Strand, London WC2R 2LS UK e-mail: [email protected]

Guy Henniart Université de Paris-Sud et umr 8628 du CNRS Département de Mathématiques Bâtiment 425 91405 Orsay France e-mail: [email protected]

Library of Congress Control Number: 2006924564 Mathematics Subject Classification (2000): Primary: i1F70, 22E50, 20G05 Secondary: i1D88, 11F27, 22C08 ISSN 0072-7830 ISBN-10 3-540-31486-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-31486-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A EX macro package Typesetting: by the author and SPi using a Springer LT Cover design: design & production GmbH, Heidelberg

Printed on acid-free paper

SPIN: 11604440

41/SPI

543210

For Elisabeth and Lesley

Foreword

This book gives a complete and self-contained proof of Langlands’ conjecture concerning the representations of GL(2) of a non-Archimedean local ﬁeld. It has been written to be accessible to a doctoral student with a standard grounding in pure mathematics and some extra facility with local ﬁelds and representations of ﬁnite groups. It had its origins in a lecture course given by the authors at the ﬁrst Beijing-Zhejiang International Summer School on p-adic methods, held at Zhejiang University Hangzhou in 2004. We hope this is found a ﬁtting response to the eﬀorts of the organizers and the enthusiastic contribution of the student participants. King’s College London and Universit´e de Paris-Sud.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes for the reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 4

1

Smooth Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. Locally Proﬁnite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. Smooth Representations of Locally Proﬁnite Groups . . . . . . . . . 13 3. Measures and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4. The Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2

Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5. Linear Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6. Representations of Finite Linear Groups . . . . . . . . . . . . . . . . . . . . 45

3

Induced Representations of Linear Groups . . . . . . . . . . . . . . . . . 7. Linear Groups over Local Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Representations of the Mirabolic Group . . . . . . . . . . . . . . . . . . . . 9. Jacquet Modules and Induced Representations . . . . . . . . . . . . . . 10. Cuspidal Representations and Coeﬃcients . . . . . . . . . . . . . . . . . . 10a. Appendix: Projectivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Intertwining, Compact Induction and Cuspidal Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

49 50 56 61 69 73 76

Cuspidal Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 12. Chain Orders and Fundamental Strata . . . . . . . . . . . . . . . . . . . . . 86 13. Classiﬁcation of Fundamental Strata . . . . . . . . . . . . . . . . . . . . . . . 95 14. Strata and the Principal Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 15. Classiﬁcation of Cuspidal Representations . . . . . . . . . . . . . . . . . . 105 16. Intertwining of Simple Strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 17. Representations with Iwahori-Fixed Vector . . . . . . . . . . . . . . . . . 115

X

Contents

5

Parametrization of Tame Cuspidals . . . . . . . . . . . . . . . . . . . . . . . . 123 18. Admissible Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 19. Construction of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 20. The Parametrization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 21. Tame Intertwining Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 22. A Certain Group Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6

Functional Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 23. Functional Equation for GL(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 24. Functional Equation for GL(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 25. Cuspidal Local Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 26. Functional Equation for Non-Cuspidal Representations . . . . . . . 162 27. Converse Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7

Representations of Weil Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 28. Weil Groups and Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 180 29. Local Class Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 30. Existence of the Local Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 31. Deligne Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 32. Relation with -adic Representations . . . . . . . . . . . . . . . . . . . . . . . 201

8

The Langlands Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 33. The Langlands Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 34. The Tame Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 35. The -adic Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

9

The Weil Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 36. Whittaker and Kirillov Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 37. Manifestation of the Local Constant . . . . . . . . . . . . . . . . . . . . . . . 230 38. A Metaplectic Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 39. The Weil Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 40. A Partial Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

10 Arithmetic of Dyadic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 41. Imprimitive Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 42. Primitive Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 43. A Converse Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 11 Ordinary Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 44. Ordinary Representations and Strata . . . . . . . . . . . . . . . . . . . . . . 267 45. Exceptional Representations and Strata . . . . . . . . . . . . . . . . . . . . 279

Contents

XI

12 The Dyadic Langlands Correspondence . . . . . . . . . . . . . . . . . . . . 285 46. Tame Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 47. Interior Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 48. The Langlands-Deligne Local Constant modulo Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 49. The Godement-Jacquet Local Constant and Lifting . . . . . . . . . . 304 50. The Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 51. Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 52. Octahedral Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 13 The Jacquet-Langlands Correspondence . . . . . . . . . . . . . . . . . . . 325 53. Division Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 54. Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 55. Functional Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 56. Jacquet-Langlands Correspondence . . . . . . . . . . . . . . . . . . . . . . . . 334 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Some Common Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Some Common Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Introduction

We work with a non-Archimedean local ﬁeld F which, we always assume, has ﬁnite residue ﬁeld of characteristic p. Thus F is either a ﬁnite extension of the ﬁeld Qp of p-adic numbers or a ﬁeld Fpr ((t)) of formal Laurent series, in one variable, over a ﬁnite ﬁeld. The arithmetic of F is encapsulated in the Weil group WF of F : this is a topological group, closely related to the Galois group of a separable algebraic closure of F , but with rather more sensitive properties. One investigates the arithmetic via the study of continuous (in the appropriate sense) representations of WF over various algebraically closed ﬁelds of characteristic zero, such as the complex ﬁeld C or the algebraic closure Q of an -adic number ﬁeld. Sticking to the complex case, the one-dimensional representations of WF are the same as the characters (i.e., continuous homomorphisms) F × → C× : this is the essence of local class ﬁeld theory. The n-dimensional analogue of a character of F × = GL1 (F ) is an irreducible smooth representation of the group GLn (F ) of invertible n × n matrices over F . As a speciﬁc instance of a wide speculative programme, Langlands [55] proposed, in a precise conjecture, that such representations should parametrize the n-dimensional representations of WF in a manner generalizing local class ﬁeld theory and compatible with parallel global considerations. The excitement provoked by the local Langlands conjecture, as it came to be known, stimulated a period of intense and widespread activity, reﬂected in the pages of [8]. The ﬁrst case, where n = 2 and F has characteristic zero, was started in Jacquet-Langlands [46]; many hands contributed but Kutzko, bringing two new ideas to the subject, completed the proof in [52], [53]. Subsequently, the conjecture has been proved in all dimensions, ﬁrst in positive characteristic by Laumon, Rapoport and Stuhler [58], then in characteristic zero by Harris and Taylor [38], also by Henniart [43] on the basis of an earlier paper of Harris [37].

2

Introduction

Throughout the period of this development, the subject has largely remained conﬁned to the research literature. Our aim in this book is to provide a navigable route into the area with a complete and self-contained account of the case n = 2, in a tolerable number of pages, relying only on material readily available in standard courses and texts. Apart from a couple of unavoidable caveats concerning Chapter VII, we assume only the standard representation theory for ﬁnite groups, the beginnings of the theory of local ﬁelds and some very basic notions from topology. In consequence, our methods are entirely local and elementary. Apart from Chapter I (which could equally serve as the start of a treatise on the representation theory of p-adic reductive groups) and some introductory material in Chapter VII, we eschew all generality. Whenever possible, we exploit special features of GL(2) to abbreviate or simplify the arguments. The desire to be both compact and complete removes the option of appealing to results derived from harmonic analysis on ad`ele groups (“base change” [57], [1]) which originally played a determining rˆ ole. This particular constraint has forced us to give the ﬁrst proof of the conjecture that can claim to be completely local in method. There is an associated loss, however. The local Langlands Conjecture is just a speciﬁc instance of a wide programme, encompassing local and global issues and all connected reductive algebraic groups in one mighty sweep. Beyond the minimal gesture of Chapter XIII, we can give the reader no idea of this. Nor have we mentioned any of the geometric methods currently necessary to prove results in higher dimensions. Fortunately, the published literature contains many ﬁne surveys, from Gelbart’s book [32], which still conveys the breadth and excitement of the ideas, to the new directions described in [4]. The approach we take is guided by [46] and [50–53], but we have rearranged matters considerably. We have separated the classiﬁcation of representations from the functional equation. We have imported ideas of Bernstein and Zelevinsky into the discussion of non-cuspidal representations. While the treatment of cuspidal representations is essentially that of Kutzko, it is heavily informed by hindsight. We have given precedence to the Godement-Jacquet version of the functional equation and so had to treat the Converse Theorem in a novel manner, owing something to ideas of G´erardin and Li. There is also some degree of novelty in our treatment of the Kirillov model and the relation between the functional equation it gives and that of Godement and Jacquet. We have given a quick and explicit proof of the existence of the Langlands correspondence, in the case p = 2, at an early stage. The case p = 2 has many pages to itself. The method is essentially that of Kutzko, but we have had to bring a new idea to the closing pages (the treatment of the so-called octahedral representations) to avoid an appeal to

Notation

3

base change. We regard this case as being particularly important. It remains the one instance of the local conjecture in which the detail is suﬃciently complex to be interesting, yet suﬃciently visible to illuminate the miracle that is the Langlands correspondence. Even after 25 years, it stands as a sturdy corrective to over-optimistic attitudes to more general problems. As light relief, we have broadened the picture with some discussion of -adic representations, since these provide a forum in which the correspondence ﬁnds much of its application. The ﬁnal Chapter XIII stands outside the main sequence. There, D is the quaternion division algebra over F . The irreducible representations of D× = GL1 (D) can be classiﬁed by a method parallel to that used for GL2 (F ). The Jacquet-Langlands correspondence provides a canonical connection between the representation theories of D× and GL2 (F ). We include it as an indication of further dimensions in the subject. Given the experience of GL2 (F ), it is a fairly straightforward matter which we have left as a sequence of exercises. Acknowledgement. The ﬁnal draft was read by Corinne Blondel, whose acute comments led us to remove a large number of minor errors and obscurities, along with a couple of more signiﬁcant lapses. It is a pleasure to record our debt to her. Notation We list some standard notations which we use repeatedly, without always recalling their meaning. F o p k UF

= a non-Archimedean local ﬁeld; = the discrete valuation ring in F ; = the maximal ideal of o; = o/p; p = the characteristic of k; q = |k|; = the group of units of o; UFn = 1+pn , n 1.

(Thus the characteristic of F is 0 or p: we never need to impose any further restriction.) In addition, υF : F × → Z is the normalized (surjective) additive valuation and x = q −υF (x) . We denote by µF the group of roots of unity in F of order prime to p. If E/F is a ﬁnite ﬁeld extension, we use the analogous notations oE , pE , etc. The norm map E × → F × is denoted NE/F , and the trace E → F is TrE/F . The ramiﬁcation index and the residue class degree are e(E|F ), f (E|F ) respectively. The discriminant is dE/F = pd+1 , d = d(E|F ). The symbol tr is reserved for the trace of an endomorphism, such as a matrix or a group representation, and det is invariably the determinant. If R is a ring with 1, R× is its group of units and Mn (R) is the ring of n × n matrices over R. When R is commutative, GLn (R) (resp. SLn (R)) is the

4

Introduction

group of n × n matrices over R which are invertible (resp. of determinant 1). We use the notation B, T , N , Z for the subgroups of GL2 of matrices of the form ∗∗ ∗ 0 1∗ a 0 , , , 0 ∗

0 ∗

0 1

0 a

respectively. Unless otherwise speciﬁed, A = M2 (F ) and G = GL2 (F ). Notes for the reader Prerequisites. We assume the beginnings of the representation theory of ﬁnite groups, including Mackey theory: the ﬁrst 11 sections of [77] cover it all, bar a couple of results requiring reference to [26]. Of non-Archimedean local ﬁelds, we need general structure theory as far as the discriminant and structure of tame extensions, plus behaviour of the norm in tame or quadratic extensions. Practically everything can be found in [30] or the ﬁrst two parts of [76], while [87] is the source of many of the ideas here. From topology and measure theory, beyond the most elementary concepts, we cover practically everything we need. All this material is commonly available in many books: we mention only personal favourites. From Chapter VII onwards, we rely on local class ﬁeld theory. No detail is involved, so we have been able to take an axiomatic approach. The reader might consult the compact [68] or [74], [76]. More serious is the treatment in §30 of the existence of the Langlands-Deligne local constant. This depends on an interplay between local and global ﬁelds using some deep (but classical) theorems. The reader could again take an axiomatic approach. We have included a brief account which is complete modulo the classical background. (The requisite material is in [68] or [54].) Navigation. Sections are numbered consecutively throughout the book. Each section is divided into (usually) short paragraphs, numbered in the form y.z. A reference y.z Proposition means the (only) proposition in paragraph y.z. Chapter I stands alone, and could serve as an introduction to much wider areas. Chapter II is elementary, and could be read ﬁrst. Parts of Chapters VII and X can be read independently. Chapter XIII could be read directly after Chapter VI. Otherwise, the logical dependence is linear and fairly rigid. Principal series (or non-cuspidal) representations form a distinct subtheme. At a ﬁrst reading, this could be edited out or pursued exclusively, according to taste. (For a diﬀerent approach, emphasizing non-cuspidal representations and their importance for L-functions, see Bump’s book [10].) Another “short course” option would be to stop at the end of Chapter VIII, by which stage the argument is complete for all but dyadic ﬁelds F .

Notes for the reader

5

Exercises. A few exercises are scattered through the text. These are intended to illuminate, entertain, or to indicate directions we do not follow. Only the simpler ones ever make a serious contribution to the main argument. Notes. We have appended brief notes or comments to some chapters, to indicate further reading or wider perspectives. They tend to pre-suppose greater experience than the main text. History. We have written an account of the subject, not its history: that would be a separate project of comparable scope. We have made no attempt at a complete bibliography. We have cited sources of major importance, and those we have found helpful in the preparation of this volume. We have also mentioned a number of recent works, along with older ones that, in our opinion, remain valuable to one learning the subject.

1 Smooth Representations

1. 2. 3. 4.

Locally profinite groups Smooth representations of locally profinite groups Measures and duality The Hecke algebra

This chapter is introductory and foundational in nature. We deﬁne a class of topological groups, the locally proﬁnite groups, and study their smooth representations on complex vector spaces. These representations are often inﬁnite-dimensional, but smoothness imposes a drastic continuity condition. Nontheless, this class of objects is quite wide: it includes, for example, all representations of discrete groups. We start by recalling some standard facts. We then develop the elementary aspects of smooth representation theory, very much guided by the ordinary representation theory of ﬁnite groups. We occasionally turn to nonArchimedean local ﬁelds as a source of examples. The topic of Haar measure and integration on topological groups necessarily enters the picture. Since we have only to deal with locally proﬁnite groups, this is a straightforward matter of which we give just as much as we need. While we will ultimately be concerned only with non-Archimedean local ﬁelds F and associated groups like GL2 (F ), there is nothing to be gained from specialization at this stage. Looking beyond the conﬁnes of the present book, there is much to be lost. We therefore work, throughout this chapter, in quite extreme generality.

8

1 Smooth Representations

1. Locally Proﬁnite Groups In this section, we introduce and brieﬂy discuss the notion of a locally proﬁnite group. We concentrate on showing how this framework accommodates the non-Archimedean local ﬁelds and some associated groups and rings. We give very few proofs in the ﬁrst four paragraphs: it is more a case of gathering together the pre-requisite threads. We conclude the section with a couple of paragraphs about various characters associated with a non-Archimedean local ﬁeld F . We make unceasing use of this material in the later chapters. More immediately, it gives us some examples to illuminate the general theory of the following sections.

1.1. Deﬁnition. A locally proﬁnite group is a topological group G such that every open neighbourhood of the identity in G contains a compact open subgroup of G. For example, any discrete group is locally proﬁnite. A closed subgroup of a locally proﬁnite group is locally proﬁnite. The quotient of a locally proﬁnite group by a closed normal subgroup is locally proﬁnite. A locally proﬁnite group is locally compact. If it is compact, it is proﬁnite in the usual sense, that is, the limit of an inverse system of ﬁnite discrete groups. In fact, if G is a compact locally proﬁnite group, it is not hard to show directly that the obvious map G −→ lim G/K ←−

is a topological isomorphism, where K ranges over the open normal subgroups of G. In general, any open neighbourhood of 1 in a locally proﬁnite group G contains a compact open subgroup K of G. As we have just seen, K is proﬁnite: the terminology is therefore apt. Remark. A locally proﬁnite group is locally compact and totally disconnected. In the converse direction, it is known that a compact, totally disconnected topological group is proﬁnite. Likewise, a locally compact, totally disconnected group is locally proﬁnite, but we shall make no use of that fact. 1.2. Let F be a non-Archimedean local ﬁeld. Thus F is the ﬁeld of fractions of a discrete valuation ring o. Let p be the maximal ideal of o and k = o/p the residue class ﬁeld. We will always assume that k is ﬁnite, and we will generally denote the cardinality |k| by q.

1. Locally Proﬁnite Groups

9

Let be a prime element of F , that is, an element such that o = p. Every element x ∈ F × can be written uniquely as x = un , for some unit u ∈ o× = UF , and some n ∈ Z. (We use the notation n = υF (x).) The ﬁeld F carries an absolute value x = q −n = q −υF (x) ,

0 = 0,

giving a metric on F , relative to which F is complete. In the metric space topology, F is a topological ﬁeld. The fractional ideals pn = n o = {x ∈ F : x q −n },

n ∈ Z,

are open subgroups of F and give a fundamental system of open neighbourhoods of 0 in F . Combining the deﬁnition of the topology with the completeness property, one sees that the canonical map o −→ lim

←− n1

o/pn

is a topological isomorphism. Since k is ﬁnite, each group o/pn is ﬁnite, and the limit is compact. Each fractional ideal pn , n ∈ Z, is isomorphic to o and so is compact. We conclude: Proposition. The additive group F is locally proﬁnite, and F is the union of its compact open subgroups. 1.3. The multiplicative group F × is likewise a locally proﬁnite group: the congruence unit groups UFn = 1+pn , n 1, are compact open, and give a fundamental system of open neighbourhoods of 1 in F × . 1.4. Let n 1 be an integer. The vector space F n = F × · · · × F carries the product topology, relative to which it is a locally proﬁnite group. As a special case, the matrix ring Mn (F ) is a locally proﬁnite group under addition, in which multiplication of matrices is continuous. The group G = GLn (F ) is an open subset of Mn (F ); inversion of matrices is continuous, so G is a topological group. The subgroups K = GLn (o),

Kj = 1 + pj Mn (o),

j 1,

are compact open, and give a fundamental system of open neighbourhoods of 1 in G. Thus G = GLn (F ) is a locally proﬁnite group. More generally, let V be an F -vector space of ﬁnite dimension n. The choice of a basis gives an isomorphism V ∼ = F n , which we use to impose a topology on V . This topology is independent of the choice of basis. The remarks above apply equally to the algebra EndF (V ) and the group AutF (V ).

10

1 Smooth Representations

1.5. Let V be an F -vector space of ﬁnite dimension n. An o-lattice in V is a ﬁnitely generated o-submodule L of V such that the F -linear span F L of L is V . Proposition. Let L be an o-lattice in V . There is an F -basis {x1 , x2 , . . . , xn } n of V such that L = i=1 oxi . Proof. By deﬁnition, L has a ﬁnite o-generating set. Choose a minimal such set {y1 , y2 , . . . , ym }: we show that this is an F -basis of V . It certainly spans V . Suppose it is linearly dependent over F : ai yi = 0, 1im

with ai ∈ F , not all zero. We can multiply through by an element of F × and assume that all ai ∈ o and that at least one of them, aj say, is a unit of o. Thus yj is an o-linear combination of the other yi , contrary to the minimality hypothesis. In particular, an o-lattice L is a compact open subgroup of V . The o-lattices in V give a fundamental system of open neighbourhoods of 0 in V . More generally, a lattice in V is a compact open subgroup of V . Here we have: Lemma. Let L be a subgroup of V ; then L is a lattice in V if and only if there exist o-lattices L1 , L2 in V such that L1 ⊂ L ⊂ L2 . Proof. Suppose L1 ⊂ L ⊂ L2 , where the Li are o-lattices. Since L contains L1 , it is open and hence closed. Since L is contained in L2 , it is compact. Conversely, if L is a lattice in V , it must contain an o-lattice (since L is an open neighbourhood of 0), and so F L = V . In the opposite direction, we choose a basis {x1 , . . . , xn } of V . The image of L under the obvious projection V → F xi is a compact open subgroup of F xi . It is therefore contained in a group of the form ai xi , for some fractional ideal ai = pai of o. Thus L ⊂ oL ⊂ i ai xi , and this is an o-lattice. 1.6. Let G be a locally proﬁnite group. Proposition. Let ψ : G → C× be a group homomorphism. The following are equivalent: (1) ψ is continuous; (2) the kernel of ψ is open. If ψ satisﬁes these conditions and G is the union of its compact open subgroups, then the image of ψ is contained in the unit circle |z| = 1 in C.

1. Locally Proﬁnite Groups

11

Proof. Certainly (2) ⇒ (1). Conversely, let N be an open neighbourhood of 1 in C. Thus ψ −1 (N ) is open and contains a compact open subgroup K of G. However, if N is chosen suﬃciently small, it contains no non-trivial subgroup of C× and so K ⊂ Ker ψ. The unit circle S 1 is the unique maximal compact subgroup of C× . If K is a compact subgroup of G, then ψ(K) is compact, and so it is contained in S 1 . The ﬁnal assertion follows. We deﬁne a character of a locally proﬁnite group G to be a continuous homomorphism G → C× . We usually write 1G , or even just 1, for the trivial (constant) character of G. We call a character of G unitary if its image is contained in the unit circle. 1.7. We will later make frequent use of another property of the local ﬁeld F . The set of characters of F is a group under multiplication; we denote it F. Since F is the union of its compact open subgroups pn , n ∈ Z, all characters of F are unitary (1.6 Proposition). If ψ is a character and ψ = 1, there is a least integer d such that pd ⊂ Ker ψ. Deﬁnition. Let ψ ∈ F, ψ = 1. The level of ψ is the least integer d such that pd ⊂ Ker ψ. If we ﬁx d, the set of characters of F of level d is the subgroup of ψ ∈ F such that ψ | pd = 1. Proposition (Additive duality). Let ψ ∈ F, ψ = 1, have level d. (1) Let a ∈ F . The map aψ : x → ψ(ax) is a character of F . If a = 0, the character aψ has level d−υF (a). (2) The map a → aψ is a group isomorphism F ∼ = F. Proof. Part (1) is immediate, and a → aψ is an injective group homomorphism F → F. Let θ ∈ F, θ = 1, and let l be the level of θ. Let be a prime element of F , and u ∈ UF . The character ud−l ψ has level l, and so agrees with θ on pl . The characters ud−l ψ, u d−l ψ, u, u ∈ UF , agree on pl−1 if and only if u ≡ u (mod p). The group pl−1 has q−1 non-trivial characters which are trivial on pl . As u ranges over UF /UF1 , the q−1 characters ud−l ψ | pl−1 are distinct, non-trivial, but trivial on pl . Therefore one of them, say u1 d−l ψ | pl−1 , equals θ | pl−1 . Iterating this procedure, we ﬁnd a sequence of elements un ∈ UF such that un d−l ψ agrees with θ on pl−n and un+1 ≡ un (mod pn ). The Cauchy sequence {un } converges to some u ∈ UF and we have θ = ud−l ψ. Exercise. Let L be a lattice in F and let χ be a character of L (in the sense of 1.6). Show there exists a character ψ of F such that ψ | L = χ.

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1.8. We turn to the multiplicative group F × . Let χ be a character of F × . By 1.6 Proposition, χ is trivial on UFm , for some m 0. Deﬁnition. Let χ be a non-trivial character of F × . The level of χ is deﬁned to be the least integer n 0 such that χ is trivial on UFn+1 . We use the same terminology for characters of open subgroups of F × . Observe that a character of F × need not be unitary: for example, the map x → x is a character. Note also that, in a related contrast to the additive case, F × has a unique maximal compact subgroup, namely UF . The structure of the group of characters of F × is more subtle than that of F. However, we shall make frequent use of a partial description in additive terms. Let m, n be integers, 1 m < n 2m. The map x → 1+x gives an isomorphism pm /pn ∼ = UFm /UFn . This gives an isomorphism of character groups m n m n ∼ (p /p ) = (UF /UF ), and we can use 1.7 to describe the group (pm /pn ). For this purpose, it is convenient to ﬁx a character ψF ∈ F of level 1. For a ∈ F , we deﬁne a function

ψF,a : F −→ C× , ψF,a (x) = ψF (a(x−1)).

(1.8.1)

Proposition 1.7 then yields: Proposition. Let ψ ∈ F have level 1. Let m, n be integers, 0 m < n 2m+1. The map a → ψF,a | UFm+1 induces an isomorphism p−n /p−m

≈ −−−− −→

UFm+1 /UFn+1 ).

Observe that, viewed as a character of UFm+1 , the function ψF,a has level −υF (a). Also, the condition relating m and n can be re-formulated as n2 m < n, where x → [x] denotes the greatest integer function. Terminology. We will use analogues of the notion of level, as deﬁned in this paragraph, in many contexts where we study representations of groups with a ﬁltration indexed by the non-negative integers. As we shall see, it is very convenient. From this more general viewpoint, the deﬁnition in 1.7 for characters of F appears anomalous. This version is forced on us by a variety of historical conventions: the only point on which these agree is that a character of F of level zero must be trivial on o but not on p−1 . This is so ﬁrmly established that it would be confusing to change it now.

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13

2. Smooth Representations of Locally Proﬁnite Groups In this section, we introduce the notion of a smooth representation of a locally proﬁnite group G. We develop the basic theory, along lines familiar from the ordinary representation theory of ﬁnite groups. New phenomena do arise, but the general outline is very similar to that of the standard theory. 2.1. Let G be a locally proﬁnite group, and let (π, V ) be a representation of G. Thus V is a complex vector space and π is a group homomorphism G → AutC (V ). The representation (π, V ) is called smooth if, for every v ∈ V , there is a compact open subgroup K of G (depending on v) such that π(x)v = v, for all x ∈ K. Equivalently, if V K denotes the space of π(K)-ﬁxed vectors in V , then V K, V = K

where K ranges over the compact open subgroups of G. In practice, we will usually have to deal with representations of inﬁnite dimension. A smooth representation (π, V ) is called admissible if the space V K is ﬁnite-dimensional, for each compact open subgroup K of G. Let (π, V ) be a smooth representation of G; then any G-stable subspace of G provides a further smooth representation of G. Likewise, if U is a G-subspace of V , the natural representation of G on the quotient V /U is smooth. One says that (π, V ) is irreducible if V = 0 and V has no G-stable subspace U , 0 = U = V . For smooth representations (πi , Vi ) of G, the set HomG (π1 , π2 ) is just the space of linear maps f : V1 → V2 commuting with the G-actions: f ◦ π1 (g) = π2 (g) ◦ f,

g ∈ G.

(2.1.1)

With this deﬁnition, the class of smooth representations of G forms a category Rep(G). We remark that the category Rep(G) is abelian. We say that two smooth representations (π1 , V1 ), (π2 , V2 ) of G are isomorphic, or equivalent, if there exists a C-isomorphism f : V1 → V2 satisfying (2.1.1). Example 1. A character χ of G (1.6) can be viewed as a representation χ : G → C× = AutC (C). The representation (χ, C) is smooth. A one-dimensional representation of G is smooth if and only if it is equivalent to a representation deﬁned by a character of G. Indeed, the set of isomorphism classes of onedimensional smooth representations of G is in canonical bijection with the group of characters of G.

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Example 2. Suppose that G is compact, hence proﬁnite. Let (π, V ) be an irreducible smooth representation of G. The space V is then ﬁnite-dimensional. For, if v ∈ V , v = 0, then v ∈ V K , for an open subgroup K of G. The index (G : K) is necessarily ﬁnite, and the set {π(g)v : g ∈ G/K} spans V . Further, if K = g∈G/K gKg −1 , then K is an open normal subgroup of G of ﬁnite index, acting trivially on V . Thus V is eﬀectively an irreducible representation of the ﬁnite discrete group G/K . In this more general context, one can still deﬁne the group ring C[G] as the algebra of ﬁnite formal linear combinations of elements of G. A smooth representation V of G is then a C[G]-module. However, an arbitrary C[G]module need not provide a smooth representation of G, and so the group ring is not an eﬀective tool for analyzing smooth representations. For this purpose, one has to replace C[G] by a diﬀerent algebra. We discuss this in §4 below. 2.2. We recall a standard concept in the present context. Proposition. Let G be a locally proﬁnite group, and let (π, V ) be a smooth representation of G. The following conditions are equivalent: (1) V is the sum of its irreducible G-subspaces; (2) V is the direct sum of a family of irreducible G-subspaces; (3) any G-subspace of V has a G-complement in V . Proof. We start with the implication (1) ⇒ (2). We take a family {Ui : i ∈ I} of irreducible G-subspaces Ui of V such that V = i∈I Ui . We consider the set I of subsets J of I such that the sum i∈J Ui is direct. The set I is nonempty; we show it is inductively ordered by inclusion. For, suppose we have a totally ordered set {Ja : a ∈ A} of elements of I. Put J = a∈A Ja . If the sum j∈J Uj is not direct, there is a ﬁnite subset S of J for which j∈S Uj is not direct. Since S must be contained in some Ja , this is impossible. Therefore J ∈ I. We can now apply Zorn’s Lemma to get a maximal element J0 of I. For this set, we have

Ui , V = a∈J0

as required for (2). In (3), let W be a G-subspace of V . By (2), we can assume that V = of V . We consider the i∈I Ui , for a family (Ui ) of irreducible G-subspaces set J of subsets J of I for which W ∩ i∈J Ui = 0. Again, the set J is nonempty and inductively ordered by inclusion. If J is a maximal element of J , the sum X = W + j∈J Uj is direct. If X = V , there is i ∈ I with Ui ⊂ X, so the sum X + Ui is direct, and J ∪ {i} ∈ J , contrary to hypothesis. Thus (2) ⇒ (3). Suppose now that (3) holds. Let V0 be the sum of all irreducible G-subspaces of V and write V = V0 ⊕ W , for some G-subspace W of V .

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15

Assume for a contradiction that W = 0. By its deﬁnition, the space W has no irreducible G-subspace. However, there is a non-zero G-subspace W1 of W which is ﬁnitely generated over G. By Zorn’s Lemma, W1 has a maximal G-subspace W0 , and then W1 /W0 is irreducible. We have V = V0 ⊕ W0 ⊕ U , for some G-subspace U of V , and hence a G-projection V → U . The image of W1 in U is an irreducible G-subspace of U , hence an irreducible subspace of V which is not contained in V0 . This is nonsense, so V = V0 and (3) ⇒ (1). One says that (π, V ) is G-semisimple if it satisﬁes the conditions of the proposition. Interesting locally proﬁnite groups G usually have many representations which are not semisimple. However, the property can be employed in a slightly diﬀerent context: Lemma. Let G be a locally proﬁnite group, and let K be a compact open subgroup of G. Let (π, V ) be a smooth representation of G. The space V is the sum of its irreducible K-subspaces. Proof. Let v ∈ V . As in 2.1 Example 2, v is ﬁxed by an open normal subgroup K of K, and it generates a ﬁnite-dimensional K-space W on which K acts trivially. Thus W is eﬀectively a ﬁnite-dimensional representation of the ﬁnite group K/K and so is the sum of its irreducible K-subspaces. Since v ∈ V was chosen at random, the lemma follows. The lemma says that V is K-semisimple. 2.3. Again let G be a locally proﬁnite group and K a compact open subgroup of G. denote the set of equivalence classes of irreducible smooth represenLet K and (π, V ) is a smooth representation of G, we deﬁne tations of K. If ρ ∈ K ρ V to be the sum of all irreducible K-subspaces of V of class ρ. We call V ρ the ρ-isotypic component of V . In particular, V K is the isotypic subspace for the class of the trivial representation 1 of K. Proposition. Let (π, V ) be a smooth representation of G and let K be a compact open subgroup of G. (1) The space V is the direct sum of its K-isotypic components:

V = V ρ. ρ∈K (2) Let (σ, W ) be a smooth representation of G. For any G-homomorphism we have f : V → W and ρ ∈ K, f (V ρ ) ⊂ W ρ

and

W ρ ∩ f (V ) = f (V ρ ).

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Proof. We use 2.2 to write V =

Ui ,

i∈I

for a family of irreducible K-subspaces Ui of V . We let U (ρ) be the sum of those Ui of class ρ. We then have

V = U (ρ). ρ∈K If W is an irreducible K-subspace of V of class ρ, then W ⊂ U (ρ): otherwise, there would be a non-zero K-homomorphism W → Ui , for some Ui of some class τ = ρ. We deduce that V ρ = U (ρ) and (1) follows. In (2), the image of V ρ is a sum of irreducible K-subspaces of W , all of class ρ and therefore contained in W ρ . Moreover, f (V ) is the sum of the and f (V τ ) ⊂ W τ . Since the sum of the W τ is direct, images f (V τ ), τ ∈ K, f (V ) is the direct sum of the f (V τ ) and the second assertion follows. We frequently use part (2) of the Proposition in the following context: Corollary 1. Let a : U → V , b : V → W be G-homomorphisms between smooth representations U , V , W of G. The sequence U−−−a−−→ V−−−b−−→ W is exact if and only if U K−−−a−−→ V K−−−b−−→ W K is exact, for every compact open subgroup K of G. If H is a subgroup of G, we deﬁne V (H) = the linear span of {v−π(h)v : v ∈ V, h ∈ H}.

(2.3.1)

In particular, V (H) is an H-subspace of V . Corollary 2. Let G be a locally proﬁnite group, and let (π, V ) be a smooth representation of G. Let K be a compact open subgroup of G. Then

V (K) = V ρ, V = V K ⊕ V (K), ρ ∈ K, ρ = 1

and V (K) is the unique K-complement of V K in V .

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ρ Proof. The sum W = V , with ρ not trivial, is a K-complement of V K in V . There is therefore a K-surjection V → V K with kernel W . Clearly, V (K) is contained in the kernel of any K-homomorphism V → V K ; we conclude that W contains V (K). On the other hand, if U is an irreducible K-space of class ρ = 1, then U (K) = U , so V ρ ⊂ V (K). Exercises. (1) Let (π, V ) be an abstract (i.e., not necessarily smooth) representation of G. Deﬁne V K, V∞ = K

where K ranges over the compact open subgroups of G. Show that V ∞ is a G-stable subspace of V . Deﬁne a homomorphism π ∞ : G −→ AutC (V ∞ ) by π ∞ (g) = π(g) | V ∞ . Show that (π ∞ , V ∞ ) is a smooth representation of G. (2) Let (π, V ) be a smooth representation of G and (σ, W ) an abstract representation. Let f : V → W be a G-homomorphism. Show that f (V ) ⊂ W ∞ , and hence HomG (V, W ) = HomG (V, W ∞ ). (3) Let 0 → U−−−a−−→ V−−−b−−→ W → 0 be an exact sequence of G-homomorphisms of abstract G-spaces. Show that the induced sequence 0 → U ∞−−−a−−→ V ∞−−−b−−→ W ∞ is exact. Show by example that the map b : V ∞ → W ∞ need not be surjective. 2.4. We now consider the notion of an induced representation. Let G be a locally proﬁnite group, and let H be a closed subgroup of G. Thus H is also locally proﬁnite. Let (σ, W ) be a smooth representation of H. We consider the space X of functions f : G → W which satisfy (1) f (hg) = σ(h)f (g), h ∈ H, g ∈ G; (2) there is a compact open subgroup K of G (depending on f ) such that f (gx) = f (g), for g ∈ G, x ∈ K.

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We deﬁne a homomorphism Σ : G → AutC (X) by Σ(g)f : x −→ f (xg),

g, x ∈ G.

The pair (Σ, X) provides a smooth representation of G. It is called the representation of G smoothly induced by σ, and is usually denoted (Σ, X) = IndG H σ. The map σ → IndG H σ gives a functor Rep(H) → Rep(G). There is a canonical H-homomorphism ασ : IndG H σ −→ W, f −→ f (1). The pair (IndG H , α) has the following fundamental property: Frobenius Reciprocity. Let H be a closed subgroup of a locally proﬁnite group G. For a smooth representation (σ, W ) of H and a smooth representation (π, V ) of G, the canonical map HomG (π, IndG H σ) −→ HomH (π, σ), φ −→ ασ ◦ φ, is an isomorphism that is functorial in both variables π, σ. Proof. Let f : V → W be an H-homomorphism. We deﬁne a G-homomorphism f : V → Ind σ by letting f (v) be the function g → f (π(g)v). The map f → f is then the inverse of (2.4.2). A simple, but very useful, consequence is that ασ (V ) = 0, for any non-zero G-subspace V of IndG H σ. We will also need a less formal property: Proposition. The functor IndG H : Rep(H) → Rep(G) is additive and exact. Proof. For a smooth representation (σ, W ) of H, temporarily let I(σ) denote the space of functions G → W satisfying the ﬁrst condition f (hg) = σ(h)f (g) of the deﬁnition above. Thus I is a functor to the category of abstract rep∞ resentations of G; it is clearly additive and exact, while IndG H (σ) = I(σ) . G Thus IndH is surely additive, and 2.3 Exercise (3) shows it to be left-exact. To prove it is right-exact, let (σ, W ), (τ, U ) be smooth representations of H and let f : W → U be an H-surjection. Take φ ∈ I(τ )∞ , and choose a compact open subgroup K of G which ﬁxes φ. The support of φ is a union of cosets HgK, and the value φ(g) ∈ U must be ﬁxed by τ (H ∩ gKg −1 ). By 2.3 Corollary 1 (applied to the group H and the trivial representation

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of its compact open subgroup H ∩ gKg −1 ), there exists wg ∈ W , ﬁxed by σ(H ∩ gKg −1 ), such that f (wg ) = φ(g). We deﬁne a function Φ : G → W to have the same support as φ and Φ(hgk) = σ(h)wg , for each g ∈ H\supp φ/K. The function Φ is ﬁxed by K, and hence lies in I(σ)∞ . Its image in I(τ )∞ is φ, as required. 2.5. There is a variation on this theme. With (σ, W ) and X as in 2.4, consider the space Xc of functions f ∈ X which are compactly supported modulo H: this means that the image of the support supp f of f in H\G is compact or, equivalently, supp f ⊂ HC, for some compact set C in G. The space Xc is stable under the action of G and provides another smooth representation of G. It is denoted c-IndG H σ, and gives a functor c-IndG H : Rep(H) −→ Rep(G). One calls it compact induction, or smooth induction with compact supports. Exercise 1. Show that the functor c-IndG H is additive and exact. G In all cases, there is a canonical G-embedding c-IndG H σ → IndH σ. Put G G another way, there is a morphism of functors c-IndH → IndH . This is an isomorphism if and only if H\G is compact. On the other hand, for speciﬁc G G, H, σ, the map c-IndG H σ → IndH σ can be an isomorphism even when H\G is not compact. Signiﬁcant examples of this phenomenon arise in 11.4 below. This construction is mainly (but not exclusively) of interest when the subgroup H is open in G. In this case, there is a canonical H-homomorphism

ασc : W −→ c-Ind σ, w −→ fw ,

(2.5.1)

where fw ∈ Xc is supported in H and fw (h) = σ(h)w, h ∈ H. Exercise 2. Suppose H is open in G. Let φ : G → W be a function, compactly supported modulo H, such that φ(hg) = σ(h)φ(g), h ∈ H, g ∈ G. Show that φ ∈ Xc . Lemma. Let H be an open subgroup of G, and let (σ, W ) be a smooth representation of H. (1) The map ασc : w → fw is an H-isomorphism of W with the space of functions f ∈ c-IndG H σ such that supp f ⊂ H. (2) Let W be a C-basis of W and G a set of representatives for G/H. The set {gfw : w ∈ W, g ∈ G} is a C-basis of c-Ind σ.

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Proof. In (1), surely ασc is an H-homomorphism to the space of functions supported in H; the inverse map is f → f (1). −1 , The support of a function f ∈ c-IndG H σ is a ﬁnite union of cosets Hg for various g ∈ G, and the restriction of f to any one of these also lies in c-Ind σ. If supp f = Hg −1 , then g −1 f has support contained in H, and so is a ﬁnite linear combination of functions fw , w ∈ W. Clearly, the set of functions gfw , w ∈ W, g ∈ G, is linearly independent, and the proof is complete. For open subgroups, compact induction has its own form of Frobenius Reciprocity property: Proposition. Let H be an open subgroup of G, let (σ, W ) be a smooth representation of H and (π, V ) a smooth representation of G. The canonical map HomG (c-Ind σ, π) −→ HomH (σ, π), f −→ f ◦ ασc , is an isomorphism which is functorial in both variables. Proof. Let φ be an H-homomorphism W → V . There is a unique G-homomorphism φ∗ : c-Ind σ → V such that φ∗ (fw ) = φ(w), w ∈ W . The map φ → φ∗ is then inverse to (2.5.2). Remark . Suppose for the moment that G is ﬁnite. The coincident deﬁnitions of induction above are then equivalent to the standard one: this is easily proved directly. Alternatively, the version (2.5.2) of Frobenius Reciprocity is the same as the usual one for ﬁnite groups and one can use uniqueness of adjoint functors. 2.6. It is convenient to introduce a technical restriction on the group G. From now on, we assume that: Hypothesis. For any compact open subgroup K, the set G/K is countable. We remark that, if G/K is countable for one compact open subgroup K of G, then G/K is countable for any compact open subgroup K of G. For, K ∩ K is compact, open, and of ﬁnite index in K. Thus the surjection G/(K ∩ K ) → G/K has ﬁnite ﬁbres and G/(K ∩ K ), hence also G/K , is countable. Certain of the things we do in this section do not require the property, but it holds for every concrete group in which we shall be interested. The main eﬀect of the hypothesis is: Lemma. Let (π, V ) be an irreducible smooth representation of G. The dimension dimC V is countable.

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21

Proof. Let v ∈ V , v = 0, and choose a compact open subgroup K of G such that v ∈ V K . Since V is irreducible, the countable set {π(g)v : g ∈ G/K} spans V . This enables us to generalize a familiar result, as follows. Schur’s Lemma. If (π, V ) is an irreducible smooth representation of G, then EndG (V ) = C. Proof. Let φ ∈ EndG (V ), φ = 0. The image and the kernel of φ are both G-subspaces of V , so φ is bijective and invertible. Therefore EndG (V ) is a complex division algebra. If we ﬁx v ∈ V , v = 0, the G-translates of v span V so an element φ ∈ EndG (V ) is determined uniquely by the value φ(v). We deduce that / C, is EndG (V ) has countable dimension. However, any φ ∈ EndG (V ), φ ∈ transcendental over C, and generates a ﬁeld C(φ) ⊂ EndG (V ). The subset {(φ−a)−1 : a ∈ C} of C(φ) is linearly independent over C, so the C-dimension of C(φ) is uncountable, and this is impossible. The only conclusion is that EndG (V ) = C, as required. Corollary 1. Let (π, V ) be an irreducible smooth representation of G. The centre Z of G then acts on V via a character ωπ : Z → C× , that is, π(z)v = ωπ (z)v, for v ∈ V and z ∈ Z. Proof. By Schur’s Lemma, there is surely a homomorphism ωπ : Z → C× such that π(z)v = ωπ (z)v, z ∈ Z, v ∈ V . If K is a compact open subgroup of G such that V K = 0, then ωπ is trivial on the compact open subgroup K ∩ Z of Z. Thus ωπ is a character of Z. One calls ωπ the central character of π. Corollary 2. If G is abelian, any irreducible smooth representation of G is one-dimensional. Remark . If G is compact, the converse of Schur’s Lemma holds: a smooth representation (π, V ) of G is a direct sum of irreducible representations, so EndG (V ) is one-dimensional if and only if π is irreducible. This is false for smooth representations of locally proﬁnite groups in general: see 9.10 below for a signiﬁcant example. 2.7. We will sometimes need a more general version of the preceding machinery. Before dealing with this, however, it is convenient to interpolate a general lemma. Lemma. Let G be a locally proﬁnite group, and let H be an open subgroup of G of ﬁnite index.

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1 Smooth Representations

(1) If (π, V ) is a smooth representation of G, then V is G-semisimple if and only if it is H-semisimple. (2) Let (σ, W ) be a semisimple smooth representation of H. The induced representation IndG H σ is G-semisimple. Proof. Suppose that V is H-semisimple, and let U be a G-subspace of V . By hypothesis, there is an H-subspace W of V such that V = U ⊕ W . Let f : V → U be the H-projection with kernel W . Consider the map π(g) f (π(g)−1 v), v ∈ V. f G : v −→ (G:H)−1 g∈G/H

The deﬁnition is independent of the choice of coset representatives and it follows that f G is a G-projection V → U . We then have V = U ⊕ Ker f G and Ker f G is a G-subspace of V . Thus V is G-semisimple (cf. 2.2 Proposition). Conversely, suppose that V is G-semisimple. Thus G is a direct sum of irreducible G-subspaces (2.2), and it is enough to treat the case where V is irreducible over G. As representation of H, the space V is ﬁnitely generated and so admits an irreducible H-quotient U . Suppose for the moment that H is a normal subgroup of G. By Frobenius Reciprocity (2.4.2), the H-map V → U gives a non-trivial, hence injective, G-map V → IndG H U . As represenG U = c-Ind U is a direct sum of tation of H, the induced representation IndG H H G-conjugates of U (cf. 2.5 Lemma). These are all irreducible over H, so Ind U is H-semisimple. Proposition 2.2 then implies that V ⊂ Ind U is H-semisimple. In general, we set H0 = g∈G/H gHg −1 . This is an open normal subgroup of G of ﬁnite index. We have just shown that the G-space V is H0 -semisimple; the ﬁrst part of the proof shows it is H-semisimple. This completes the proof of (1), and (2) follows readily from the same arguments. We ﬁrst apply this in the following context. Let Z be the centre of G, and ﬁx a character χ of Z. We consider the class of smooth representations (π, V ) of G which admit χ as central character, that is, π(z)v = χ(z)v,

v ∈ V, z ∈ Z.

Proposition. Let (π, V ) be a smooth representation of G, admitting χ as a central character. Let K be an open subgroup of G such that KZ/Z is compact. (1) Let v ∈ V . The KZ-space spanned by v is of ﬁnite dimension, and is a sum of irreducible KZ-spaces. (2) As representation of KZ, the space V is semisimple.

2. Smooth Representations of Locally Proﬁnite Groups

23

Proof. The vector v is ﬁxed by a compact open subgroup K0 of K. The set KZ/K0 Z is ﬁnite, so the space W spanned by π(KZ)v has ﬁnite dimension. Surely W is K0 Z-semisimple; the lemma implies it is KZ-semisimple. Since v was chosen arbitrarily, assertion (2) follows. In practice, the open subgroup K will contain Z, with K/Z compact. The discussion is equally valid if Z is a closed subgroup of the centre Z(G) of G. 2.8. The notion of duality, for smooth representations of a locally proﬁnite group, is both more subtle and more signiﬁcant than it is for representations of ﬁnite groups. We examine it in some detail. Let (π, V ) be a smooth representation of the locally proﬁnite group G. Write V ∗ = HomC (V, C), and denote by V ∗ × V −→ C, (v ∗ , v) −→ v ∗ , v, the canonical evaluation pairing. The space V ∗ carries a representation π ∗ of G deﬁned by π ∗ (g)v ∗ , v = v ∗ , π(g −1 )v,

g ∈ G, v ∗ ∈ V ∗ , v ∈ V.

This is not, in general, smooth. We accordingly deﬁne (V ∗ )K , Vˇ = (V ∗ )∞ = K

where K ranges over the compact open subgroups of G. Thus (cf. 2.3 Exercise (1)) Vˇ is a G-stable subspace of V ∗ , and provides a smooth representation π ˇ = (π ∗ )∞ : G −→ AutC (Vˇ ). The representation (ˇ π , Vˇ ) is called the contragredient, or smooth dual, of (π, V ). We continue to denote the evaluation pairing Vˇ × V → C by (ˇ v , v) → ˇ v , v. Therefore ˇ π (g)ˇ v , v = ˇ v , π(g −1 )v,

g ∈ G, vˇ ∈ Vˇ , v ∈ V.

(2.8.1)

Let K be a compact open subgroup of G. We recall that V K has a unique K-complement V (K) in V (2.3). If vˇ ∈ Vˇ is ﬁxed under K, we have ˇ v , V (K) = 0, by the deﬁnition of V (K). Thus vˇ ∈ Vˇ K is determined by its eﬀect on V K . Proposition. Restriction to V K induces an isomorphism Vˇ K ∼ = (V K )∗ . Proof. One can extend a linear functional on V K to an element of Vˇ K by deeming that it be trivial on V (K).

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1 Smooth Representations

Corollary. Let (π, V ) be a smooth representation of G, and let v ∈ V , v = 0. There exists vˇ ∈ Vˇ such that ˇ v , v = 0. Remark . Let (π, V ) be a smooth representation of G. Although undeclared in the notation, the subspace Vˇ of V ∗ does depend on G, in the following sense. Let H be a closed subgroup of G, and let V denote the space of H-smooth vectors in V ∗ . Certainly Vˇ ⊂ V , but it is easy to produce examples where V = Vˇ . 2.9. Let (π, V ) be a smooth representation of the locally proﬁnite group G. ˇ , Vˇ ) of (ˇ We can form the smooth dual (π π , Vˇ ). There is a canonical G-map ˇ δ : V → V given (in the obvious notation) by v ∈ V, vˇ ∈ Vˇ .

δ(v), vˇVˇ = ˇ v , vV , It is injective (2.8 Corollary). Proposition. Let (π, V ) be a smooth group G. The canonical map δ : V → π is admissible.

representation of a locally proﬁnite Vˇ is an isomorphism if and only if

Proof. The map δ induces a map δ K : V K → Vˇ K for each compact open subgroup K of G, and δ is surjective if and only if δ K is surjective for all K (2.3 Corollary). However, δ K is the canonical map V K → (V K )∗∗ , which is surjective if and only if dim V K < ∞. 2.10. Let (π, V ), (σ, W ) be smooth representations of G, and let f : V → W ˇ → Vˇ by the relation be a G-map. We deﬁne a map fˇ : W fˇ(w), ˇ v = w, ˇ f (v),

ˇ , v ∈ V. w ˇ∈W

The map fˇ is a G-homomorphism, and (π, V ) → (ˇ π , Vˇ ) gives a contravariant functor of Rep(G) to itself. Lemma. The contravariant functor Rep(G) −→ Rep(G), (π, V ) −→ (ˇ π , Vˇ ), is exact. Proof. If we have an exact sequence of smooth representations (πi , Vi ) of G: 0 → V1 −→ V2 −→ V3 → 0,

3. Measures and Duality

25

the sequence 0 → V1K −→ V2K −→ V3K → 0 is exact (2.3). The sequence of dual spaces (ViK )∗ is then exact, that is, 0 → Vˇ1K −→ Vˇ2K −→ Vˇ3K → 0 is exact. The result now follows from 2.3 Corollary 1. We deduce: Proposition. Let (π, V ) be an admissible representation of G. Then (π, V ) is irreducible if and only if (ˇ π , Vˇ ) is irreducible. Remark . For certain locally proﬁnite groups G, there do exist examples of irreducible smooth representations π for which π ˇ is not irreducible: see 8.2 Remark for an example. Exercise. Let (π, V ), (σ, W ) be smooth representations of G. Let ℘(π, σ) be the space of G-invariant bilinear pairings V × W → C. Show that there are canonical isomorphisms ˇ) ∼ ˇ ). HomG (π, σ = ℘(π, σ) ∼ = HomG (σ, π

3. Measures and Duality We now give a quick, but essentially complete, account of some invariant measures on a locally proﬁnite group and associated homogeneous spaces. Locally proﬁnite groups are such that their measure theory is eﬀectively algebraic in nature, and can be treated as an episode in their representation theory. 3.1. Let G be a locally proﬁnite group. Let Cc∞ (G) be the space of functions f : G → C which are locally constant and of compact support. Let f ∈ Cc∞ (G). Local constancy and compactness of support together imply that there exist compact open subgroups K1 , K2 of G such that f (k1 g) = f (g) = f (gk2 ), for all g ∈ G, ki ∈ Ki . Taking K = K1 ∩ K2 , one sees that f is a ﬁnite linear combination of characteristic functions of double cosets KgK. The group G acts on Cc∞ (G) by left translation λ and by right translation ρ: λg f : x −→ f (g −1 x), (3.1.1) x, g ∈ G, f ∈ Cc∞ (G). ρg f : x −→ f (xg), Both of the G-representations (Cc∞ (G), λ), (Cc∞ (G), ρ) are smooth.

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1 Smooth Representations

Deﬁnition. A right Haar integral on G is a non-zero linear functional I : Cc∞ (G) −→ C such that (1) I(ρg f ) = I(f ), g ∈ G, f ∈ Cc∞ (G), and (2) I(f ) 0 for any f ∈ Cc∞ (g), f 0. One deﬁnes a left Haar integral similarly, using left translation λ instead of right translation. We now show that G possesses a right Haar integral, and essentially only one. Proposition. There exists a right Haar integral I : Cc∞ (G) → C. Moreover, a linear functional I : Cc∞ (G) → C is a right Haar integral if and only if I = cI, for some constant c > 0. ∞

Proof. Let K be a compact open subgroup of G; we denote by K C c (G) the ∞ space of functions in Cc∞ (G) ﬁxed by λ(K). We view K C c (G) as G-space via right translation. It is then identical to the representation of G compactly induced from the trivial representation 1K of K: K

∞

C c (G) = c-IndG K 1K .

Lemma. Viewing C as the trivial G-space, we have ∞

dimC HomG (K C c (G), C) = 1. ∞

There exists a non-zero element IK ∈ HomG (K C c (G), C) such that IK (f ) 0 whenever f 0. If hK is the characteristic function of K, then IK (hK ) > 0. Proof. The ﬁrst assertion is given by 2.5 Proposition. For g ∈ G, let fg denote the characteristic function of Kg. The set of functions fg , g ∈ K\G, then forms ∞ a C-basis of the space K C c (G) (2.5 Lemma). The functional IK : fg → 1 has the required properties, noting that hK = f1 . We choose a descending sequence {Kn }n1 of compact open subgroups Kn of G such that n Kn = 1. We then have Kn ∞ Cc∞ (G) = C c (G). n1 ∞

For each n 1, there is a unique right G-invariant functional In on Kn C c (G) which maps the characteristic function of Kn to (K1 : Kn )−1 . We have In+1 | Kn ∞ C c (G) = In , and so the family {In } gives a functional on Cc∞ (G) of the required kind. The uniqueness statement is immediate.

3. Measures and Duality

27

Remark . The lemma also implies that, if we view Cc∞ (G) as a smooth representation of G under right translation, then dim HomG (Cc∞ (G), C) = 1.

(3.1.2)

The functional I of the Proposition is a right Haar integral on G. One can produce a left Haar integral in exactly the same way. Alternatively, one can proceed as follows. Corollary. For f ∈ Cc∞ (G), deﬁne fˇ ∈ Cc∞ (G) by fˇ(g) = f (g −1 ), g ∈ G. The functional I : Cc∞ (G) −→ C, I (f ) = I(fˇ), is a left Haar integral on G. Moreover, any left Haar integral on G is of the form cI , for some c > 0. The uniqueness statement follows from the proposition on observing that, if J is a left Haar integral, then f → J(fˇ) is a right Haar integral. Let I be a left Haar integral on G. Let S = ∅ be a compact open subset of G and let ΓS be its characteristic function. We deﬁne µG (S) = I(ΓS ). Then µG (S) > 0 and the measure µG satisﬁes µG (gS) = µG (S), g ∈ G. One refers to µG as a left Haar measure on G. The relation with the integral is expressed via the traditional notation

f (g) dµG (g), f ∈ Cc∞ (G). I(f ) = G

Further traditional abbreviations are frequently permitted, in particular,

ΓS (x)f (x) dµG (x) = f (x) dµG (x). G

S

Deﬁnition. The group G is unimodular if any left Haar integral on G is a right Haar integral. 3.2. One can extend the domain of Haar integration, much as in the classical case of the Lebesgue measure. We outline a few examples of what we will need. First, one can integrate more general functions. For example, let f be a function on G invariant under left translation by a compact open subgroup

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1 Smooth Representations

K of G. Let µG be a left Haar measure on G. If the series |f (x)| dµG (x) g∈K\G

Kg

converges, so does the series without the absolute value, and we put

f (x) dµG (x) = f (x) dµG (x). G

g∈K\G

Kg

The result does not depend on the choice of K, and this extended Haar integral has the translation invariance property of the original. Next, let G1 , G2 be locally proﬁnite groups, and set G = G1 × G2 . Then G is locally proﬁnite. An element 1ir fi1 ⊗fi2 of the tensor product Cc∞ (G1 )⊗ Cc∞ (G2 ) gives a function on G by Φ(g1 , g2 ) =

fi1 (g1 )fi2 (g2 ).

i

Then Φ ∈ Cc∞ (G), and this process gives an isomorphism Cc∞ (G1 )⊗Cc∞ (G2 ) → Cc∞ (G). Let µj be a left Haar measure on Gj , j = 1, 2. There is then a unique left Haar measure µG on G such that

f1 ⊗ f2 (g) dµG (g) = f1 (g1 ) dµ1 (g1 ) f2 (g2 ) dµ2 (g2 ), G

G1

G2

for fi ∈ Cc∞ (Gi ). One writes µG = µ1 ⊗ µ2 . For a general f ∈ Cc∞ (G), the function

f1 (g1 ) = f (g1 , g2 ) dµ2 (g2 ) G2

lies in Cc∞ (G1 ). We have

f (g) dµG (g) = G

f1 (g1 ) dµ1 (g1 ),

G1

and symmetrically: if f is of the form f1 ⊗f2 , this is obvious, and such functions span Cc∞ (G). Next, let V be a complex vector space, and consider the space Cc∞ (G; V ) of locally constant, compactly supported functions f : G → V . This space is isomorphic to Cc∞ (G) ⊗ V in the obvious way: a tensor i fi ⊗ vi gives the function fi (g)vi . g −→ i

3. Measures and Duality

29

If µG is a left Haar measure on G, there is a unique linear map IV : Cc∞ (G; V ) → V such that

IV (f ⊗ v) = f (g) dµG (g) · v. G

We write IV (φ) =

φ(g) dµG (g),

φ ∈ Cc∞ (G; V ).

G

This has the same invariance properties as the Haar integral on scalar-valued functions. 3.3. Let µG be a left Haar measure on G. For g ∈ G, consider the functional Cc∞ (G) −→ C,

f (xg) dµG (x). f −→ G

This is a left Haar integral on G, so there is a unique δG (g) ∈ R× + such that

δG (g) f (xg) dµG (x) = f (x) dµG (x), G

G

for all f ∈ Cc∞ (G). The function δG is a homomorphism G → R× + . It is trivial if G is abelian; more generally, δG is trivial if and only if G is unimodular. Taking f to be the characteristic function of a compact open subgroup K of G, we see that δG is trivial on K and hence a character of G. One calls δG the module of G. The functional

δG (x)−1 f (x) dµG (x), f ∈ Cc∞ (G), f −→ G

is a right Haar integal on G. The mnemonic dµG (xg) = δG (g) dµG (x) may be found helpful. Remark . We have already observed that δG is trivial on any compact open subgroup of G. In particular, if G is compact, then δG = 1 and G is unimodular. In the general case, any character G → R× + is trivial on compact × subgroups, since R+ has only the trivial compact subgroup.

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3.4 Let H be a closed subgroup of G, with module δH . Let θ : H → C× be a character of H. We consider the space of functions f : G → C which are G-smooth under right translation, compactly supported modulo H, and satisfy f (hg) = θ(h)f (g), h ∈ H, g ∈ G. We call this space Cc∞ (H\G, θ), and view it as a smooth G-space via right translation ρ. (Note that Cc∞ (H\G, θ) = c-IndG H θ, but this characterization is not helpful.) Proposition. Let θ : H → C× be a character of H. The following are equivalent: (1) There exists a non-zero linear functional Iθ : Cc∞ (H\G, θ) → C such that Iθ (ρg f ) = Iθ (f ), for all g ∈ G. (2) θδH = δG | H. When these conditions hold, the functional Iθ is uniquely determined up to constant factor. Proof. Let µG , µH be left Haar measures on G, H respectively. We deﬁne a G-map Cc∞ (G) → Cc∞ (H\G, θ), denoted f → f , by

θδH (h)−1 f (hg) dµH (h). f (g) = H

This map satisﬁes

−1 f, λ k f = δH θ(k)

for k ∈ H and f ∈ Cc∞ (G). We prove it is surjective. If K is a compact open subgroup of G, the space Cc∞ (G)K is spanned by the characteristic functions of cosets gK, g ∈ G/K. On the other hand, each coset HgK supports, at most, a one-dimensional space of functions in Cc∞ (H\G, θ)K . These subspaces span Cc∞ (H\G, θ)K , so the map f → f is surjective on K-ﬁxed functions. It is therefore surjective. Suppose that the space Cc∞ (H\G, θ) admits a functional Iθ of the required kind. The map f → Iθ (f ) is then a non-trivial G-homomorphism ∞ Cc (G), ρ −→ C. However, the space HomG (Cc∞ (G), C) has dimension 1 (3.1.2), and is spanned by a right Haar integral. The condition (1) holds, therefore, if and only if the right Haar integral Cc∞ (G) → C factors through the quotient map Cc∞ (G) → Cc∞ (H\G, θ). When it does hold, Iθ is determined up to a constant factor. The kernel of the map f → f contains all functions λh f − δH θ(h)−1 f , for h ∈ H and f ∈ Cc∞ (G). Applying the right Haar integral on G to such

3. Measures and Duality

31

functions, we get

λh f (g) − δH θ(h−1 ) f (g) δG (g)−1 dµG (g) G

= δG (h−1 ) − δH θ(h−1 )

f (g)δG (g)−1 dµG (g).

G

This vanishes identically if and only if δG (h−1 ) = δH θ(h−1 ), for all h ∈ H. Thus (1) ⇒ (2). For the converse, we take a function f ∈ Cc∞ (G) such that f = 0. The function f is ﬁxed by a compact open subgroup K, and it is enough to treat the case where supp f ⊂ HgK, for some g ∈ K. Thus f is a ﬁnite linear combination of the characteristic function of cosets hi gK, hi ∈ H. The condition f = 0 then amounts to θδH (hi )−1 f (hi g) = 0, µH (H ∩ gKg −1 ) i

since θδH is trivial on the compact subgroup H ∩ gKg −1 of H. On the other hand,

f (x)δG (x)−1 dµG (x) = µG (K)δG (g)−1 δG (hi )−1 f (hi g). G

i

If (2) holds, this surely vanishes as required. When the conditions of the proposition hold, the character θ takes only positive real values. Let f ∈ Cc∞ (G) satisfy f (g) 0, for all g ∈ G; we then have f (g) 0 for all g. Consequently: Corollary. Suppose that the conditions of the proposition hold. There is then a non-zero linear functional Iθ on Cc∞ (H\G, θ) such that: (1) Iθ (ρg f ) = Iθ (f ), for f ∈ Cc∞ (H\G, θ), g ∈ G; (2) Iθ (f ) 0, for f ∈ Cc∞ (H\G, θ), f 0. These conditions determine Iθ uniquely, up to a positive constant factor. One habitually uses a notation like

Iθ (f ) = f (g) dµH\G (g),

f ∈ Cc∞ (H\G, θ),

H\G −1 and calls µH\G a positive semi-invariant measure on H\G. (Since θ = δH δG | H is uniquely determined, there is no real need to refer to it again.)

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3.5. Let G be a locally proﬁnite group and H a closed subgroup of G. Put −1 δH\G = δH δG | H : H −→ R× +.

Duality Theorem. Let µ˙ be a positive semi-invariant measure on H\G. Let (σ, W ) be a smooth representation of H. There is a natural isomorphism ∨ ∼ c-IndG ˇ, = IndG Hσ H δH\G ⊗ σ depending only on the choice of µ. ˙ ˇ as σ Proof. We view δH\G ⊗ σ ˇ as acting on the same space W ˇ . Let (w, ˇ w) → ˇ w, ˇ w be the evaluation pairing W ×W → C. Let φ ∈ c-Ind σ, Φ ∈ Ind δH\G ⊗ σ ˇ , and consider the function f : g −→ Φ(g), φ(g),

g ∈ G.

This lies in Cc∞ (H\G, δH\G ), so we have a G-invariant pairing Ind δH\G ⊗ σ ˇ × c-Ind σ −→ C,

Φ(g), φ(g) dµ(g). ˙ (Φ, φ) −→ H\G

∨ This induces a G-homomorphism Ind δH\G ⊗ σ ˇ → c-Ind σ . It is clearly natural in σ; we show it is an isomorphism. To do this, we take a compact open subgroup K of G and describe the space (c-Ind σ)K of K-ﬁxed functions in c-Ind σ. The support of such a function, f say, is a ﬁnite union of cosets HgK. The value f (g) ∈ W is then ﬁxed by the compact open subgroup H ∩gKg −1 of H. We choose a set G of representatives for the coset space H\G/K. For each g ∈ G, we choose a basis Wg of the space −1 W H∩gKg . For each g ∈ G and each w ∈ Wg , there is a unique function fg,w ∈ (c-Ind σ)K , supported on HgK, such that fg,w (g) = w. The set {fg,w : g ∈ G, w ∈ Wg } is then a basis of (c-Ind σ)K . That is: Lemma 1. The space (c-Ind σ)K consists of the ﬁnite linear combinations of functions fg,w , g ∈ G, w ∈ Wg . −1

ˇ g = HomC (W H∩gKg , C) dual to Wg . For Let Wg∗ denote the basis of W ∗ ˇ )K in the same way as before. g ∈ G, w ˇ ∈ Wg , we deﬁne fg,wˇ ∈ (Ind δH\G ⊗ σ We then have: ˇ )K consists of all functions f such that, Lemma 2. The space (Ind δH\G ⊗ σ for any g ∈ G, the restriction f | HgK is a ﬁnite linear combination of ˇ ∈ Wg∗ . functions fg,wˇ , w

4. The Hecke Algebra

33

For g1 , g2 ∈ G, w ∈ Wg1 , w ˇ ∈ Wg∗2 , we have fg2 ,wˇ , fg1 ,w =

ˇ w µ(Hg ˙ 1 K)w,

if Hg1 K = Hg2 K,

0 otherwise,

for some µ(Hg ˙ ˇ )K with the 1 K) > 0. The pairing thus identiﬁes (Ind δH\G ⊗ σ K ∨ ˇ with (c-Ind σ) , as required. linear dual of (c-Ind σ) , and hence Ind δH\G ⊗ σ

4. The Hecke Algebra If G is a ﬁnite group, the concept of a representation of G is essentially identical to that of a module over the group algebra C[G]. This relation extends to smooth representations of a locally proﬁnite group, but only with a suitable deﬁnition of ‘group algebra’. 4.1. To avoid pointless technical complications, we impose the following: Hypothesis. Unless otherwise stated, we assume that G is unimodular. We ﬁx a Haar measure µ on G. For f1 , f2 ∈ Cc∞ (G), we deﬁne

f1 ∗ f2 (g) = f1 (x)f2 (x−1 g) dµ(x). G

The function (x, g) → f1 (x)f2 (x−1 g) lies in Cc∞ (G × G) so (cf. 3.2) f1 ∗ f2 ∈ Cc∞ (G). Similarly, for fi ∈ Cc∞ (G), the integral expressing f1 ∗ (f2 ∗ f3 )(g) is that of a function from Cc∞ (G × G), so we can manipulate formally:

f1 ∗ (f2 ∗ f3 )(g) = f1 (x)f2 (y)f3 (y −1 x−1 g) dµ(y)dµ(x)

= f1 (x)f2 (x−1 y)f3 (y −1 g) dµ(y)dµ(x)

= f1 (x)f2 (x−1 y)f3 (y −1 g) dµ(x)dµ(y) = (f1 ∗ f2 ) ∗ f3 (g). The binary operation ∗, called convolution, is thus associative. The pair H(G) = Cc∞ (G), ∗ is an associative C-algebra called the Hecke algebra of G. In general, H(G) has no unit element; it is commutative if G is commutative.

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Remark 1. The algebra structure on H(G), that is, the convolution operation, clearly depends on the choice of Haar measure µ. However, suppose we have two Haar measures µ, ν, giving rise to algebra structures Hµ (G), Hν (G) on Cc∞ (G). There is a constant c > 0 such that ν = cµ. The map f → c−1 f is then an algebra isomorphism Hµ (G) → Hν (G). Remark 2. Suppose that G is discrete. For Haar integral, we may take

f (g) dµ(g) = f (g). G

g∈G

The map f −→

f (g)g

g∈G

is then an algebra isomorphism of H(G) with the group algebra C[G]. Observe that in this case (and only this case — exercise) H(G) has a unit element. While, in general, H(G) has no unit element, it does have a copious supply of idempotent elements. For example, let K be a compact open subgroup of G, and deﬁne a function eK ∈ H(G) by µ(K)−1 if x ∈ K, eK (x) = (4.1.1) 0 if x ∈ / K. Proposition. (1) The function eK satisﬁes eK ∗ eK = eK . (2) A function f ∈ H(G) satisﬁes eK ∗ f = f if and only if f (kg) = f (g), for all k ∈ K, g ∈ G. (3) The space eK ∗ H(G) ∗ eK is a sub-algebra of H(G), with unit element eK . Proof. First consider the integral

eK ∗ eK (g) = eK (x) eK (x−1 g) dµ(x). G

If g ∈ / K, the integrand is identically zero. If g ∈ K, it vanishes for x ∈ / K while, for x ∈ K, it takes the value µ(K)−2 . Integrating, we get the ﬁrst part. Taking f ∈ H(G), k ∈ K, g ∈ G, we have

eK ∗ f (kg) = eK (x)f (x−1 kg) dµ(x)

G = eK (kx)f (x−1 g) dµ(x) G

= eK (x)f (x−1 g) dµ(x) = eK ∗ f (g). G

4. The Hecke Algebra

35

If f is left K-invariant, the last integral reduces to f (g), while the function eK ∗ f is visibly left K-invariant. Part (3) is now obvious. We note that eK ∗ H(G) ∗ eK is the space of f ∈ H(G) satisfying f (k1 gk2 ) = f (g), for g ∈ G, k1 , k2 ∈ K. We often write H(G, K) = eK ∗ H(G) ∗ eK . 4.2. Let M be a left H(G)-module: it will be convenient to denote the module action by (f, m) → f ∗ m, for f ∈ H(G), m ∈ M . We say that M is smooth if H(G) ∗ M = M . Since H(G) is the union of its sub-algebras eK ∗ H(G) ∗ eK , the module M is smooth if and only if, for every m ∈ M , there is a compact open subgroup K of G such that eK ∗ m = m. Let M1 , M2 be smooth H(G)-modules; we deﬁne HomH(G) (M1 M2 ) to be the space of all H(G)-homomorphisms M1 → M2 . With this deﬁnition, the class of smooth H(G)-modules forms a category, which we denote H(G)-Mod. Let (π, V ) be a smooth representation of G. For f ∈ H(G), v ∈ V , we set

f (g)π(g)v dµ(g).

π(f )v =

(4.2.1)

G

The integrand lies in Cc∞ (G; V ) (notation of 3.2), so the integral (4.2.1) deﬁnes an element of V . Alternatively, we can choose a compact open subgroup K of G which ﬁxes v and f (under right translation); we may then interpret the integral as the ﬁnite sum π(f )v = µ(K)

f (g)π(g)v.

(4.2.2)

g∈G/K

Immediately from (4.2.2), we get π(eK )v = v,

v ∈ V K.

Proposition 1. Let (π, V ) be a smooth representation of G. The operation (f, v) −→ π(f )v,

v ∈ V, f ∈ H(G),

gives V the structure of a smooth H(G)-module. If (π , V ) is another smooth representation of G and φ : V → V is a G-homomorphism, then φ is also an H(G)-homomorphism: φ ◦ π(f ) = π (f ) ◦ φ,

f ∈ H(G).

(4.2.3)

36

1 Smooth Representations

Proof. Let v ∈ V , f1 , f2 ∈ H(G); as in 4.1, we can compute formally:

π(f1 ∗ f2 )v = f1 ∗ f2 (g)π(g)v dµ(g) G

= f1 (h)f2 (h−1 g)π(g)v dµ(h)dµ(g)

= f1 (h)f2 (h−1 g)π(g)v dµ(g)dµ(h)

= f1 (h)f2 (g)π(hg)v dµ(g)dµ(h)

= f1 (h)π(h) f2 (g)π(g)v dµ(g) dµ(h)

= f1 (h)π(h) π(f2 )v dµ(h) = π(f1 )π(f2 )v. So, in EndC (V ), we have π(f1 ∗ f2 ) = π(f1 ) π(f2 ). Thus f → π(f ) is a homomorphism of C-algebras H(G) → EndC (V ), and gives V the structure of an H(G)-module. If K is a compact open subgroup of G, we have π(eK )v = v, v ∈ V K , so V has become a smooth H(G)-module. The relation (4.2.3) follows from (4.2.2). Example. Take (π, V ) = (λ, H(G)) (notation of (3.1.1)). For φ, f ∈ H(G), ˇ λ(φ)f = φ ∗ f . If, on the other hand, (π, V ) = (H(G), ρ), then ρ(φ)f = f ∗ φ, −1 ˇ where φ is the function g → φ(g ). One can work the process in the other direction. Let M be a smooth H(G)-module, and denote the action of H(G) on M by (f, m) −→ f ∗ m,

f ∈ H(G), m ∈ M.

With this notation: Proposition 2. There is a unique G-homomorphism π : G → AutC (M ) such that (π, M ) is a smooth representation of G and π(f )m = f ∗ m,

f ∈ H(G), m ∈ M.

(4.2.4)

Moreover, if M is a smooth H(G)-module with associated G-representation (π , M ), then any H(G)-homomorphism M → M is a G-homomorphism π → π . Proof. Consider the canonical map H(G) ⊗H(G) M −→ M.

(4.2.5)

4. The Hecke Algebra

37

This is surjective, by the deﬁnition of smoothness. Let 1ir fi ⊗ mi lie in the kernel. We choose a compact open subgroup K of G which ﬁxes each fi on either side and such that mi ∈ eK ∗ M for all i. In particular, eK ∗ mi = mi for all i. We then have r i=1

fi ⊗ mi = eK ⊗

fi ∗ mi = 0,

i

so the map (4.2.5) is injective. Thus (4.2.5) is an H(G)-isomorphism H(G) ⊗H(G) M ∼ = M . The tensor product carries a smooth representation of G, via left translation on the ﬁrst factor. The isomorphism transfers the G-action to M , to give a smooth representation (π, M ), say. One can write this action more explicitly. Take m ∈ M , and choose a compact open subgroup K of G such that eK ∗ m = m. For g ∈ G, let f denote the characteristic function of gK. We then have π(g)m = µ(K)−1 f ∗ m.

(4.2.6)

The property (4.2.4) now follows from the example above. The ﬁnal assertion is clear from (4.2.6). The concepts “smooth representation of G” and “smooth H(G)-module” are thus identical and interchangeable. For example, if (π, V ) is a smooth representation of G, the G-subspaces of V are just the H(G)-submodules of V , the G-homomorphisms V → V are the H(G)-homomorphisms V → V , and so on. In particular, the categories Rep(G), H(G)-Mod are equivalent (but this statement is weaker than what we have just proved). 4.3. There are some associated algebra structures of interest. We ﬁx a compact open subgroup K of G, and deﬁne eK ∈ H(G) as in (4.1.1). We consider the sub-algebra H(G, K) = eK ∗ H(G) ∗ eK of H(G). Since eK is idempotent, this algebra has a unit element, namely eK . Using the notation of 2.3, we have: Lemma. Let (π, V ) be a smooth representation of G. The operator π(eK ) is the K-projection V → V K with kernel V (K). The space V K is an H(G, K)module on which eK acts as the identity. Proof. For v ∈ V and k ∈ K, we have (by a simple calculation) π(k) π(eK ) v = π(eK ) π(k) v = π(eK ) v. Thus π(eK ) is a K-map V → V K . As π(eK )v = v for v ∈ V K , the image of π(eK ) is V K . Since eK is idempotent, so is π(eK ). Certainly π(eK ) annihilates the unique K-complement V (K) of V K , so π(eK ) is the unique K-projection V → V K.

38

1 Smooth Representations

The H(G, K)-module V K encapsulates signiﬁcant information about the representation (π, V ). Proposition. (1) Let (π, V ) be an irreducible smooth representation of G. The space V K is either zero or a simple H(G, K)-module. (2) The process (π, V ) → V K induces a bijection between the following sets of objects: (a) equivalence classes of irreducible smooth representations (π, V ) of G such that V K = 0; (b) isomorphism classes of simple H(G, K)-modules. Proof. Let (π, V ) be an irreducible smooth representation of G; suppose that V K = 0, and let M be a non-zero H(G, K)-submodule of V K . The space π(H(G))M is a non-zero G-subspace of V , hence equal to V itself. We therefore have V K = π(eK )V = π(eK )π(H(G))M = π(H(G, K))M = M. This proves (1). So, V → V K gives a map from equivalence classes of irreducible representations (π, V ) with V K = 0 to isomorphism classes of simple H(G, K)-modules. In the opposite direction, let M be a simple H(G, K)-module and consider the G-space U = H(G) ⊗H(G,K) M , where G acts on the ﬁrst tensor factor by left translation. The space U satisﬁes U K = eK ∗ H(G) ⊗H(G,K) M = eK ⊗ M ∼ = M. Zorn’s Lemma gives the existence of a G-subspace X of U which is maximal for the property X ∩ eK ⊗ M = 0 or, equivalently, X K = 0. Given any other G-subspace Y of U with Y K = 0, we have (X+Y )K = X K +Y K = 0. So, by the maximality of X, we have Y ⊂ X and X is the unique G-subspace maximal for the property X K = X ∩ eK ⊗ M = 0. A G-subspace W of U strictly containing X must meet, and therefore contain, the simple H(G, K)-module eK ⊗ M , with the result that W = U . Thus X is a maximal G-subspace of U and V = U/X is irreducible. It satisﬁes VK ∼ = M as H(G, K)-module. It remains to show that the isomorphism class of V depends only on that of M . Let f : M → M be an isomorphism of H(G, K)-modules. The map f extends uniquely to an H(G)-isomorphism f : U = H(G) ⊗H(G,K) M −→ H(G) ⊗H(G,K) M = U , φ ⊗ m −→ φ ⊗ f (m).

4. The Hecke Algebra

39

The image X = f (X) is then the unique maximal H(G)-subspace of U such that X K = 0. The map f therefore induces an isomorphism U/X ∼ = U /X , as required. Corollary. Let (π, V ) be a smooth representation of G, V = 0; then (π, V ) is irreducible if and only if, for any compact open subgroup K of G, the space V K is either zero or H(G, K)-simple. Proof. One implication is given by the proposition. Conversely, suppose (π, V ) is not irreducible, and let U V be a non-zero G-subspace. Set W = V /U . There exists a compact open subgroup K of G such that both spaces U K , W K are non-zero. The sequence 0 → U K −→ V K −→ W K → 0 is exact, and is an exact sequence of H(G, K)-modules. Thus V K is non-zero and not simple over H(G, K). 4.4. There is a more general version of 4.3 which we outline. We start with We consider the function a compact open subgroup K of G and ρ ∈ K. eρ ∈ H(G), with support contained in K, and given by eρ (x) =

dim ρ tr ρ(x−1 ), µ(K)

x ∈ K.

(4.4.1)

If K is the kernel of ρ, then eρ is constant on cosets gK and K g , so eK ∗ eρ = eρ ∗ eK = eρ . Thus eρ lies in the subalgebra H(K, K ) of H(G, K ). However, eK → 1 induces an algebra isomorphism H(K, K ) → C K/K . This isomorphism takes eρ to the idempotent of the group algebra corresponding to the irreducible representation ρ of the ﬁnite group K/K . Therefore: Proposition. (1) The function eρ ∈ H(G) is idempotent. (2) If (π, V ) is a smooth representation of G, then π(eρ ) is the K-projection V → V ρ. In particular, V ρ = π(eρ )V , and the isotypic space V ρ is a module over the subalgebra eρ ∗ H(G) ∗ eρ of H(G). The exact analogue of 4.3 Proposition holds, with eρ replacing eK and V ρ replacing V K . Example. Suppose that dim ρ = 1, that is, ρ is a character of K. The algebra eρ ∗H(G)∗eρ is then the space of f ∈ H(G) such that f (kgk ) = ρ(kk )−1 f (g), k, k ∈ K, g ∈ G.

40

1 Smooth Representations

4.5. If G is a ﬁnite group, a key property of the group algebra C[G] is that it is semisimple. Since C[G] has ﬁnite dimension over C, this is equivalent to the Jacobson radical of C[G] being trivial or, again, to the property that if x ∈ C[G], x = 0, then there exists a simple C[G]-module M such that xM = 0. For a locally proﬁnite group G satisfying 2.6 Hypothesis, there is a corresponding property for the Hecke algebra H(G). Although we will not use this result, we include a proof to give a satisfying completeness to the generalization. Separation property. Let f ∈ H(G), f = 0. There exists an irreducible smooth representation (π, V ) of G such that π(f ) = 0. Proof. We shall need: Lemma. Let A be a C-algebra with 1, of countable dimension. Let a ∈ A be non-nilpotent. (1) There exists λ ∈ C× such that a−λ1 is not invertible in A. (2) There exists a simple A-module M such that aM = 0. Proof. Suppose, for a contradiction, that a−µ1 ∈ A× , for all µ ∈ C× . In particular, a = µ1, for any µ ∈ C× . The set {(a−µ1)−1 : µ ∈ C} is uncountable, and so linearly dependent over C. A non-trivial linear dependence relation n

αi (a−µi 1)−1 = 0

i=1

yields a non-trivial polynomial φ(t) ∈ C[t] such that φ(a) =

r

(a−λj 1) = 0,

j=1

for some r 1 and certain λj ∈ C. Not all λj are zero, since a is not nilpotent. If λj = 0, then b = a−λj 1 is a divisor of zero in A and so not invertible. Consequently, Ab = A and there exists a maximal left ideal I of A such that b ∈ I. The A-module A/I is then simple, and a(1+I) = λj +I = 0. Let f ∈ H(G), f = 0. Deﬁne f ∗ ∈ H(G) by f ∗ (g) = f (g −1 ), where the bar denotes complex conjugation. We set h = f ∗ ∗ f . The function h satisﬁes h∗ = h and h(1G ) > 0. In particular, h = 0. Applying the same argument to h, we have h ∗ h = h∗ ∗ h = 0. Inductively, we conclude that h is not nilpotent. There exists a compact open subgroup K of G such that h ∈ H(G, K), and H(G, K) is a C-algebra with 1, of countable dimension. By the lemma, there is a simple H(G, K)-module M such that hM = 0. By 4.3 Proposition, there is an irreducible smooth representation (π, V ) of G such that V K is H(G, K)isomorphic to M . We have π(h) = 0, and so π(f ) = 0, as required.

4. The Hecke Algebra

41

Further reading. For foundational matters concerning locally proﬁnite groups, or locally compact, totally disconnected groups, see [63]. The various topologies associated with local ﬁelds are discussed systematically in [87]. Measure theory is available in many texts but, particularly as regards Haar measure, [84] remains a classic.

2 Finite Fields

5. Linear groups 6. Representations of finite linear groups

In this brief chapter, we work out the irreducible representations of the group GL2 (k) of invertible 2 × 2 matrices over a ﬁnite ﬁeld k. This anticipates some of the phenomena which arise in the representation theory of GL2 (F ), where F is a non-Archimedean local ﬁeld. Indeed, if k is the residue ﬁeld of the non-Archimedean local ﬁeld F , the representation theory of GL2 (k) actually governs a part of that of GL2 (F ): see 11.5 below for a ﬁrst example of this.

5. Linear Groups In this section only, F denotes an arbitrary ﬁeld. We brieﬂy recall some basic facts about the group G = GL2 (F ) of 2 × 2 invertible matrices over F . The aim is to ﬁx some notation and terminology for use in later sections. We give no proofs: these are at the level of elementary linear algebra. 5.1. The group G possesses some particularly important algebraic subgroups: B = {( a0 cb ) ∈ G} , N = {( 10 1b ) ∈ G} . T = {( a0 0b ) ∈ G} .

(5.1.1)

The group B is called the standard Borel subgroup of G = GL2 (F ), and N is the unipotent radical of B. The group T is the standard split maximal torus in G. We have a semi-direct product decomposition B = T N . We also put Z = {( a0 a0 ) ∈ G} . Thus Z is the centre of G, and Z is canonically isomorphic to F × .

44

2 Finite Fields

5.2. In the notation (5.1.1), elementary methods of linear algebra give the Bruhat decomposition G = B ∪ BwB, where w denotes the permutation matrix 01 . w= 10 That is, {1, w} is a set of representatives for the coset space B\G/B. The coset BwB consists of the matrices with non-zero (2, 1)-entry. Since B = N T = T N and w normalizes T , we have BwB = N wB = BwN . Moreover, the map B × N −→ BwN, (b, n) −→ bwn, is bijective. 5.3. We recall the conjugacy class structure of the group G. For g ∈ G, let chg (t) = det(tI−g) denote the characteristic polynomial of g. Thus chg (t) is a monic quadratic polynomial with coeﬃcients in F , and chg (0) = det g = 0. Proposition. Let g ∈ G and set f (t) = chg (t). (1) Suppose that f (t) is irreducible over F . The sub-algebra F [g] of A = M2 (F ) is a ﬁeld and the G-centralizer of g is F [g]× . If f (t) = t2 +at+b, then g is G-conjugate to the matrix 0 −b . 1 −a An element h ∈ G is G-conjugate to g if and only if chh (t) = f (t). (2) Suppose f (t) has distinct roots a, b ∈ F × . Then g is G-conjugate to the matrix a0 . 0b The G-centralizer of g is F [g]× . (3) Suppose that f (t) has a repeated root a ∈ F × . Then g is conjugate to exactly one of the matrices a0 a1 , . 0a 0a In the ﬁrst case, g lies in the centre of G. In the second, the G-centralizer of g is F [g]× .

6. Representations of Finite Linear Groups

45

6. Representations of Finite Linear Groups In this section, k denotes a ﬁnite ﬁeld with q elements. We classify the irreducible (complex) representations of the ﬁnite group G = GL2 (k). We use the notation B, N , T , Z , as in §5. 6.1. The group G has order (q 2 −1)(q 2 −q). Further: Lemma. The group G has exactly q 2 −1 conjugacy classes. Proof. The ﬁeld k has exactly (q 2 −q)/2 irreducible, monic polynomials of degree 2. Thus there are (q 2 −q)/2 conjugacy classes as in 5.3 Proposition (1). The second case of 5.3 Proposition gives (q−1)(q−2)/2 classes, while the third case gives 2(q−1). The lemma follows immediately. Thus G has precisely q 2 −1 irreducible representations, up to isomorphism. 6.2. The group N of upper triangular unipotent matrices in G is isomorphic to the additive group of k, via the map x → ( 10 x1 ). If we ﬁx a non-trivial character ψ of k, the function aψ : x → ψ(ax), x ∈ k, ranges over the characters of k as a ranges over k. Under the natural action of T (or B) on N , the characters of N thus fall into two conjugacy classes, consisting respectively of the trivial character and all of the non-trivial ones. 6.3. We ﬁrst consider a straightforward method of constructing irreducible representations of G. Let χ1 , χ2 be characters of k× . We form the character χ = χ1 ⊗ χ2 :

a 0 0 b

−→ χ1 (a)χ2 (b)

of T : we regard this as a character of B, trivial on N , via the quotient B → B/N ∼ = T . We form the induced representation IndG B χ of G, and consider its irreducible components. These are characterized independently as follows: Lemma. Let π be an irreducible representation of G. The following conditions are equivalent: (1) π is equivalent to a G-subspace of IndG B χ, for some character χ of T ; (2) π contains the trivial character of N . Proof. The representation π contains the trivial character of N if and only if it contains an irreducible representation σ of B containing the trivial character of N . This condition is equivalent to σ being the inﬂation of a character of T . The lemma now follows from Frobenius Reciprocity.

46

2 Finite Fields

We have to analyze the induced representation IndG B χ. To do this, we form the character a0 χw : 0 b −→ χ2 (a)χ1 (b) of T . We abuse notation and view this as a character of B trivial on N . We then have: Proposition. Let χ, ξ be characters of T , viewed as characters of B which are trivial on N . G w (1) The space HomG IndG B (χ), IndB (ξ) is trivial unless χ = ξ or χ = ξ . (2) The spaces G G w HomG IndG HomG IndG B (χ), IndB (χ) , B (χ), IndB (χ ) , have the same dimension. This dimension is 2 if χ = χw , and otherwise is 1. Proof. We use the canonical isomorphism of Frobenius Reciprocity G G G ∼ HomG IndG B (χ), IndB (ξ) = HomB rB IndB (χ), ξ

(6.3.1)

where rG B denotes the functor of restriction of representations from G to B. The restriction-induction formula of Mackey gives By G y ∼ IndB (6.3.2) rG B∩B y rB∩B y (χ ) , B IndB (χ) = y∈B\G/B

where χy denotes the character b → χ(yby −1 ) of the group B y = y −1 By. By 5.2, we only have to consider the cases y = 1 and y = w. The term in (6.3.2) corresponding to y = 1 is just χ, and so contributes a factor HomT (χ, ξ) to (6.3.1). The term corresponding to w contributes w w ∼ HomB (IndB T (χ ), ξ) = HomT (χ , ξ).

Altogether, (6.3.1) reduces to G w ∼ HomB (IndG B (χ), IndB (ξ)) = HomT (χ, ξ) ⊕ HomT (χ , ξ),

whence follows the result. We deduce: Corollary 1. Let χ be a character of T , viewed as a character of B which is trivial on N . w (1) The representation IndG B χ is irreducible if and only if χ = χ . χ has length 2, with distinct compo(2) If χ = χw , the representation IndG B sition factors.

6. Representations of Finite Linear Groups

47

Example. Consider the trivial character 1B of B. The trivial character 1G occurs in IndG B 1B , so IndG B 1B = 1G ⊕ StG , for a unique irreducible representation StG , called the Steinberg representation. Clearly, dim StG = q. The representations IndG B χ, for characters χ of T , are called the principal series. The terminology is often extended to include their irreducible components, that is, all irreducible representations of G containing the trivial character of N . Counting, we get: Corollary 2. Up to isomorphism, the group G has precisely 12 (q 2 +q) − 1 irreducible representations which contain the trivial character of N . 6.4. An irreducible representation of G not containing the trivial character of N is called cuspidal. Such a representation σ must contain some non-trivial character of N . Any two non-trivial characters of N are B-conjugate (6.2), so σ contains all non-trivial characters of N . The cuspidal representations cannot be constructed directly by induction. Let l/k be a quadratic ﬁeld extension: thus l is uniquely determined, up to k-isomorphism. The non-trivial k-automorphism of the ﬁeld l is x → xq . A character θ of l× is called regular if θq = θ. Choosing a k-basis of l identiﬁes l with the vector space k ⊕ k and G with Autk (l). The natural action of l× on l thus gives an embedding of l× in G, the G-conjugacy class of which is uniquely determined (5.3 Proposition (1)). We henceforward identify l× with a subgroup E of G. Note that any element of G with irreducible characteristic polynomial is conjugate to an element of E. Let θ be a regular character of E and ψ a non-trivial character of N . We deﬁne a character θψ of ZN by θψ : ( a0 a0 ) u −→ θ(a)ψ(u),

a ∈ k× , u ∈ N.

We observe that, by (6.2), the representation IndG ZN θψ is, up to equivalence, independent of the choice of ψ. Theorem. Let θ be a regular character of E and ψ a non-trivial character of N . (1) The virtual representation G πθ = IndG ZN θψ − IndE θ

is an irreducible cuspidal representation of G, of dimension q−1. (2) Let θ1 , θ2 be regular characters of E; then πθ1 ∼ = πθ2 if and only if θ2 = θ1 or θ2 = θ1q . (3) Every irreducible cuspidal representation of G is of the form πθ , for some regular character θ of E.

48

2 Finite Fields

Proof. We observe that part (3) follows from parts (1) and (2) on counting up the dimensions. One may prove the ﬁrst two parts directly, by computing characters. One ﬁnds: tr πθ (z) = (q−1) θ(z), z ∈ Z; tr πθ (zu) = −θ(z), z ∈ Z, u ∈ N, u = 1; tr πθ (y) = −(θ(y)+θq (y)), y ∈ E Z.

(6.4.1)

The character value tr πθ (g) is zero if g is not conjugate to an element of ZN ∪ E. The character table (6.4.1) gives part (2) straightaway. One checks that 1 |tr πθ (g)|2 = 1, |G| g∈G

so πθ is an irreducible representation of G. Finally, tr πθ (u) = 0, u∈N

so πθ | N does not contain the trivial character of N , and πθ is therefore cuspidal. Exercise. Let ψ be a non-trivial character of N , and consider the representation = IndG N ψ. Let σ be an irreducible representation of G. Show that: (1) If σ = φ ◦ det, for a character φ of k× , then σ does not occur in . (2) Otherwise, σ occurs in with multiplicity one. Further reading. The complex representation theory of the group GLn (k), for a ﬁnite ﬁeld k, was originally worked out by J.A. Green [36]. For a systematic development, emphasizing the related combinatorics, see [60]. The representation theory of reductive algebraic groups over ﬁnite ﬁelds has been intensively studied: see [22], [23], [29] for introductions.

3 Induced Representations of Linear Groups

7. Linear groups over local fields 8. Representations of the mirabolic group 9. Jacquet modules and induced representations 10. Cuspidal representations and coefficients 10a. Appendix: projectivity theorem 11. Intertwining, compact induction and cuspidal representations From now on, we concentrate on the group G = GL2 (F ) over a nonArchimedean local ﬁeld F . The group G inherits a locally proﬁnite topology from the base ﬁeld F , as in 1.4. Our objective is the classiﬁcation of the irreducible smooth representations of G, although we shall not achieve it until the end of Chapter IV. In this chapter, the algebraic subgroups B, N , T of G (as in 5.1) play a pivotal rˆ ole. In parallel with the representation theory of the ﬁnite group GL2 (k) worked out in §6, the irreducible smooth representations of G fall into two broad classes. First, there is a “principal series” of representations: these are the composition factors of representations obtained from characters of T by a process of inﬂation to B and then induction to G. Frobenius Reciprocity characterizes them, among the irreducible smooth representations of G, as those admitting a non-trivial quotient on which N acts trivially. The irreducible smooth representations not obtainable this way are called “cuspidal”. The main result of this chapter (9.11) gives a complete classiﬁcation of the principal series representations. There is a further subgroup of G which plays a surprisingly important part in the classiﬁcation process. This is the “mirabolic subgroup” M of matrices xij ∈ B with x22 = 1. The group M has a very simple representation theory: besides an obvious family of characters, it has a unique irreducible smooth representation. Further, irreducible representations of G decompose very little

50

3 Induced Representations of Linear Groups

when restricted to M . This is the basis of our detailed analysis of the principal series representations of G. In this chapter, we make only rather general remarks about the irreducible cuspidal representations of G. We give a characterization of them more helpful than that of not being in the principal series, and a speculative method for constructing them. This prepares the ground for the analysis in Chapter IV. Some of the arguments and results here, particularly in §11, apply to quite arbitrary locally proﬁnite groups: we point these out as they arise. Most of the time, we work exclusively with GL2 (F ) or its subgroups, and exploit this restriction as much as we can to simplify and abbreviate the treatment. We are rarely unwilling to substitute an explicit matrix calculation for a more general abstract argument.

7. Linear Groups over Local Fields As noted in §1, the group G = GL2 (F ) has many compact open subgroups, of which a small number are of particular importance. This is expressed ﬁrst via various coset decompositions of G, beyond the universal Bruhat decomposition of 5.2. Using these decompositions, one can turn the general measure theory of §3 into an eﬀective computational tool, necessary for handling the integrals arising within the representation theory of G. This section thus amounts to a course of calculus on GL2 (F ), which can be skimmed at ﬁrst reading and referred back to at need. The only result to which we will return is the Duality Theorem at the end, but the general techniques developed here are used frequently. 7.1. For this section, we set V = F ⊕F , and think of it as the space of column vectors with G acting on the left. The standard subgroups B, N , T , Z are as in §5. These are all closed subgroups of G. The group isomorphisms B/N ∼ =T and B ∼ = T N are homeomorphisms. 7.2. Reﬂecting the special nature of the base ﬁeld F , the group G admits decompositions besides the Bruhat decomposition of 5.2. The ﬁrst of these is: (7.2.1)Iwasawa decomposition. Let B be the standard Borel subgroup of G and set K = GL2 (o); then G = BK. Proof. Take g ∈ G; if the (2, 1)-entry of g is zero, then g ∈ B. Otherwise, postmultiplying by the permutation matrix w ∈ K if necessary, we can assume υF (g21 ) υF (g22 ). We can then post-multiply by a lower triangular matrix in K to achieve g21 = 0. Consequently, the quotient space B\G is a continuous image of the compact group K = GL2 (o), and so:

7. Linear Groups over Local Fields

51

Corollary. The quotient space B\G is compact. Continuing with the notation K = GL2 (o), we also have: (7.2.2) Cartan decomposition. Let be a prime element of F . The matrices a 0 , a, b ∈ Z, a b, 0 b form a set of representatives for the coset space K\G/K. Proof. Permuting rows and columns, using the permutation matrix in K, we can arrange for the largest entry of g (in absolute value) to be in the 1, 1 place. Multiplying by elementary matrices from K, we can then arrange for g to be diagonal, and unit factors can be absorbed into K. This gives a G= K 0 0b K. ab

We have to prove this union is disjoint. That is, we have to recover the integers a, b from the coset KgK, where a g = 0 0b . First, we have a+b = υF (det h), for any h ∈ KgK. Next, the group index (K : K ∩ hKh−1 ) depends only on the coset KhK, and (K : K ∩ gKg −1 ) = 1 if b = a, or (q+1)q b−a−1 if b > a. Corollary. If K is a compact open subgroup of G, the set G/K is countable. Proof. As observed in 2.6, it is enough to show that G/K is countable for one choice of K: we take K = GL2 (o). The space K\G/K is certainly countable, and each double coset KgK contains only ﬁnitely many cosets g K. That is, G satisﬁes the countability hypothesis of 2.6. Exercise. Let K be a compact subgroup of G. Show that gKg −1 ⊂ GL2 (o), for some g ∈ G. Deduce that, up to G-conjugacy, GL2 (o) is the unique maximal compact subgroup of G. Hint. There are two steps. One ﬁrst shows that there exists a K-stable olattice in V : consider the o-span of KL, for a randomly chosen o-lattice L. The second consists of showing that the only GL2 (o)-stable lattices in V are the obvious ones pj ⊕ pj , j ∈ Z. 7.3. The standard Iwahori subgroup of G is the compact open subgroup I = ac db : a, d ∈ UF , b ∈ o, c ∈ p . Let N = N w denote the group of lower triangular unipotent matrices in G.

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(7.3.1) Iwahori decomposition. We have I = (I ∩ N )(I ∩ T )(I ∩ N ). More precisely, the product map I ∩ N × I ∩ T × I ∩ N −→ I is bijective, and a homeomorphism, for any ordering of the factors on the left hand side. Proof. The product map is certainly continuous. It is elementary to write down its inverse and observe that it is continuous. Set K = GL2 (o); under the canonical surjection K → GL2 (k), the image of I is the standard Borel subgroup of GL2 (k). The Bruhat decomposition for GL2 (k) implies K = I ∪ IwI. (7.3.2) Combining (7.3.2) with the Iwasawa decomposition (7.2.1) for G, we obtain the more symmetric double-coset decomposition G = BI ∪ BwI = B(I ∩ N ) ∪ Bw(I ∩ N ).

(7.3.3)

The cosets BwI, BI are both open in G. Remark. Let L = o ⊕ o, L = o ⊕ p. The Iwahori subgroup I is then the common G-stabilizer of the two lattices L, L . 7.4. We now describe the Haar measures attached to the various locally proﬁnite groups under discussion. We start with the basic example of the ﬁeld F itself. Lemma. The vector space Cc∞ (F ) is spanned by the characteristic functions of sets a+pm , a ∈ F , m ∈ Z. Proof. Surely the characteristic function of a+pm lies in Cc∞ (F ). Conversely, let Φ ∈ Cc∞ (F ). Since Φ has compact support, there exists n ∈ Z such that supp Φ ⊂ pn . Also, Φ is ﬁxed under translation by a compact open subgroup of F , hence by pm , for some m ∈ Z. Thus Φ is a linear combination of char acteristic functions of sets a+pm , a ∈ pn /pm . If Φ0 denotes the characteristic function of o and µ is a Haar measure on F , we have Φ0 (x) dµ(x) = c0 , F

for some c0 > 0. If Φ1 is the characteristic function of a coset a+pb , a ∈ F , b ∈ Z, then Φ1 (x) dµ(x) = c0 q −b . F

7. Linear Groups over Local Fields

53

This identity suﬃces for integrating any function Φ ∈ Cc∞ (F ). Now take Φ ∈ Cc∞ (F ) and y ∈ F × . Using the identity above, we ﬁnd Φ(xy) dµ(x) = y −1 Φ(x) dµ(x), F

F −υF (y)

where, we recall, y = q . We accordingly deﬁne a measure µ× on F × by dµ× (x) = dµ(x)/ x , meaning the following. If Φ ∈ Cc∞ (F × ), the function x → x −1 Φ(x) (vanishing at 0) lies in Cc∞ (F ), so we can put Φ(x) dµ× (x) = Φ(x) x −1 dµ(x), Φ ∈ Cc∞ (F × ). (7.4.1) F×

F

A simple manipulation shows that (7.4.1) deﬁnes a Haar integral on F × . 7.5. The matrix ring A = M2 (F ) is (as additive group) a product of 4 copies of F and a Haar measure is obtained by taking a (tensor) product of 4 copies of a Haar measure on F . Proposition. Let µ be a Haar measure on A. For Φ ∈ Cc∞ (G), the function x → Φ(x) det x −2 (vanishing on A G) lies in Cc∞ (A). The functional Φ(x) det x −2 dµ(x), Φ ∈ Cc∞ (G), Φ → A

is a left and right Haar integral on G. In particular, G is unimodular. Proof. Let g ∈ G and consider the functionals ⎧ ⎪ ⎪ Φ(gx) dµ(x), ⎨ A Φ ∈ Cc∞ (A). Φ −→ ⎪ ⎪ ⎩ Φ(xg) dµ(x), A

Each is a Haar integral on A and diﬀers from the initial one by a positive constant (depending on g). To evaluate this constant, we take Φ to be the characteristic function of m = M2 (o). In the ﬁrst instance, the function x → Φ(gx) is the characteristic function of the lattice m = g −1 m. Thus

Φ(gx) dµ(x) = µ(m ) = µ(m) (m : m ∩ m )/(m : m ∩ m ).

A

This quotient of indices depends only on the double coset KgK, K = GL2 (o). Taking g in diagonal form (7.2.2), one gets

−2

Φ(gx) dµ(x) = det g

A

Φ(x) dµ(x). A

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The second instance is treated in same way to get −2 Φ(xg) dµ(x) = det g

Φ(x) dµ(x). A

A

The proposition then follows from simple manipulations. For example, if Φ ∈ Cc∞ (G), −2 −2 Φ(xg) det x dµ(x) = det g

Φ(x) det xg −1 −2 dµ(x) A A = Φ(x) det x −2 dµ(x), A

as required. 7.6. We turn to the subgroups B, N , T of G. Since N ∼ = F and T ∼ = F × ×F × , there is nothing more to say about them. We have B = T N ; we deﬁne a linear functional on the space Cc∞ (B) = Cc∞ (T ) ⊗ Cc∞ (N ) by Φ(tn) dµT (t)dµN (n), Φ ∈ Cc∞ (B), Φ −→ T

N

where µT , µN are Haar measures on T , N respectively. One veriﬁes immediately that this functional is left B-invariant, so it is a left Haar integral on B. We are so justiﬁed in denoting it Φ(b) dµB (b). Φ −→ B

The Haar measure µB may be thought of as the tensor product, µB = µT ⊗µN , but the two factors do not commute. This reﬂects the fact that the group B is not unimodular. In the language of 3.3: Proposition. The module δB of the group B is given by δB : tn −→ t2 /t1 , n ∈ N, t = t01 t02 ∈ T. Proof. Setting c = sm, we get

m ∈ N, s =

Φ(bc) dµB (b) = B

T

s1

0 0 s2

(7.6.1)

∈ T,

Φ(ts s−1 ns m) dµT (t)dµN (n).

N

We use the obvious isomorphism N → F to identify µN with a certain Haar measure µF on F . For φ ∈ Cc∞ (N ), we then have −1 φ(s−1 ns) dµN (n) = φ 10 s1 1xs2 dµF (x) N F

= s1 s−1 2 N φ(n) dµN (n).

7. Linear Groups over Local Fields

55

By deﬁnition,

Φ(bc) dµB (b) = δB (c)−1

B

Φ(b) dµB (b), B

and the result follows.

In the notation of 3.4, we now have: −1 ) admits a positive semi-invariant meaCorollary. The space Cc∞ (B\G, δB sure µ, ˙ where δB is given by (7.6.1). If K = GL2 (o), there is a Haar measure µK on K such that f (g) dµ(g) ˙ = f (k) dµK (k), B\G

K

−1 for f ∈ Cc∞ (B\G, δB ).

Proof. The character δB is trivial on the compact group K ∩ B. Restriction −1 ) → Cc∞ (K ∩ B\K, 1), where 1 of functions is an isomorphism Cc∞ (B\G, δB denotes the trivial character of K ∩ B. The semi-invariant measure µ˙ thus restricts to a semi-invariant measure on Cc∞ (K ∩ B\K, 1), but so does any Haar measure on K. We observe that µK is eﬀectively just the restriction of a Haar measure µG on G. Comparing with the proof of 3.4 Proposition, there is a left Haar measure µB on B such that φ(g) dµG (g) = φ(bk) dµB (b)dµG (k), φ ∈ Cc∞ (G). (7.6.2) G

K

B

Exercises. (1) Let I be the standard Iwahori subgroup; let dn , dt, dn be Haar measures on the groups I ∩ N , I ∩ T , I ∩ N respectively. Show that the functional f −→ f (n tn) dn dt dn, f ∈ Cc∞ (I), is a Haar integral on I. (2) Let C = N T N . Show that C is open and dense in G, and that the product map N × T × N → C is a homeomorphism. (3) Let dg be a Haar measure on G. Show that there are Haar measures dn , dt, dn on N , T , N such that f (g) dg = f (n tn)δB (t)−1 dn dt dn, f ∈ Cc∞ (G). G

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7.7. Let σ be a smooth representation of T , viewed as representation of B trivial on N . Corollary 7.2 implies that the canonical inclusion map G c-IndG B σ −→ IndB σ

is an isomorphism. We can therefore apply the Duality Theorem of 3.5 to get: Duality Theorem. Let σ be a smooth representation of T , viewed as representation of B trivial on N , and ﬁx a positive semi-invariant measure µ˙ on −1 ). There is a canonical isomorphism Cc∞ (B\G, δB

IndG Bσ

∨

−1 ∼ ˇ, = IndG B δB ⊗ σ

depending only on the choice of µ. ˙

8. Representations of the Mirabolic Group Before starting on the representation theory of the group G = GL2 (F ), we study the representations of a certain subgroup of G, the so-called mirabolic subgroup M = ( a0 x1 ) : a ∈ F × , x ∈ F . Thus M is the semi-direct product of N by the group S = T ∩ M ∼ = F ×. 8.1. To start with, let (π, V ) be a smooth representation of N and let ϑ be a character of N . We denote by V (ϑ) the linear subspace of V spanned by the vectors π(n)v−ϑ(n)v, n ∈ N , v ∈ V . We set Vϑ = V /V (ϑ): this is the unique maximal N -quotient of V on which N acts via the character ϑ. If ϑ0 is the trivial character of N , we have V (ϑ0 ) = V (N ) (notation of 2.3) and we write Vϑ0 = VN . Lemma. Let µN be a Haar measure on N and ϑ a character of N . (1) Let (π, V ) be a smooth representation of N and v ∈ V . The vector v lies in V (ϑ) if and only if there is a compact open subgroup N0 of N such that ϑ(n)−1 π(n)v dµN (n) = 0.

(8.1.1)

N0

(2) The process (π, V ) → Vϑ is an exact functor from Rep(N ) to the category of complex vector spaces. Proof. We assume ﬁrst that ϑ is the trivial character of N .

8. Representations of the Mirabolic Group

57

The group N ∼ = F is the union of an ascending sequence of compact open subgroups. So, if r vi −π(ni )vi ∈ V (N ), v= i=1

there is a compact open subgroup N0 of N containing all ni . The relation (8.1.1) then holds for this choice of N0 . Conversely, let v ∈ V and suppose (8.1.1) holds. There is an open normal subgroup N1 of N0 such that v ∈ V N1 . The space V N1 carries a representation of the ﬁnite group N0 /N1 . Therefore, in the obvious notation, V N1 = V N1 (N0 /N1 ) ⊕ V N0 (cf. 2.3) and the map w −→ µN (N0 )−1 π(n)w dµN (n), w ∈ V N1 , N0

is the N0 -projection V N1 → V N0 . This has kernel V N1 (N0 /N1 ) ⊂ V (N ) and we have proved (1) for the trivial character of N . Now let ϑ be an arbitrary character of N , and consider the representation (π , V ) of N , where V = V and π (n) = ϑ(n)−1 π(n). We then have V (ϑ) = V (N ) and so (1) follows in general. Part (2) is an immediate consequence of (1). We mention some simple consequences of the lemma. If (π, V ) is a smooth representation of N (or of M ), then V (N ) is an N - (or M -) subspace of V . The exact sequence 0 → V (N ) −→ V −→ VN → 0 gives an exact sequence 0 → V (N )N −→ VN −→ VN → 0 in which the map VN → VN is the identity. Therefore V (N )N = 0

and V (N )(N ) = V (N ).

(8.1.2)

Suppose that ϑ = 1. As N acts trivially on V /V (N ), we have (V /V (N ))ϑ = 0 and so the inclusion V (N ) → V induces an isomorphism V (N )ϑ ∼ = Vϑ .

(8.1.3)

Proposition. Let (π, V ) be a smooth representation of N , and let v ∈ V , v = 0. There exists a character ϑ of N such that v ∈ / V (ϑ).

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Proof. We write

Nj =

1 pj 0 1

,

j ∈ Z.

(8.1.4)

Take v ∈ V , v = 0. We choose j0 ∈ Z such that Nj0 ﬁxes v. For j j0 , let Vj denote the Nj -space generated by v. This is the direct sum of isotypic components Vjη , as η ranges over the characters of Nj trivial on Nj0 . For each η j j0 , there exists ηj such that Vj j = 0; by deﬁnition, we have ηj (n)−1 π(n)v dn = 0. Nj η

The Nj−1 -space generated by Vj j is contained in Vj−1 , so we may choose ηj−1 such that ηj−1 | Nj = ηj . It follows (compare the argument in 1.7) that there exists a character ϑ of N such that, for all j j0 , we have ϑ(n)−1 π(n)v dn = 0. Nj

Therefore v ∈ / V (ϑ), as required. Corollary 1. Let (π, V ) be a smooth representation of N . If Vϑ = 0 for all characters ϑ of N , then V = 0. Now let (π, V ) be a smooth representation of M . The space V (N ) is then an M -subspace of V and VN carries a natural representation of M/N = S. On the other hand, π(S) permutes the subspaces V (ϑ), ϑ = 1, transitively. Explicitly, for s ∈ S, π(s)V (ϑ) = V (ϑ ), where ϑ (n) = ϑ(s−1 ns). We can therefore sharpen Corollary 1 for representations of M : Corollary 2. Let (π, V ) be a smooth representation of M . Suppose that VN = 0 and that Vϑ = 0 for some non-trivial character ϑ of N . Then V = 0. 8.2. We now ﬁx a non-trivial character ϑ of N , and consider the two M -spaces M IndM N ϑ, c-IndN ϑ. Observe that, if ϑ is some other non-trivial character of M N , then IndN ϑ is M -isomorphic to IndM N ϑ and similarly for the compactly induced representations. Proposition. Let ϑ be a non-trivial character of N and set W = IndM N ϑ, W c = c-IndM N ϑ. Let α : W → C denote the canonical map f → f (1). (1) We have W(N ) = W c (N ) = W c and (W/W c )(N ) = 0. (2) The map α induces isomorphisms Wϑ ∼ = C and Wϑc ∼ = C. Proof. Let f ∈ W and n ∈ N . For a ∈ S, we have f (an) = ϑ(ana−1 )f (a). As there is an integer j such that Nj (as in (8.1.4)) ﬁxes f , we see that f (a) = 0 if det a is suﬃciently large. On the other hand, f (an) = f (a) if det a is suﬃciently small, so nf −f vanishes at a if det a is suﬃciently small. This implies that nf −f ∈ W c . Thus W(N ) ⊂ W c and N acts trivially on W/W c .

8. Representations of the Mirabolic Group

59

We next prove that W c (N ) = W c . Let ψ be the character of F deﬁned by ψ(x) = ϑ 10 x1 . Take a ∈ F × , j ∈ Z, j 1. Let fa,j ∈ W c be the function such that au 0 −→ ψ(x), u ∈ UFj , fa,j : 10 x1 0 1 and which vanishes elsewhere. The various functions fa,j span W c over C. We have 1x b 0 b 0 f = ψ(bx)f . a,j a,j 0 1 01 01 We deduce that nfa,j has the same support as fa,j , n ∈ N . We can certainly ﬁnd x ∈ F such that the function u → ψ(aux), u ∈ UFj , is constant, equal to c say, with c = 1. If n = ( 10 x1 ), then nfa,j −fa,j = (c−1)fa,j , whence fa,j ∈ W c (N ) and so W c (N ) = W c . This implies W(N ) = W c also, and we have proved (1). The map α induces a surjection Wϑ → C. On the other hand, since N acts trivially on W/W c , the inclusion W c → W induces an isomorphism Wϑc ∼ = Wϑ . To prove (2), therefore, it is enough to show that any f ∈ W c with f (1) = 0 belongs to W c (ϑ). A function f , vanishing at 1, is a ﬁnite / UFj , so it is enough to treat such linear combination of functions fa,j with a ∈ j functions. However, as a ∈ / UF , there exists x ∈ F such that the function u → ψ(aux)−ψ(x), u ∈ UFj , is a non-zero constant. Taking n = ( 10 x1 ), the same calculation as before shows that nfa,j − ϑ(n)fa,j is a non-zero constant multiple of fa,j , whence fa,j ∈ W c (ϑ) and W c (ϑ) = W c , as required. Corollary. The representation c-IndM N ϑ is irreducible over M . c = 0, the spaces VN , (W c /V )N Proof. Let V be an M -subspace of W c . As WN are both zero. The sequence

0 → Vϑ −→ Wϑc −→ (W c /V )ϑ → 0 is exact. As dim Wϑc = 1, we conclude that dim Vϑ is 0 or 1. In the ﬁrst case, V = 0 by 8.1 Corollary 2. In the second, (W c /V )ϑ = 0 and so W c = V . We display some of the remarks made in the course of the proof of the proposition: Gloss. (1) A function f ∈ W is determined by its restriction to S ∼ = F × . The × × restriction f | F is a smooth function on F .

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(2) A smooth function φ on F × is of the form φ = f | F × , for some f ∈ W, if and only if there exists c > 0 such that φ(x) = 0 for all x satisfying

x > c. (3) A function f ∈ W lies in W c if and only if f | F × ∈ Cc∞ (F × ). Remark. Part (3) implies that the representation IndM N ϑ is never irreducible, for any non-trivial character ϑ of N . The Duality Theorem 3.5 implies M ˇ ∨ ∼ (c-IndM N ϑ) = IndN ϑ.

Thus c-IndM N ϑ provides an example of an irreducible smooth representation with reducible contragredient (cf. 2.10). 8.3. Again let ϑ be a non-trivial character of N . Let (π, V ) be a smooth representation of M . Frobenius Reciprocity (2.4) gives a canonical isomorphism HomN (V, Vϑ ) ∼ = HomM (V, IndM N Vϑ ). Let q : V → Vϑ denote the quotient map and let q be the map V → IndM N Vϑ corresponding to q under this isomorphism. Explicitly, for v ∈ V , q (v) is the function m → q(π(m)v). Theorem. Let (π, V ) be a smooth representation of M . The M -homomorphism M ∼ q : V → IndM N Vϑ induces an isomorphism V (N ) = c-IndN Vϑ . Proof. The N -space Vϑ is a direct sum of copies of ϑ. Therefore IndM N Vϑ is a ϑ. Proposition 8.2 so yields direct sum of copies of IndM N M M (IndM N Vϑ )(N ) = c-IndN Vϑ = (c-IndN Vϑ )(N ).

(8.3.1)

M The M -homomorphism q : V → IndM N Vϑ surely maps V (N ) to (IndN Vϑ )(N ) = c-IndM N Vϑ . Let W = Ker q ∩ V (N ) and C = c-IndM N Vϑ /q (V (N )). The natural map WN → V (N )N is injective, by 8.1 Lemma (2), so WN = 0. Likewise, (c-Ind Vϑ )N is zero, so CN = 0. The map q : V → c-Ind Vϑ induces a map

q,ϑ : Vϑ = V (N )ϑ −→ (c-Ind Vϑ )ϑ . By 8.2 Proposition (2), the canonical N -map Ind Vϑ → Vϑ induces an isomorphism αϑ : (c-Ind Vϑ )ϑ → Vϑ . The composite map αϑ ◦ q,ϑ : Vϑ → Vϑ is the identity. However, the kernel of this map is Wϑ and its cokernel is isomorphic to Cϑ . We have shown that the spaces WN , Wϑ , CN , Cϑ are all zero. By 8.1 Corollary 2, therefore, both W and C are trivial and q : V (N ) → c-Ind Vϑ is an isomorphism, as desired.

9. Jacquet Modules and Induced Representations

61

Corollary. Let (π, V ) be an irreducible smooth representation of M . Either: ∼ F × , or (1) dim V = 1 and π is the inﬂation of a character of M/N = M (2) dim V is inﬁnite and π ∼ = c-IndN ϑ, for any character ϑ = 1 of N . In case (1), dim VN = 1 and Vϑ = 0 for ϑ = 1. In case (2), VN = 0 and dim Vϑ = 1 for all ϑ = 1. Proof. If V (N ) = 0, then N acts trivially on V . The group M/N is abelian, so Schur’s Lemma 2.6 implies dim V = 1 and we are in case (1). If V (N ) = 0, then V (N ) = V and VN = 0. Therefore Vϑ = 0 for all characters ϑ = 1 of N , and so dim V is inﬁnite. The theorem implies that V = V (N ) is M -isomorphic to c-IndM N Vϑ . The N -space Vϑ is a direct sum of copies of ϑ, so V is a direct sum of copies of c-IndM N ϑ and, since it is ϑ. irreducible, V ∼ = c-IndM N

9. Jacquet Modules and Induced Representations We start the process of classifying the irreducible smooth representations of the locally proﬁnite group G = GL2 (F ). In this section, we deal completely with those irreducible smooth representations (π, V ) of G (the “principal series”) for which VN = 0. 9.1. Let (π, V ) be a smooth representation of G. As in 8.1, V (N ) denotes the subspace of V spanned by the vectors v−π(x)v, for v ∈ V and x ∈ N . The space VN = V /V (N ) inherits a representation πN of B/N = T , which is smooth. We call (πN , VN ) the Jacquet module of (π, V ) at N . In particular, the Jacquet functor Rep(G) −→ Rep(T ), (π, V ) −→ (πN , VN ),

(9.1.1)

is exact and additive. Let (σ, W ) be a smooth representation of T . We view σ as a smooth representation of B which is trivial on N , and form the smooth induced repG resentation IndG B σ. (We sometimes abbreviate IndB σ = Ind σ, since B and G are the only groups involved for most of the time.) If (π, V ) is a smooth representation of G, Frobenius Reciprocity (2.4) gives an isomorphism HomG (π, Ind σ) ∼ = HomB (π, σ). However, σ is trivial on N so any B-homomorphism π → σ factors through the quotient map π → πN . We deduce HomG (π, Ind σ) ∼ = HomT (πN , σ).

(9.1.2)

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This has the following consequence: Proposition. Let (π, V ) be an irreducible smooth representation of G. The following are equivalent: (1) The Jacquet module VN is non-zero. (2) The representation π is isomorphic to a G-subspace of a representation IndG B χ, for some character χ of T . Proof. Suppose (2) holds. From (9.1.2) we get HomT (πN , χ) ∼ = HomG (π, Ind χ) = 0, so surely πN = 0. To prove (1)⇒(2), we have to show that the Jacquet module VN admits an irreducible T - (or B-) quotient. Choose v ∈ V , V = 0. Since V is irreducible over G, any element of V is a ﬁnite linear combination of translates π(g)v of v, for various g ∈ G. Write K = GL2 (o). The vector v is ﬁxed by a subgroup K of K of ﬁnite index; let {v1 , v2 , . . . , vr } be the distinct elements of the form π(k)v, k ∈ K. In particular, r (K:K ). Since G = BK, the elements v1 , . . . , vr generate V over B, and their images generate VN over T . Thus VN is ﬁnitely generated as a representation of T . We choose a minimal generating set {u1 , . . . , ut }, t 1, say. A standard Zorn’s Lemma argument shows that VN has a T -subspace U , containing u1 , . . . , ut−1 , and maximal for / U . Then U is a maximal T -subspace of VN and VN /U is an the property ut ∈ irreducible representation of T , hence a character (2.6 Corollary 2). An irreducible smooth representation (π, V ) of G is called cuspidal if VN is zero. In the literature, cuspidal representations are usually called supercuspidal or absolutely cuspidal. On the other hand, if VN = 0, one says that π is in the principal series. 9.2. In the case of a ﬁnite ﬁeld k, we divided the irreducible representations according to whether or not they contained the trivial character of N . For GL2 (F ) we use the existence of an N -trivial quotient, for the following reason: Exercise 1. Let (π, V ) be an irreducible smooth representation of G with a non-trivial π(N )-ﬁxed vector. Show that π = φ ◦ det, for some character φ of F × . Hint. If v ∈ V is ﬁxed by N , it is ﬁxed by the subgroup H of G generated by N and some open subgroup K. Show that, since H contains a lower triangular unipotent matrix, it also contains SL2 (F ).

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Exercise 2. Let (π, V ) be an irreducible smooth representation of G such that dim V is ﬁnite. Show that V has a non-zero π(N )-ﬁxed vector. Deduce that dim V = 1 and π is of the form φ ◦ det, for some character φ of F × . These exercises help to explain the direction we take, although they play no part in the argument to follow. (We will, however, need Exercise 1 at a later stage.) In this connection, we note: Proposition. Any character of G is of the form φ ◦ det, for some character φ of F × . Proof. If χ is a character of G, its kernel contains the commutator subgroup of G. Since F is inﬁnite, this commutator subgroup is SL2 (F ), so χ = φ ◦ det, for some homomorphism φ : F × → C× . The determinant map is surjective and open, so φ is a character. 9.3. An important fact concerns the structure of the Jacquet module (IndG B χ)N of an induced representation. Here, it is no more diﬃcult to give a very general result. Let µN be a Haar measure on N and let t ∈ T . The measure S → µN (t−1 St) is the Haar measure δB (t)µN , for δB as in (7.6.1): f (txt−1 ) dµN (x) = δB (t) f (x) dµN (x), f ∈ Cc∞ (N ). N

N

As before, let w denote the permutation matrix w = 01 10 . If σ is a smooth representation of T , we can form the representation σ w : t → σ(wtw−1 ), and view it as a representation of B which is trivial on N . As in 2.4, ασ denotes the canonical B-map Ind σ → σ given by f → f (1). It induces a canonical T -map (Ind σ)N → σ, which we continue to denote ασ . Restriction-Induction Lemma. Let (σ, U ) be a smooth representation of T and set (Σ, X) = IndG B σ. There is an exact sequence of representations of T : −1 σ −→ ΣN −−−α−−→ σ → 0. 0 → σ w ⊗ δB Proof. By deﬁnition, X is the space of G-smooth functions f : G → U such that f (bg) = σ(b)f (g), b ∈ B, g ∈ G. The canonical map ασ : X → U amounts to restriction of functions to B. Set V = Ker ασ . Thus V provides a smooth representation of B and there is an exact sequence 0 → VN −→ XN −→ U → 0. We have to identify the T -representation VN .

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We recall that G = B ∪ BwN . A function f ∈ X thus lies in V if and only if supp f ⊂ BwN . More precisely: Lemma. Let f ∈ X; then f ∈ V if and only if there is a compact open subgroup N0 of N (depending on f ) such that supp f ⊂ BwN0 . Proof. A function f ∈ X lies in V if and only if f (1) = 0. Since f is Gsmooth, such a function f vanishes on a set BN0 , where N0 is some compact open subgroup of N . The identity (for x = 0) −1 −1 −1 −1 1 0 −x 0 1x 1x 1x = w ∈ Bw x1 0

1

0

x

0

1

0

1

implies that supp f ⊂ BwN0 , for some compact open subgroup N0 of N , as required. Let f ∈ V ; in view of the lemma, we can deﬁne a function fN : T → U by fN (x) = f (xwn) dn = σ(x) fN (1), x ∈ T. N

By 8.1 Lemma, the kernel of the map f → fN is V (N ), and so f → fN (1) gives a bijective map VN → U . Taking t ∈ T and f ∈ V , we have (tf )N (x) = f (xwnt) dn N −1 = δB (t−1 ) f (xtw wn) dn = δB (t) (tw fN )(x). N

−1 Thus f → fN (1) is a B-homomorphism V → σ w ⊗ δB inducing a T −1 w ∼ isomorphism VN = σ ⊗ δB .

9.4. The irreducible representations of G exhibit a helpful ﬁniteness property: Proposition. Let (π, V ) be an irreducible smooth representation of G which is not cuspidal. The representation π is admissible. Proof. By deﬁnition, VN = 0. By 9.1 Proposition, π is equivalent to a subrepresentation of IndG B χ, for some character χ of T . It is enough, therefore, to prove that Ind χ is admissible. We ﬁx a compact open subgroup K of G; shrinking it if necessary, we may assume that K ⊂ K0 = GL2 (o). The space X K of K-ﬁxed points in Ind χ consists of the functions f : G → C satisfying f (bgk) = χ(b)f (g),

b ∈ B, g ∈ G, k ∈ K.

(9.4.1)

We have G = BK0 , so the set B\G/K is ﬁnite, and each double coset BgK supports, at most, a one-dimensional space of functions satisfying (9.4.1) (cf. 3.5). Thus X K is ﬁnite-dimensional, as required. Remark. The irreducible cuspidal representations of G are likewise admissible, but the proof requires diﬀerent techniques: see 10.2 Corollary below.

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9.5. We introduce another notation. If (π, V ) is a smooth representation of G and φ is a character of F × , we deﬁne a smooth representation (φπ, V ) of G by setting φπ(g) = φ(det g) π(g), g ∈ G. (9.5.1) One calls φπ the twist of π by φ. Similarly for characters of T : if χ = χ1 ⊗ χ2 is a character of T and φ is a character of F × , then we put φ · χ = φχ1 ⊗ φχ2 . If we inﬂate φ · χ to a representation of B trivial on N , we get φ · χ = (φ ◦ det | B) ⊗ χ. It follows immediately that G ∼ (9.5.2) IndG B (φ · χ) = φ IndB χ. This allows us to make convenient adjustments to the character χ without changing the essential structure of the induced representation. 9.6. We aim to give a precise account of the structure of representations of the form IndG B χ. The main step is: Irreducibility Criterion. Let χ = χ1 ⊗ χ2 be a character of T , and set (Σ, X) = IndG B χ. is either (1) The representation (Σ, X) is reducible if and only if χ1 χ−1 2 the trivial character or the character x → x 2 of F × . (2) Suppose that (Σ, X) is reducible. Then: (a) the G-composition length of X is 2; (b) one composition factor of X has dimension 1, the other is of inﬁnite dimension; (c) X has a 1-dimensional G-subspace if and only if χ1 χ−1 2 = 1; 2 (d) X has a 1-dimensional G-quotient if and only if χ1 χ−1 2 (x) = x , × x∈F . We will reﬁne this to a classiﬁcation of the irreducible principal series representations in 9.11 below. The proof of the theorem occupies paragraphs 9.7–9.9 to follow. 9.7. We use the notation of 9.6. Let V = {f ∈ X : f (1) = 0}. This is a B-subspace of X and we have an exact sequence 0 → V −→ X −→ C → 0, where the one-dimensional space C = X/V carries the character χ of T . By −1 w the Restriction-Induction Lemma (9.3), VN ∼ χ . = δB

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Proposition. Let W be the kernel V (N ) of the canonical map V → VN . The space W is irreducible over B. Proof. We shall actually prove that W is irreducible as a representation of the mirabolic group M of §8. We observe that, by (8.1.2), WN = 0 and W = W (N ). Lemma. For f ∈ V , deﬁne a function fN ∈ Cc∞ (N ) by fN (n) = f (wn), n ∈ N . The map V −→ Cc∞ (N ), (9.7.1) f −→ fN , is an N -isomorphism. Proof. As in 9.3 Lemma, the support of f ∈ V is contained in BwN , for some compact open subgroup N of N . The assertions follow immediately (observing that the notation fN here is not the same as that in 9.3). For φ ∈ Cc∞ (N ) and a ∈ F × , we deﬁne aφ ∈ Cc∞ (N ) by −1 aφ 10 x1 = χ2 (a) φ 1 a x . 0

×

1

Cc∞ (N )

This gives an action of F on which we regard as an action of the group S of matrices ( a0 10 ). We combine this with the natural action of N to give Cc∞ (N ) the structure of a smooth representation of M . With this structure, the map (9.7.1) is a M -isomorphism. Let ϑ be a non-trivial character of N . The map f → ϑf is a linear automorphism of V = Cc∞ (N ) carrying V (N ) to V (ϑ). Since V /V (N ) has dimension 1, we deduce that dim Vϑ = 1 also. However, since N acts trivially on VN , the inclusion W → V induces an isomorphism Wϑ ∼ = Vϑ , whence Wϑ ∼ = ϑ. We M ∼ apply 8.3 Theorem to get W = W (N ) = c-IndN ϑ which, by 8.2 Corollary, is irreducible. As a direct consequence of the Proposition, we have: Corollary. As a representation of B or of M , IndG B χ has composition length 3. Two of the composition factors have dimension one, and the third is of inﬁnite dimension. In particular, the G-composition length of the representation IndG B χ is at most 3. 9.8. We continue with the same notation and observe: Proposition. The following are equivalent: (1) χ1 = χ2 ; (2) X has a one-dimensional N -subspace. When these conditions hold, (3) X has a unique one-dimensional N -subspace X0 ; (4) X0 is a G-subspace of X, and it is not contained in V .

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Proof. If (1) holds, we may as well take χ1 = χ2 = 1 (cf. (9.5.2)). The (nonzero) constant function then spans a one-dimensional G-subspace of X. Conversely, let f ∈ X span an N -stable subspace of dimension 1. Thus N acts on f (by right translation) as a character. The support of f is leftinvariant under B, so supp f is either G or BwN . The second case is impossible: if f (1) = 0, then the support of f is conﬁned to BwN0 , for some compact open subgroup N0 of N (9.3 Lemma). Thus supp f = G and f vanishes nowhere. In particular f (1) = 0, and f ∈ / V . The canonical N -map X → C = X/V identiﬁes the N -space Cf with the trivial N -space C. It follows that N ﬁxes f under right translation. Take x ∈ F × , and consider the identity −1 −1 1 0 −x 0 w 10 x1 = 1 x . −1 x 0 1 1 0 x 1 0 . If x is suﬃciently large, then f is ﬁxed under right translation by x−1 1 So, as f is ﬁxed by N , we have f (w) = χ1 (−1) χ−1 1 χ2 (x) f (1), for all x ∈ F × of suﬃciently large absolute value. Thus χ1 = χ2 = φ, say, and f (g) = φ(det g)f (1). We have proved (1) ⇔ (2), and that the one-dimensional N -subspace is uniquely determined. We have already shown that it is not contained in V . 9.9. We now ﬁnish the proof of the Irreducibility Criterion 9.6. Assume X is reducible. Its G-length is 2 or 3, and it has either a ﬁnite-dimensional Gsubspace or a ﬁnite-dimensional G-quotient (9.7 Corollary). Assume the ﬁrst alternative. Thus X has a one-dimensional N -subspace, and we are in the situation of 9.8: X has a one-dimensional G-subspace L, and χ1 = χ2 = φ, say. Moreover, G acts on L as the character φ ◦ det and L ∩ V = 0 (notation of 9.7). The quotient Y = X/L is therefore B-isomorphic to V . If X has G-length 3, then Y has G-length 2. However, V has B-length 2 and a unique B-quotient, which is of dimension 1. This gives a G-quotient of Y on which G must act as a character φ ◦ det (9.2). This would force φ ⊗ φ −1 to appear as a factor in the Jacquet module YN ∼ , which it cannot. = φ · δB Thus X has G-length 2 and we are in case (2)(c) of 9.6. In the other alternative, X has a ﬁnite-dimensional G-quotient. The repˇ therefore has a ﬁnite-dimensional G-subspace, and we are back resentation X −1 ˇ∼ ˇ so with the ﬁrst alternative. By the Duality Theorem of 7.7, X = IndG B δB χ, we are in the case (2)(d) of 9.6. Thus, in part (1) of the theorem, we have shown that X reducible implies χ has the stated form. The converse is given by 9.8 and the dual case. We have also proved statements (a)–(d).

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9.10. To get a classiﬁcation of the irreducible, non-cuspidal representations of G, we need to investigate the homomorphisms between induced representations: G Proposition. Let χ, ξ be characters of T . The space HomG (IndG B χ, IndB ξ) w −1 has dimension 1 if ξ = χ or χ δB , zero otherwise.

Proof. We use Frobenius Reciprocity (9.1.2): G ∼ HomG (IndG B χ, IndB ξ) = HomT ((Ind χ)N , ξ).

The Jacquet module (Ind χ)N ﬁts into the exact sequence −1 −→ (Ind χ)N −→ χ → 0. 0 → χw δB −1 , then this sequence splits and the result follows If we assume that χ = χw δB −1 amounts to χ1 (x) = x χ2 (x), x ∈ F × . immediately. The equation χ = χw δB In this case, Ind χ is irreducible and the result again follows.

By way of some examples, we examine in more detail the case where IndG Bχ × is reducible. Thus there is a character φ of F such that χ = φ · 1T or χ = −1 . Twisting does not aﬀect the situation materially, so we assume φ = 1. φ · δB G Consider ﬁrst the case IndG B 1T . The irreducible G-quotient of IndB 1T is called the Steinberg representation of G, and is denoted StG : 0 → 1G −→ IndG B 1T −→ StG → 0.

(9.10.3)

−1 . Likewise, if φ is a character of Its dimension is inﬁnite and (StG )N ∼ = δB × F , we have an exact sequence

0 → φG −→ IndG B φT −→ φ · StG → 0, where φG = φ ◦ det and φT = φ ⊗ φ. (Representations of the form φ · StG are sometimes called special.) Taking the smooth dual of (9.10.3), we get an exact sequence G −1 0 → St∨ G −→ IndB δB −→ 1G → 0.

The proposition implies

StG ∼ = St∨ G.

(9.10.4)

(9.10.5)

Remark. The proposition also implies that the space EndG (Ind 1T ) has dimension 1, while Ind 1T is not irreducible. Thus the converse of Schur’s Lemma fails in this context, as remarked in 2.6. Observe also the imperfect parallelism between the proposition above and the corresponding result (6.3) for the ﬁnite ﬁeld case.

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69

9.11. We introduce a new notation. If σ is a smooth representation of T , we deﬁne G −1/2 ⊗ σ). (9.11.1) ιG B σ = IndB (δB This provides another exact functor Rep(T ) → Rep(G), known as normalized or unitary smooth induction. It gives rise to more convenient combinatorics, for example: G ∨ ιB σ ∼ ˇ. (9.11.2) = ιG Bσ In this language, the Irreducibility Criterion (9.6) and 9.10 Proposition say: Lemma. (1) Let χ = χ1 ⊗χ2 be a character of T . The representation ιG B χ is reducible ±1 is one of the characters x →

x

of F × or, if and only if χ1 χ−1 2 ±1/2 × equivalently, χ = φ · δB for some character φ of F . G (2) Let χ, ξ be characters of T . The space HomG (ιG B χ, ιB ξ) is not zero if and only if ξ = χ or ξ = χw . Gathering up our earlier arguments and results, we get: Classiﬁcation Theorem. The following is a complete list of the isomorphism classes of irreducible, non-cuspidal representations of G: ±1/2

for (1) the irreducible induced representations ιG B χ, where χ = φ · δB any character φ of F × ; (2) the one-dimensional representations φ ◦ det, where φ ranges over the characters of F × ; (3) the special representations φ · StG , where φ ranges over the characters of F × . ∼ The classes in this list are all distinct except that, in (1), we have ιG Bχ = G w ιB χ . Proof. The examples in 9.10 show that every irreducible, non-cuspidal representation of G appears in this list. The relations between the irreducibly induced ones are given by 9.10 Proposition. The same result also implies that no special representation φ · StG can be equivalent to φ · StG , φ = φ, or any irreducibly induced representation.

10. Cuspidal Representations and Coeﬃcients In 9.1, we deﬁned an irreducible smooth representation (π, V ) of GL2 (F ) to be cuspidal if its Jacquet module (πN , VN ) is trivial or, equivalently, if it is not isomorphic to a composition factor of an induced representation IndG B χ, for

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any character χ of T . Such a negative and exclusive approach yields essentially no information about this particularly important class of representations. We now give an alternative deﬁnition, valid in a much wider context, and show it is equivalent to the original one. It provides the starting point for a method of constructing cuspidal representations. From this new viewpoint, irreducible cuspidal representations have a striking algebraic property: they are projective objects in the appropriate subcategory of Rep(G). We have no direct need for this result, but we have included it as an appendix. 10.1. Let (π, V ) be a smooth representation of G = GL2 (F ); from vectors v ∈ V , vˇ ∈ Vˇ , we get a smooth function on G by γvˇ⊗v : g −→ ˇ v , π(g)v. We let C(π) be the vector space spanned by the functions γvˇ⊗v , vˇ ⊗ v ∈ Vˇ ⊗ V . The functions f ∈ C(π) are called the (matrix) coeﬃcients of π. The space Vˇ ⊗V carries a smooth representation of the group G×G, while G × G acts on the function space C(π) by translation: the ﬁrst factor acts by left translation and the second by right translation. The map vˇ ⊗ v → γvˇ⊗v is then a surjective G × G-homomorphism Vˇ ⊗ V → C(π). The space C(π) is primarily, but not exclusively, of interest in the case where π is irreducible. When π is irreducible, the centre Z of G acts on V via the central character ωπ of π and γ(zg) = ωπ (z)γ(g),

z ∈ Z, g ∈ G, γ ∈ C(π).

The support of a coeﬃcient is therefore invariant under translation by Z. Deﬁnition. Let (π, V ) be an irreducible smooth representation of G; one says that π is γ-cuspidal if every γ ∈ C(π) is compactly supported modulo Z. The term “γ-cuspidal” is a convenient, but temporary, expedient. Convention. To save adjectives, if a representation is described as cuspidal or γ-cuspidal, it is implicitly assumed to be smooth. We ﬁrst achieve some technical control: Proposition. (1) If (π, V ) is an irreducible γ-cuspidal representation of G, then π is admissible. (2) Let (π, V ) be an irreducible admissible representation of G, and suppose that some non-zero coeﬃcient of π is compactly supported modulo Z; then π is γ-cuspidal.

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71

Proof. In part (1), we suppose for a contradiction that π is not admissible. We choose a compact open subgroup K such that V K has inﬁnite dimension. This dimension, we note, is countable (2.6). The dimension of Vˇ K ∼ = HomC (V K , C) is therefore uncountable. We ﬁx a non-zero v ∈ V K and consider the map Γv : Vˇ K → C(π) given by vˇ → γvˇ⊗v . Since the translates gv, g ∈ G, span V , the map Γv is injective. Its image is a space of functions f on G, satisfying f (zkgk ) = ωπ (z)f (g),

g ∈ G, z ∈ Z, k, k ∈ K,

(10.1.1)

and supported on a ﬁnite union of cosets ZKgK. The dimension of Γv (Vˇ K ) is therefore at most countable, while Γv is injective and dim Vˇ K is uncountable. This gives the desired contradiction. We turn to part (2). The smooth dual (ˇ π , Vˇ ) is irreducible and admissible ˇ (2.10). We view the space V ⊗ V as a smooth representation of G × G, and hence as a smooth module over H(G × G) = H(G) ⊗ H(G). If K is a compact open subgroup of G, we have (Vˇ ⊗ V )K×K = (eK ⊗ eK ) ∗ (Vˇ ⊗ V ) = Vˇ K ⊗ V K . If K is suﬃciently small, the spaces V K , Vˇ K are ﬁnite-dimensional simple modules over H(G, K). The Jacobson Density Theorem implies that Vˇ K ⊗V K is a simple module over H(G, K) ⊗ H(G, K) ∼ = H(G × G, K × K). This holds for all suﬃciently small K, so Vˇ ⊗ V is an irreducible admissible G × G-space (4.3 Corollary). The surjective G × G-homomorphism γ : Vˇ ⊗ V → C(π) is therefore an isomorphism and C(π) is irreducible over G × G. If γ ∈ C(π) is non-zero, any γ ∈ C(π) is a ﬁnite linear combination of functions (g, h)γ, (g, h) ∈ G × G. If γ is compactly supported modulo Z, then so is γ . Remark 1. All of the preceding deﬁnitions and arguments apply in the general case, where G is a unimodular locally proﬁnite group satisfying 2.6 Hypothesis. Indeed, 2.6 is only used at one point, in the proof of part (1) of the proposition. Even this can be avoided by noting that the dual of a vector space W has dimension strictly greater than dim W except when dim W is ﬁnite. Remark 2. In the general context of Remark 1, part (2) of the proposition fails when the irreducible smooth representation (π, V ) is not admissible. An example is given by the representation c-IndM N ϑ considered in 8.2: see especially 8.2 Remark. 10.2. The reason for introducing the notion of γ-cuspidality is explained by: Theorem. Let (π, V ) be an irreducible smooth representation of G; then π is cuspidal if and only if it is γ-cuspidal.

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Proof. We ﬁrst assume that π is cuspidal, and show it is γ-cuspidal. Let be a prime element of F , and put 0 . t= 0 1 The set T + of powers tn , n 0, then provides a family of representatives for ZK\G/K, K = GL2 (o) (7.2.2). Lemma. Let v ∈ V , vˇ ∈ Vˇ . There exists m 0 such that γvˇ⊗v (tn ) = 0, for all n m. Proof. We choose a compact open subgroup N1 of N which ﬁxes vˇ. Since VN = 0, we have v ∈ V (N ) and (8.1) there is a compact open subgroup N2 of N such that π(x)v dx = 0. N2

We then have

π(x)v dx = 0, N0

for any compact open subgroup N0 of N containing N2 . However, there exists m 0 such that ta N2 t−a ⊂ N1 for all a m. For such a we have (for certain positive constants k1 , k2 ) a ˇ π (x−1 )ˇ v , π(ta )v dx ˇ v , π(t )v = k1 N1 ˇ π (t−a )ˇ v , π(t−a xta )v dx = k1 N1 ˇ π (t−a )ˇ v , π(x)v dx = k2 t−a N1 ta

= 0, since t−a N1 ta ⊃ N2 .

Continuing with the proof of the theorem, we ﬁx a non-zero coeﬃcient f = γvˇ⊗v of π. We write K = GL2 (o) and let K be an open normal subgroup of K ﬁxing both vˇ and v. We let k1 , k2 , . . . , kr be a set of coset representatives for K/K . Thus, if g ∈ G, there exists n 0 such that ZK ki−1 tn kj K . ZKgK = ZKtn K = i,j

It follows that supp f ⊂

1i,jr

ZK (supp fij ∩ T + ) K ,

10a. Appendix: Projectivity Theorem

73

where fij denotes the coeﬃcient function x → f (ki xkj−1 ). This set is compact modulo Z, by the lemma. It follows that all coeﬃcients γvˇ⊗v of π are compactly supported modulo Z, and π is therefore γ-cuspidal. Combining this argument with 9.4 Proposition and 10.1 Proposition, we have shown: Corollary. Every irreducible smooth representation of G = GL2 (F ) is admissible. We now prove the converse statement in the theorem. Let (π, V ) be an irreducible γ-cuspidal representation of G. In particular, (π, V ) is admissible. By 2.10 Proposition, the dual (ˇ π , Vˇ ) is irreducible and admissible. Let Kn n denote the group 1+p M2 (o), n 1. We take v ∈ V and choose n 1 so that v is ﬁxed by π(Kn ). We take t as before. v , π(g)v is compactly supported modulo For vˇ ∈ Vˇ Kn , the function g → ˇ Z; we deduce that ˇ v , π(ta )v = 0 for all a ∈ Z suﬃciently large. Since Vˇ Kn is of ﬁnite dimension there is a constant c such that ˇ v , π(ta )v = 0 for all Kn a ˇ and all a c. This implies π(eKn )π(t )v = 0 for a c. We write, vˇ ∈ V for j ∈ Z, j Nj = 10 p1 , Nj = p1j 01 , Tn = Kn ∩ T, so that Kn = Nn Tn Nn . Set Kn for a c,

(a)

= t−a Kn ta = Nn−a Tn Nn+a ; we then have,

0 = π(eKn )π(ta )v = π(ta )π(eK (a) )v n π(x)π(eK (a) ∩Kn )v. = π(ta ) n

x∈Nn−a /Nn (a)

However, v is ﬁxed by Kn ∩ Kn , so this equation reduces to 0 = k π(ta ) π(x)v dx, Nn−a

for a constant k > 0 depending on the choice of a Haar measure dx on N . We deduce that v ∈ V (N ) (8.1 Lemma). This applies to all v ∈ V , so π is cuspidal, as required.

10a. Appendix: Projectivity Theorem We give another property of γ-cuspidal representations. We will not use this result, but we have included it for its power and beauty. It also holds in a very general context.

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10a.1. Let G be a locally proﬁnite group with centre Z. Let χ be a character of Z and let (π, V ) be a smooth representation of G. We recall that π admits χ as central character if π(z)v = χ(z)v, for z ∈ Z, v ∈ V . Projectivity Theorem. Let G be a unimodular locally proﬁnite group, satisfying (2.6) and with centre Z. Assume that any character χ : Z → R× + extends . to a character G → R× + Let (π, V ) be an irreducible γ-cuspidal representation of G, and let (τ, U ) be a smooth representation of G admitting ωπ as central character. Let f : U → V be a surjective G-homomorphism. There exists a G-homomorphism φ : V → U such that f ◦ φ = 1V . 10a.2. The group G/Z is locally proﬁnite. One sees easily that it is unimodular. Indeed, let µG , µZ be Haar measures on G, Z respectively. By 3.4 Proposition, there is a unique right Haar measure µ˙ on G/Z such that f (g) dµG (g) = f (zg) dµZ (z) dµ(g), ˙ f ∈ Cc∞ (G). G

G/Z

Z

Symmetrically, µ˙ is also a left Haar measure on G/Z. Schur’s orthogonality relation. Let dg˙ be a Haar measure on G/Z, and let v1 , v2 ∈ V , vˇ1 , vˇ2 ∈ Vˇ . The function g −→ ˇ π (g)ˇ v1 , v1 ˇ v2 , π(g)v2 ,

g ∈ G,

is invariant under translation by Z and ˇ π (g)ˇ v1 , v1 ˇ v2 , π(g)v2 dg˙ = d(π)−1 ˇ v1 , v2 ˇ v2 , v1 , G/Z

for a constant d(π) > 0 depending only on π and the measure dg. ˙ Proof. Since π is γ-cuspidal, the integrand has compact support in G/Z and the integral converges. If we ﬁx, say, vˇ1 and v2 , the integral determines a Ginvariant pairing Vˇ × V → C. Such a pairing is given by a G-homomorphism Θ : Vˇ → Vˇ (cf. Exercise 2.10). Since V is admissible (10.1 Proposition) and irreducible, the same applies to Vˇ (2.10) and Schur’s Lemma (2.6) implies that any G-invariant pairing Vˇ × V → C is a scalar multiple of the standard one. Therefore, there is a constant cvˇ1 ,v2 such that ˇ π (g)ˇ v1 , v1 ˇ v2 , π(g)v2 dg˙ = cvˇ1 ,v2 ˇ v2 , v1 . G/Z

The function (ˇ v1 , v2 ) → cvˇ1 ,v2 is again a G-invariant bilinear pairing Vˇ × V → C, so v1 , v2 , cvˇ1 ,v2 = cπ ˇ for a constant cπ .

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It remains only to prove that cπ > 0. The assumption on G allows us to replace π by a twist and assume that |ωπ | = 1. The space V then admits a positive deﬁnite, G-invariant Hermitian form h, constructed as follows. One chooses a nonzero element vˇ ∈ Vˇ and sets ˇ v , π(g)v1 ˇ v , π(g)v2 dg. ˙ h(v1 , v2 ) = G/Z

There is then a complex anti-linear G-isomorphism Θ : (π, V ) → (ˇ π , Vˇ ) such that h(v1 , v2 ) = Θv1 , v2 . Schur’s Lemma again implies that h is the unique G-invariant, positive deﬁnite Hermitian form on V , up to a positive constant factor. Going through the same argument, one sees that h(π(g)v1 , v2 ) h(v3 , π(g)v4 ) dg˙ = bπ h(v1 , v4 ) h(v3 , v2 ), G/Z

for a constant bπ . On taking v1 = v2 = v3 = v4 = 0, one sees that bπ > 0. On the other hand, h(π(g)v1 , v2 ) h(v3 , π(g)v4 ) dg˙ = ˇ π (g)Θ(v1 ), v2 )Θ(v3 ), π(g)v4 dg˙ G/Z

G/Z

= cπ Θ(v1 ), v4 Θ(v3 ), v2 = cπ h(v1 , v4 ) h(v3 , v2 ). Therefore cπ = bπ > 0, as required. Remark. Let (π, V ) be an irreducible smooth representation of G such that |ωπ | = 1. One says that π is square-integrable modulo Z if |ˇ v , π(g)v|2 dg˙ < ∞ G/Z

for all vˇ ∈ Vˇ , v ∈ V . The orthogonality relation then holds for π, with exactly the same proof. The positive constant d(π) is called the formal degree of π, relative to the measure dg. ˙ (For a full discussion of square-integrable representations of GL2 (F ), see 17.4 et seq. below.) 10a.3. We now prove the Projectivity Theorem. First, we need to generalize the constructions of 4.1, 4.2. Let χ be a character of F × . Let Hχ (G) be the space of locally constant functions f : G → C, which are compactly supported modulo Z, such that f (zg) = χ(z)−1 f (g), z ∈ Z, g ∈ G. Using a Haar measure on G/Z, we deﬁne convolution on Hχ (G) as in (4.1). If (σ, W ) is a smooth representation of G admitting χ as central character, we extend the action of G on W to one of Hχ (G), just as in 4.2.

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We let (π, V ) be an irreducible γ-cuspidal representation of G, as in the theorem. We abbreviate ω = ωπ . We take a smooth representation (τ, U ) of G, admitting ω as central character, and a G-surjection f : U → V . If u ∈ U satisﬁes f (u) = 0, the restriction of f to the G-space τ (Hω (G))u generated by u is still surjective. Composing it with the obvious G-surjection Hω (G) → τ (Hω (G))u, we get a G-surjection Π : Hω (G) −→ V, φ −→ f (τ (φ)u) = π(φ)v0 , where v0 = f (u). It is enough to show that Π splits over G. v0 , v0 = 1. The function We choose a vector vˇ0 ∈ Vˇ such that d(π)−1 ˇ π (g)ˇ v0 , v lies in Hω (G) and the map φv : g → ˇ Φ : V −→ Hω (G), v −→ φv , is a G-homomorphism. The composite map Π ◦ Φ is given by π(g)φw (g)v0 dg, ˙ w ∈ V. w −→ π(φw )v0 = G/Z

For w ˇ ∈ Vˇ , this gives

w, ˇ π ˇ (g)v0 ˇ π (g)ˇ v0 , w dg˙ = w, ˇ w,

w, ˇ ΠΦ(w) = G/Z

whence ΠΦ(w) = w, as required.

11. Intertwining, Compact Induction and Cuspidal Representations We describe a method for constructing irreducible cuspidal representations of G = GL2 (F ), using compact induction from open subgroups. At this stage, it is a purely formal matter: it is not clear that the necessary hypotheses are satisﬁed suﬃciently often to give useful results. Such issues are the subject of the next chapter. Here, we have to be content with one interesting example. 11.1. We start with general considerations so, for the time being, G is a unimodular locally proﬁnite group with the countability property 2.6. Throughout, Z denotes the centre of G. If K is a compact open subgroup of G, we for the set of isomorphism classes of irreducible smooth representawrite K tions of K.

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Deﬁnition 1. For i = 1, 2, let Ki be a compact open subgroup of G and let i . Let g ∈ G. The element g intertwines ρ1 with ρ2 if ρi ∈ K HomK1g ∩K2 (ρg1 , ρ2 ) = 0, where ρg1 denotes the representation x → ρ1 (gxg −1 ) of the group K1g = g −1 K1 g. As a property of g, this depends only on the double coset K1 gK2 . The deﬁnition applies equally if the Ki are just closed subgroups of G; we will often need it in the case where the Ki are open and compact modulo the centre of G. Deﬁnition 2. Let K be a compact open subgroup of G, and let (π, V ) be a smooth representation of G. We say that π contains ρ, or ρ occurs in π, if HomK (ρ, π) = 0. Again, we can use the same deﬁnition in more general contexts, for example, if K is open and compact modulo the centre of G and π admits a central character (see 2.7). We also use it when G is compact and K is a closed subgroup of G. Remaining with the compact open case for the time being, the signiﬁcance of the concept of intertwining is ﬁrst indicated by the following. Proposition 1. For i = 1, 2, let Ki be a compact open subgroup of G and i . Let (π, V ) be an irreducible smooth representation of G which let ρi ∈ K contains both ρ1 and ρ2 . There then exists g ∈ G which intertwines ρ1 with ρ2 . Proof. For each i, we have the decomposition of V into Ki -isotypic components (2.3 Proposition). The hypothesis is equivalent to V ρi = 0, i = 1, 2. Let e2 denote the K2 -projection V → V ρ2 . Since π is irreducible and g ρ1

= 0, the spaces π(g −1 )V ρ1 = V ρ1 , g ∈ G, span V . We can therefore V choose g ∈ G such that e2 ◦ π(g −1 ) induces a non-zero map V ρ1 → V ρ2 : this is the required element g. Take (Ki , ρi ) as in the Proposition. The representations ρg1 , ρ2 of K1g ∩ K2 are semisimple, so the spaces HomK1g ∩K2 (ρg1 , ρ2 ),

HomK1g ∩K2 (ρ2 , ρg1 ) ∼ = HomK

−1

g −1 1 ∩K2

(ρg2 , ρ1 )

have the same dimension. Therefore g intertwines ρ1 with ρ2 if and only if g −1 intertwines ρ2 with ρ1 .

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Language. (1) We say that the ρi intertwine in G if there exists g ∈ G which intertwines ρ1 with ρ2 . The relation of G-intertwining between pairs (Ki , ρi ) is therefore symmetric and reﬂexive; it is not transitive. (2) If we have a single pair (K, ρ), we say that g intertwines ρ if it intertwines ρ with itself. Remark. We will often wish to use this approach when K is not compact, but only an open subgroup of G which is compact modulo Z. One cannot, in general, decompose a smooth representation (π, V ) of G into a direct sum of K-isotypic components. Such a decomposition does exist (2.7) if (π, V ) admits a central character ωπ , in particular, if π is irreducible. With this caveat, we can treat open, compact modulo centre subgroups of G in the same way as compact open subgroups. We will later (in Chapter VI) need another intertwining criterion. (We use the notation of 4.4 here.) Proposition 2. Let K be a compact open subgroup of G, let g ∈ G, and The following are equivalent: ρ ∈ K. (1) there exists f ∈ eρ ∗ H(G) ∗ eρ such that f | KgK = 0; (2) g intertwines ρ. Proof. Consider the space C ∞ (KgK) of G-smooth functions on the coset KgK. This carries a smooth representation of K × K by (k1 , k2 )f : x −→ f (k1−1 xk2 ). Let H denote the group of pairs (k, g −1 kg) ∈ K × K, k ∈ K ∩ gKg −1 . The map f → f (g) is then an H-homomorphism C ∞ (KgK) → C (with H acting trivially). By Frobenius Reciprocity (2.4), this induces a K × Khomomorphism (1H ). (11.1.1) C ∞ (KgK) −→ IndK×K H We show this is an isomorphism. The space V = IndK×K (1H ) consists, by deﬁnition, of smooth functions H φ : K×K → C such that φ(hk1 , g −1 hgk2 ) = φ(k1 , k2 ), ki ∈ K, h ∈ K∩gKg −1 . Given such a function φ, we can deﬁne fφ ∈ C ∞ (KgK) by setting fφ (k1 gk2 ) = φ(k1−1 , k2 ); the map φ → fφ is the inverse of the map (11.1.1). In these terms, condition (1) amounts to eρ ∗ C ∞ (KgK) ∗ eρ ∼ = V ρ⊗ρˇ = 0. Equivalently, HomK×K (ρ ⊗ ρˇ, V ) ∼ = HomH (ρ ⊗ ρˇ, 1H ) = 0. The last relation is equivalent to the representation k → ρ(k) ⊗ ρˇ(g −1 kg) −1 of K ∩ gKg −1 having a ﬁxed vector, that is, HomK∩gKg−1 (ρ, ρg ) = 0, as required.

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11.2. Let K be an open subgroup of G, containing and compact modulo the centre Z of G. Let (ρ, W ) be an irreducible smooth representation of K. Let H(G, ρ) be the space of functions f : G → EndC (W ) which are compactly supported modulo Z and satisfy f (k1 gk2 ) = ρ(k1 )f (g)ρ(k2 ),

ki ∈ K, g ∈ G.

Observe that the support of any f ∈ H(G, ρ) is a ﬁnite union of double cosets KgK. Let µ˙ be a Haar measure on G/Z. For φ1 , φ2 ∈ H(G, ρ), we set φ1 (x)φ2 (x−1 g) dµ(x), ˙ g ∈ G. φ1 ∗ φ2 (g) = G/Z

The function φ1 ∗ φ2 lies in H(G, ρ) and, under this operation of convolution, the space H(G, ρ) is an associative C-algebra with 1. Remark. The algebra H(G, ρ) is called the ρ-spherical Hecke algebra of G, or the intertwining algebra of ρ in G. It is closely related to the algebra eρ ∗ H(G)∗eρ : there is a canonical algebra isomorphism eρ ∗H(G)∗eρ ∼ = H(G, ρ)⊗ EndC (W ). (In the literature, the algebra we have deﬁned is sometimes denoted H(G, ρˇ).) Lemma. Let g ∈ G; there exists φ ∈ H(G, ρ) with support KgK if and only if g intertwines ρ. Proof. Let f ∈ EndC (W ); for a ﬁxed g ∈ G, the assignment kgk → ρ(k)f ρ(k ), k, k ∈ K, gives an element of H(G, ρ) if and only if, for k ∈ K g ∩ K, we have f ◦ ρ(k) = ρg (k) ◦ f . That is, if and only if f ∈ HomK g ∩K (ρ, ρg ). The representations ρ, ρg of K ∩K g are semisimple, so the spaces HomK g ∩K (ρ, ρg ), HomK g ∩K (ρg , ρ) have the same dimension. The Lemma now follows. We have actually shown that the space of functions f ∈ H(G, ρ) supported on KgK is canonically isomorphic to HomK g ∩K (ρ, ρg ). 11.3. With (K, ρ) as in 11.2, we consider the compactly induced representation c-IndG K ρ, as in 2.5. The space X underlying this representation consists of the functions f : G → W , which are compactly supported modulo Z, and satisfy f (kg) = ρ(k)f (g), k ∈ K, g ∈ G. The group G acts by right translation. (All functions f ∈ X are G-smooth for this action, since K is open: see 2.5 Exercise 2.) For φ ∈ H(G, ρ) and f ∈ c-Ind ρ, we deﬁne φ(x)f (x−1 g) dµ(x), ˙ g ∈ G. φ ∗ f (g) = G/Z

Clearly, φ ∗ f ∈ X, and this action gives a homomorphism of C-algebras H(G, ρ) −→ EndG (c-Ind ρ).

(11.3.1)

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Proposition. The map (11.3.1) is an isomorphism of C-algebras. Proof. We use the relation EndG (c-Ind ρ) ∼ = HomK (ρ, c-Ind ρ) of (2.5.2). Let φ0 : w → φ0w be the canonical map W → c-Ind ρ, corresponding to the identity endomorphism of c-Ind ρ: the function φ0w has support K and φ0w (k) = ρ(k)w. The isomorphism EndG (c-Ind ρ) → HomK (ρ, c-Ind ρ) is composition with φ0 . Composing (11.3.1) with φ0 , we get a map H(G, ρ) → HomK (W, c-Ind ρ). We write down its inverse. Let φ : W −→ c-Ind ρ, w

−→ φw ,

be a K-homomorphism. We deﬁne a function Φ : G → EndC (W ) by Φ(g) : w −→ φw (g). For k ∈ K, we have Φ(kg) : w → φw (kg) = ρ(k)φw (g), so Φ(kg) = ρ(k)Φ(g). Also, Φ(gk) : w → φw (gk) = φρ(k)w (g), since φ is a K-map. Therefore Φ ∈ −1 Φ is the required inverse map. H(G, ρ) and φ → µ(K/Z) ˙ 11.4. The central result of this section is: Theorem. Let K be an open subgroup of G = GL2 (F ), containing and compact modulo Z. Let (ρ, W ) be an irreducible smooth representation of K and suppose that an element g ∈ G intertwines ρ if and only if g ∈ K. The compactly induced representation c-IndG K ρ is then irreducible and cuspidal. Proof. We write (Σ, X) = c-IndG K ρ. We ﬁrst show that the representation Σ has a non-zero coeﬃcient which is compactly supported modulo Z. To see this, we use the canonical K-embedding φ0 : W → X of the preceding proof, which identiﬁes W with the space of functions in X that are supported in K (2.5 Lemma). The groups K, G are unimodular, so the Duality Theorem of 3.5 implies ˇ ∼ ˇ. The induced representation IndG ˇ contains c-IndG ˇ as that X = IndG Kρ Kρ Kρ G ˇ ˇ G-subspace. The canonical K-embedding W → c-IndK ρˇ identiﬁes W with ˇ with support contained in K. We take non-zero the space of functions in X ˇ ˇ is then non-zero functions w ˇ ∈ W ⊂ X and w ∈ W ⊂ X: the coeﬃcient γw⊗w ˇ and supported in K. Consequently, we need only prove that X is irreducible: it is then admissible (10.2 Corollary) and we can apply 10.1 Proposition (2) to show it is γ-cuspidal, hence cuspidal. The centre Z of G acts on X via the character ωρ , where ρ(z)w = ωρ (z)w, z ∈ Z, w ∈ W . Therefore X is the direct sum of its K-isotypic components (2.7). Any K-map W → X has image contained in X ρ , so:

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81

HomK (W, X ρ ) = HomK (W, X) ∼ = EndG (X) ∼ = H(G, ρ). However, the intertwining condition implies that dim H(G, ρ) = 1. The space HomK (W, X ρ ) therefore has dimension 1, and we conclude that W = X ρ . Let Y be a non-zero G-subspace of X. Therefore ∼ 0 = HomG (Y, X) ⊂ HomG (Y, IndG K ρ) = HomK (Y, ρ). Since Y is semisimple over K (2.7), we have Y ρ = 0. Thus Y ⊃ W = X ρ , since W is irreducible over K. As W generates X over G, we conclude that Y = X. Thus X is irreducible, as required. Remark 1. The theorem holds (with the conclusion that c-Ind ρ is γ-cuspidal), with the same proof, in considerable generality. It is valid for a unimodular locally proﬁnite group G, satisfying 2.6 Hypothesis, and such that any irreducible smooth representation of G is admissible. Remark 2. The converse of the theorem also holds. If ρ is intertwined by some g ∈ G K, then H(G, ρ) ∼ = EndG (c-Ind ρ) has dimension > 1. Thus c-Ind ρ has a non-scalar endomorphism and cannot be irreducible. Remark 3. In the situation of the theorem, the smooth dual (c-Ind ρ)∨ is irreducible. It is, however, isomorphic to Ind ρˇ. We deduce that Ind ρˇ = c-Ind ρˇ ∼ = ∨ (c-Ind ρ) . Since these representations are all admissible, we can dualize again to get c-Ind ρ = Ind ρ. 11.5. We give an example illustrative of the above procedures. Let G = GL2 (F ), K = GL2 (o) and K1 = 1+pM2 (o). Thus K1 is an open normal subgroup of K and K/K1 ∼ = GL2 (k). We also let I1 denote the group of matrices po I1 = 1 + . pp Thus I1 is the inverse image in K of the standard group N (k) of upper triangular unipotent matrices in GL2 (k). Theorem. Let (π, V ) be an irreducible smooth representation of G, and suppose that π contains the trivial character of K1 . Exactly one of the following holds: (1) π contains a representation λ of K, inﬂated from an irreducible cuspidal ˜ of GL2 (k); representation λ (2) π contains the trivial character of I1 . In the ﬁrst case, π is cuspidal, and there exists a representation Λ of ZK such that Λ | K ∼ = λ and π∼ = c-IndG ZK Λ.

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Proof. The group K stabilizes the space V K1 , which is therefore a direct sum of irreducible representations of K which are trivial on K1 , that is, they are ˜ Either λ ˜ is inﬂated from GL2 (k). Let λ be one of these, inﬂated from λ. cuspidal (in the sense of §6), or it is not. In the latter case, it contains the trivial character of N (k), whence λ contains the trivial character of I1 . We have to show that the two cases cannot occur together. To do this, we interpolate a useful general lemma. Lemma. For i = 1, 2, let ρ˜i be an irreducible representation of GL2 (k), and let ρi denote the inﬂation of ρ˜i to a representation of K. Suppose that ρ˜1 is cuspidal. (1) The representations ρi intertwine in G if and only if ρ˜1 ∼ = ρ˜2 . (2) An element g ∈ G intertwines ρ1 if and only if g ∈ ZK. Proof. Let g ∈ G intertwine ρ2 with ρ1 . It is only the coset KgZK which intervenes, so we can take g of the form a g = 0 10 , for some a 0. If a = 0, we have g = 1 and there is nothing to do. We therefore assume a 1. The group K1g ∩ K contains the group g 1p N0 = 10 o1 ⊂ 0 1 on which ρg2 is trivial. Since ρ˜1 is cuspidal, ρ1 does not contain the trivial character of N0 , so g cannot intertwine the ρi . All assertions now follow. It follows from 11.1 Proposition 1 that, in the theorem, the two cases ˜ is cuspidal. Surely π contains cannot occur together. We now assume that λ some representation Λ of ZK extending λ. Thus we have a non-trivial ZKhomomorphism Λ → π, giving a non-trivial G-homomorphism c-IndG ZK Λ → π. However, by part (2) of the lemma and 11.4 Theorem, the representation c-Ind Λ is irreducible, so π ∼ = c-Ind Λ, as desired. Remark. We will eventually see (14.5) that the theorem has a kind of converse. If (π, V ) is an irreducible representation of G containing the trivial character of K1 , then it is cuspidal if and only if it satisﬁes condition (1) of the theorem.

Further reading. Although we have focused exclusively on G = GL2 (F ), many elements reﬂect the much more general discussions in the papers [5,6] of Bernstein and Zelevinsky. These apply in the context of connected reductive algebraic groups over

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83

F and centre on general versions of the Restriction-Induction Lemma (9.3) and the homomorphism theorem in the form 9.11 Lemma 2. That programme culminates in a classiﬁcation of the non-cuspidal representations of GLn (F ), [90]. Rodier’s report [71] is a helpful introduction. The eternal pre-print [25] is also written in these terms, from a slightly diﬀerent point of view. Only in very few cases, however, does one have a good command of the non-cuspidal representations of groups besides GLn (F ). The initial analysis of cuspidal representations in this chapter is quite general in tone, and holds very widely. Even 11.5 and its converse have close analogues for completely general reductive groups [67], [64].

4 Cuspidal Representations

12. 13. 14. 15. 16. 17.

Chain orders and fundamental strata Classification of fundamental strata Strata and the principal series Classification of cuspidal representations Intertwining of simple strata Representations with Iwahori-fixed vector

The objective of this chapter is to classify the irreducible cuspidal representations of G = GL2 (F ). The method is diﬀerent from that of Chapter III, being based on the process of restricting representations of G to certain, rather special, compact open subgroups. The maximal compact subgroup K = GL2 (o) and the standard Iwahori subgroup I of G have already appeared in incidental rˆ oles, but they now move centre-stage. Each of these groups has a canonical ﬁltration by open normal subgroups {Kn }n0 , {In }n0 respectively. Given an irreducible smooth representation (π, V ) of G, one looks for the largest proper subgroup in the ﬁltration (there is some subtlety as to whether one chooses K or I) for which V has a ﬁxed vector. The representations of the next largest ﬁltration subgroup occurring in π encapsulate a signiﬁcant amount of information concerning π. The example of 11.5 gives an idea of the method. There, one knows that π has a ﬁxed vector for the group K1 . The natural representation of K on the space of K1 -ﬁxed vectors in π reveals whether π is cuspidal or not. If it is cuspidal, one rapidly gets an explicit description of it as an induced representation. The general case is slightly more involved. It is essential to operate, for some of the time, in an intrinsic manner but we reduce to matrix calculations whenever possible. The groups K, I and their ﬁltrations have ring-theoretic descriptions which are particularly convenient. We start the chapter by

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developing these. We then have to examine the restrictions of representations of G to the ﬁltration subgroups and interpret our ﬁndings. This results in a clear characterization of the cuspidal representations, which we then reﬁne to a classiﬁcation. The ﬁnal section looks in a diﬀerent direction. Irreducible representations of G containing the trivial character of the Iwahori subgroup I play a special rˆ ole in representation theory: this is true in a much wider context than the present one. We need to consider them for two reasons. First, we have to exclude the possibility that any of them is cuspidal. Second, we have to give an account of the non-cuspidal, square-integrable representations (10a.2 Remark) to facilitate some calculations in Chapter VI. Representations with Iwahoriﬁxed vector are pivotal in this matter.

12. Chain Orders and Fundamental Strata In this section, we construct a family of compact open subgroups of G, and then a family of characters of these subgroups. The key point is that an irreducible smooth representation of G (not already covered by 11.5) contains one, and essentially only one, of these characters. 12.1. We need a special family of compact open subgroups of G = GL2 (F ). These are groups of units of certain rings, which we now deﬁne. Temporarily write V = F ⊕ F , so that G = AutF (V ) and A = EndF (V ). An o-lattice chain in V is a non-empty set L of o-lattices in V , linearly ordered under inclusion and stable under multiplication by F × . If L is an o-lattice chain in V , we can number its elements: L = {Li : i ∈ Z},

Li Li+1 .

Stability under translation by F × implies the existence of a positive integer e = eL such that xLi = Li+eυF (x) , for all i ∈ Z and x ∈ F × . Lattice chains are easy to describe completely: Proposition. Let L = {Li }i∈Z be an o-lattice chain in V = F ⊕ F . Then eL = 1 or 2. Moreover: (1) Suppose eL = 1. There exists g ∈ G such that gLi = pi ⊕ pi ,

i ∈ Z.

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(2) Suppose that eL = 2. There exists g ∈ G such that gL2i = pi ⊕ pi , gL2i+1 = pi ⊕ pi+1 ,

i ∈ Z.

(12.1.1)

Proof. Set eL = e. The quotient L0 /Le = L0 /pL0 is a vector space over k = o/p of dimension 2. The quotients Li /Le , 0 i e, form a ﬂag of subspaces of L0 /Le , whence 1 e 2, as required. The lattice L0 is the o-span of an F -basis of V (1.5 Proposition), so there exists g ∈ G such that gL0 = o ⊕ o. Hence gLei = pi ⊕ pi , i ∈ Z, and this deals completely with the case e = 1. If e = 2, we have o ⊕ o ⊃ gL1 ⊃ p ⊕ p, and U = gL1 /(p ⊕ p) is a one¯ ∈ GL2 (k) dimensional k-subspace of k ⊕ k = (o ⊕ o)/(p ⊕ p). There exists h ¯ in GL2 (k), then hgL1 = o⊕p, ¯ = k⊕0. If h ∈ GL2 (o) has image h such that hU and the result follows. Let L = {Li }i∈Z be a lattice chain in V , and deﬁne AL = Endo (Li ) = Endo (Li ) i∈Z

0ie−1

= {x ∈ A : xLi ⊂ Li , i ∈ Z}. Thus AL is a ring (with 1) and also an o-lattice in A. (Such rings are called o-orders in A.) The o-lattices L ∈ L are AL -modules. Interpreting the proposition in terms of AL , we get: Corollary. Let L be an o-lattice chain in V , and set A = AL . There exists g ∈ G such that o o ⎧ if eL = 1, ⎨M = o o (12.1.2) gAg −1 = ⎩ I= oo if eL = 2. po 12.2. An AL -lattice in V is, by deﬁnition, an o-lattice in V which is also an AL -module. Proposition 1. Let L be a lattice chain in V and let L be an AL -lattice in V . Then L ∈ L. Proof. It is enough to treat the cases (12.1.2). We start with AL = I. The order I contains the idempotent matrices e1 = 10 00 , e2 = 00 01 . It follows that L ⊃ e1 L + e2 L, whence L = e1 L ⊕ e2 L. That is, L = pa ⊕ pb , for certain integers a, b. The lattice L is invariant under the group of upper

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4 Cuspidal Representations

triangular unipotent matrices in I, so a b. It is also invariant under the group of lower triangular unipotent matrices in I, so b a+1. The only possibilities, therefore, are L = pa ⊕ pa or L = pa ⊕ pa+1 , for some a ∈ Z. All of these lie in L. The case AL = M is similar but easier, so we omit the details. In other words, L is the set of all AL -lattices in V , and so we can recover L (up to its numbering) from the ring AL . It is usually more convenient to work directly with the ring AL than via the lattice chain L. We therefore introduce the following terminology. A chain order in A = M2 (F ) is a ring of the form A = AL , for some o-lattice chain L in V . We set eA = eL . A special rˆole is played by the Jacobson radical P of the chain order A: we use the notation P = rad A. Proposition 2. Let A be a chain order in A with P = rad A, and set e = eA . Then Pe = pA, and there exists Π ∈ G such that P = ΠA = AΠ. Proof. It is enough to treat the standard examples of (12.1.2). For these, we have: 0 1 I, (12.2.1) rad M = M, rad I = 0 for any prime element of F . Such an element Π is called a prime element of A. We have Pn = Π n A = AΠ n ,

n 0.

We take this as the deﬁnition of Pn when n −1. Observe that if A = AL , for a lattice chain L = {Li }, then ΠLi = Li+1 for all i and: Pn = Homo (Li , Li+n ) = {x ∈ A : xLi ⊂ Li+n , i ∈ Z}, n ∈ Z. i∈Z

Remark. There is a purely ring-theoretic description of the class of rings we have called “chain orders”. The chain orders in A are conventionally known as the hereditary o-orders in A. 12.3. Let A be a chain order in A, with P = rad A. We put UA0 = UA = A× , UAn = 1+Pn ,

n 1.

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89

For example, observe that UM = GL2 (o) and UI = I, the standard Iwahori subgroup of G (7.3). In general, the groups UAn , n 0, are compact open subgroups of G, and each UAn is a normal subgroup of UA . If 2m n > m 1, the map x → 1+x induces an isomorphism ≈

Pm /Pn −→ UAm /UAn .

(12.3.1)

We shall also need the group KA = {g ∈ G : gAg −1 = A}.

(12.3.2)

If A = AL , for some lattice chain L, we also have KA = Auto (L) = {g ∈ G : gL ∈ L, ∀L ∈ L}. Either way, KA is the semi-direct product Π UA of UA and the cyclic group generated by a prime element Π of A. In particular, KA is open in G. It contains the centre Z of G, and KA /Z is compact. It normalizes all of the groups UAj , j 0. Exercises. (1) Let g ∈ G; show that g ∈ KA if and only if gA = Pm , for some m ∈ Z. (2) Show that KA is the normalizer NG (UA ) of UA in G. (3) Show that KA is the G-normalizer of UAm , for any m 0. Hint. In (2), the o-algebra generated by UA is A except in the case e = 2 and q = 2. 12.4 Example. There is one important abstract context in which chain orders arise. Let E be an F -subalgebra of A which is a quadratic ﬁeld extension of F . Thus V = F ⊕ F is an E-vector space of dimension one. Proposition. Let E be an F -subalgebra of A such that E/F is a quadratic ﬁeld extension. (1) The set of oE -lattices in V forms an o-lattice chain L, with the property eL = e(E|F ). Further, L is the unique lattice chain in V which is stable under translation by E × . (2) The order A = AL is the unique chain order in A such that E × ⊂ KA . (3) If P = rad A, then xA = PυE (x) , x ∈ E × , and KA = E × UA . Proof. Let v ∈ V , v = 0; then L = {pjE v : j ∈ Z} is the set of all oE -lattices in V , and is deﬁned independently of v. We have L = E × L = {xL : x ∈ E × }, for any L ∈ L. If L is a lattice chain stable under E × , each L ∈ L is stable under the action of UE ; we have oE = o[UE ], so L is an oE -lattice and L ⊂ L. But L = E × L ⊂ L , for any L ∈ L ⊂ L, so L = L. The remaining assertions follow readily.

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denote the group of characters 12.5. We take a character ψ ∈ F , ψ = 1. Let A of A and deﬁne ψA ∈ A by ψA (x) = ψ(trA x), where trA denotes the trace map A → F . For a ∈ A, we can deﬁne a character aψA of A by aψA (x) = ψA (ax) = ψA (xa), x ∈ A. Lemma. The map a → aψA gives an isomorphism A ∼ = A. Proof. Working explicitly, A = M2 (F ) ∼ = F ⊕ F ⊕ F ⊕ F (as additive group), = F × F × F × F ∼ so A = F ⊕ F ⊕ F ⊕ F , via the isomorphism F ∼ = F of 1.7. A character of A is therefore of the form a b −→ ψ(αa)ψ(βb)ψ(γc)ψ(δd), c d for uniquely determined elements α, β, γ, δ ∈ F . This expression, however, is just α γ a b ψA , β δ c d and the lemma follows.

It is now convenient to ﬁx ψ ∈ F of level 1 (cf. 1.7). Let L be some o-lattice in A; we deﬁne L∗ = {x ∈ A : ψA (xy) = 1, y ∈ L}.

If {ai }1i4 is an o-basis of L, then L∗ = pai , where {ai } is the dual basis of A deﬁned by trA (ai aj ) = δij (Kronecker delta). Thus L∗ is an o-lattice and L∗∗ = L. Because of this “bi-duality” property, the obvious relation (L1 +L2 )∗ = L∗1 ∩ L∗2 implies that (L1 ∩ L2 )∗ = L∗1 + L∗2 , for o-lattices Li in A. given by ψA , we can identify L∗ with the Via the isomorphism A ∼ = A group of characters of A which are trivial on L. Moreover, if we have o-lattices L1 ⊂ L2 , then L∗1 /L∗2 is identiﬁed with the group of characters of L2 which are trivial on L1 . Proposition. Let A be a chain order in A with radical P, and let ψ ∈ F have level one.

∗ (1) For n ∈ Z, we have Pn = P1−n . (2) Let m, n be integers such that 2m+1 n > m 0. Let (UAm+1 /UAn+1 ) denote the group of characters of the ﬁnite abelian group UAm+1 /UAn+1 . The map P−n /P−m −→ UAm+1 /UAn+1 , a+P−m −→ ψA,a | UAm+1 , is an isomorphism, where ψA,a denotes the function x → ψA (a(x−1)).

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91

Proof. The explicit form (12.1.2) gives A∗ = P. If L is an o-lattice in A and g ∈ G, then (gL)∗ = L∗ g −1 . So, if Π is a prime element of A, then (Pn )∗ = (Π n A)∗ = A∗ Π −n = P1−n , as required for (1). The second assertion then follows from (12.3.1). We normally abbreviate ψA,a = ψa . Remark 1. Let η ∈ F , η = 1. We may again form the character ηA = η ◦ trA ∈ and the function ηa : x → ηA (a(x−1)). There exists c ∈ F × such that A η = cψ; the level of η is 1−υF (c) and ηa = ψca . The map a → ηa induces an isomorphism c−1 P−n /c−1 P−m ∼ = UAm+1 /UAn+1 . Taking our basic character to have level one thus gives the simplest numerology. Remark 2. The focus of interest here is on the characters χ of the ﬁnite groups UAm+1 /UAn+1 , particularly their intertwining properties. If we choose ψ ∈ F (of level 1, for convenience) and write χ = ψa , these properties of χ are reﬂected in properties of the coset a+P−m , and are hence describable in terms of matrices. As always, there are disadvantages to making a non-canonical “choice of coordinates” to describe a mathematical object. However, it is easy to work out the eﬀect of a change of coordinates: replacing ψ by another character , for some u ∈ UF . The relevant ψ = u−1 ψ of level one, gives χ = ψa = ψua −m will be unaﬀected by such a change, and there is properties of cosets a+P no apparent way of proceeding otherwise. 12.6. We start to apply these concepts to the analysis of representations of G. Let (π, V ) be an irreducible smooth representation of G. Let S(π) denote the set of pairs (A, n), where A is a chain order in A and n 0 is an integer, subject to the condition that π contains the trivial character of UAn+1 . We deﬁne the normalized level (π) of π by (π) = min {n/eA : (A, n) ∈ S(π)}. Clearly, 2 (π) is a non-negative integer. 1 ⊂ UI1 , so In the notation (12.1.2), we have UM Proposition. Let π be an irreducible smooth representation of G; then (π) = 1 0 if and only if π contains the trivial character of UM . We have given a preliminary analysis of such representations in 11.5 (using 1 the notation UM = K1 ). 12.7. To deal with the representations π for which (π) > 0, we introduce a new concept. For the remainder of this chapter, we work relative to a ﬁxed choice of character ψ ∈ F of level one.

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A stratum in A is a triple (A, n, a), where A is a chain order in A (with radical P, say), n is an integer and a ∈ P−n . We say that strata (A, n, a1 ), (A, n, a2 ) are equivalent if a1 ≡ a2 (mod P1−n ). If n 1, we can associate to a stratum (A, n, a) the character ψa of UAn , which is trivial on UAn+1 . This character depends only on the equivalence class of the stratum (and the choice of ψ). Proposition. Let (Ai , ni , ai ), i = 1, 2, be strata in A, let Pi = rad Ai , and let g ∈ G. Assume that ni 1, i = 1, 2. The following are equivalent: (1) The element g intertwines the character ψa1 of UAn11 with the character ψa2 of UAn22 . 1 2 )g ∩ (a2 +P1−n ) is non-empty. (2) The intersection g −1 (a1 +P1−n 1 2 Proof. Consider the chain order A3 = g −1 A1 g; this has radical rad A3 = g −1 P1 g. The character (ψa1 )g of the group (UAn11 )g = UAn31 is associated to the stratum (A3 , n1 , g −1 a1 g). It is therefore enough to consider the case g = 1. If (2) holds, we take an element a of the intersection. We then have ψa = ψai on UAnii , so ψa1 = ψa2 = ψa on UAn11 ∩ UAn22 . Conversely, suppose that the ψai agree on UAn11 ∩ UAn22 : ψA (a1 x) = ψA (a2 x),

x ∈ Pn1 1 ∩ Pn2 2 .

In the notation of 12.5, we have n1 ∗ ∗ ∗ 1 2 P1 ∩ Pn2 2 = Pn1 1 + Pn2 2 = P1−n + P1−n , 1 2 and so a1 ≡ a2

1 2 (mod P1−n + P1−n ). 1 2

i That is, there exist xi ∈ P1−n such that a2 = a1 +x1 +x2 , or i 1 2 a2 −x2 = a1 +x1 ∈ (a1 +P1−n ) ∩ (a2 +P1−n ), 1 2

as required. When the element g satisﬁes condition (2) of the Proposition, we say that it intertwines (A1 , n1 , a1 ) with (A2 , n2 , a2 ). 12.8. Not all strata are of equal interest: we have to distinguish a particular class of them. Deﬁnition. Let A be a chain order in A, and set P = rad A. A stratum (A, n, a) in A is called fundamental if the coset a+P1−n contains no nilpotent element of A.

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93

This property depends only on the equivalence class of the stratum. It is indeed a property of the character ψa of UAn (assuming n 1): changing our choice of ψ changes a+P1−n to ua+P1−n , for some u ∈ UF , and this does not aﬀect the deﬁning property. We ﬁrst need an eﬀective method of recognizing fundamental strata: Proposition. Let (A, n, a) be a stratum in A, and put P = rad A. The following are equivalent: (1) The coset a+P1−n contains a nilpotent element of A. (2) There is an integer r 1 such that ar ∈ P1−rn . Proof. Condition (2) is equivalent to saying that x1 x2 . . . xr ∈ P1−rn , for any choice of elements x1 , x2 , . . . , xr ∈ a+P1−n . Thus condition (2) is a property of the coset a+P1−n , rather than just the element a. That said, the implication (1) ⇒ (2) is clear. We assume (2) holds. Nothing is changed if we replace (A, n, a) by a Gconjugate. This reduces us to the cases A = M or I (12.1.2). Likewise nothing changes if we replace (A, n, a) by (A, n−e, a), for a prime element of F , e = eA . This reduces us to the cases (A, n) = (M, 0), (I, 0) or (I, −1). In the ﬁrst case, we have a ∈ M; let a ˜ be the image of a in M2 (k). Condition ˜ being (2) holds if and only if a ˜r = 0, for some r 1. This is equivalent to a 0 1 zero (that is, 0 ∈ a+M) or GL2 (k)-conjugate to ( 0 0 ). In the latter case, a+M contains a GL2 (o)-conjugate of ( 00 10 ) and (1) holds. Now let P = rad I; let m = 0, 1. We then have m a1 0 0 1 a+Pm+1 = + Pm+1 , 0 a 0 2

for ai ∈ o. The coset a+Pm+1 only determines the ai modulo p. If m = 0, condition (2) holds if and only if a1 ≡ a2 ≡ 0 (mod p). This clearly implies (1). In the other case m = 1, (2) is equivalent to a1 a2 ≡ 0 (mod p) and again (1) follows. Using the calculations in the last proof, we can list the equivalence classes of non-fundamental strata, up to conjugation in G. Call a stratum (A, n, a) trivial if a ∈ P1−n , where P = rad A. Gloss. Let be a prime element of F . A non-trivial, non-fundamental stratum in A is equivalent to a G-conjugate of one of the following: (12.8.1) (M, n, −n a), a = 00 10 , 0 1 −n (12.8.2) (I, 2n−1, α), α = 0 0 , for some n ∈ Z.

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12.9. Let (π, V ) be an irreducible smooth representation of G. We say that π contains the stratum (A, n, a) if n 1 and π contains the character ψa of UAn . Observe that, if this is the case, then n/eA (π) by deﬁnition. The main result here is: Theorem. Let π be an irreducible smooth representation of G and let (A, n, a) be a stratum in A, contained in π. The following are equivalent: (1) (A, n, a) is fundamental; (2) (π) = n/eA . In particular, π contains a fundamental stratum if and only if (π) > 0. Proof. The ﬁrst step is: Lemma 1. (1) Let (A, n, a) be a non-fundamental stratum in A, and let P be the radical of A. There is a chain order A1 in A, with radical P1 , and an integer n1 , such that 1 a+P1−n ⊂ P−n , 1

and n1 /eA1 < n/eA .

(2) Let π be an irreducible smooth representation of G, containing a nonfundamental stratum (A, n, a). We then have (π) < n/eA . Proof. Part (1) is trivial if the stratum (A, n, a) is trivial, so we assume otherwise. The issue is unchanged if we replace (A, n, a) by a G-conjugate. Similarly if we replace (A, n, a) by (A, n−eA r, r a), for an integer r and a prime element of F . This reduces us to the cases (12.8.1), (12.8.2) with n = 0. In the ﬁrst of these, we have 01 + PM ⊂ PI , 00 and in the second

01 00

+ P2I ⊂ P1 ,

where P1 = rad A1 and

A1 =

o p−1 p o

,

which is conjugate to M. 1 For the second part, we apply (1); the relation a+P1−n ⊂ P−n implies 1 −n1 n1 +1 n1 +1 1−n n ⊂ P1 and, dualizing, P1 ⊂ P . This further gives UA1 ⊂ UAn , P and the character ψa of UAn is trivial on UAn11 +1 . Therefore (π) n1 /eA1 < n/eA , as required.

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95

If (π) > 0 then, by deﬁnition, π contains a stratum (A, n, a) with n/eA = (π). By Lemma 1, this stratum must be fundamental. If π contains another stratum (B, m, b), then (12.7, 11.1) it must intertwine with (A, n, a). Lemma 2. Let (A, n, a) be a fundamental stratum in A. Let (B, m, b) be a stratum in A which intertwines with (A, n, a). Then m/eB n/eA , with equality if and only if (B, m, b) is fundamental. Proof. Let P = rad A, Q = rad B. The property in question depends only on the coset b+Q1−m ; replacing (B, m, b) by a G-conjugate, we can arrange that b ∈ a+P1−n . Assume, for a contradiction, that m/eB < n/eA . Thus there is an integer r 1 such that p−rmeA B ⊂ p1−rneB A. Consider the element x = breA eB . Since (A, n, a) is fundamental, we have x∈ / p1−rneB A, but certainly x ∈ p−rmeA B. This contradiction proves m/eB n/eA . If (B, m, b) is fundamental, symmetry implies m/eB n/eA , and hence m/eB = n/eA . Conversely, suppose that m/eB = n/eA but that (B, m, b) is not fundamental. We have b2r ∈ p−2rn/eA A p−2rn/eA P for all integers r 1, while b2r ∈ p−2rm/eB +jr B, for a sequence of integers jr tending to inﬁnity as r → ∞. For r suﬃciently large, therefore, pjr B ⊂ P which implies b2r ∈ p−2rn/eA P. This is impossible, so (B, m, b) is fundamental. In the case (π) > 0, the proof of the theorem is complete. It remains only to show that, if (π) = 0, then π cannot contain a fundamental stratum (A, n, a), n 1. However, by deﬁnition, π contains the trivial character of 1 1 UM . It therefore contains some character of UI1 which is trivial on UM ⊃ UI2 . That is, π contains a stratum (I, 1, a) in which the element a takes the form 0 0 (mod I), a ≡ a0 0 for some a0 ∈ o. Thus (I, 1, a) is non-fundamental. If π contains a stratum (A, n, b), then n/eA 1/2; since this stratum must intertwine with (I, 1, a), it cannot be fundamental by Lemma 2.

13. Classiﬁcation of Fundamental Strata Given that there are, in eﬀect, only two chain orders in A (12.1.2), it is not hard to describe the fundamental strata completely, up to G-conjugacy. We carry out this process, to obtain a preliminary classiﬁcation of the irreducible representations of G in terms of strata. We continue to work relative to a ﬁxed ψ ∈ F of level one.

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13.1. The ﬁrst case is: Proposition 1. (1) Let (A, n, a) be a stratum in which eA = 2 and n is odd. Let P = rad A. The stratum (A, n, a) is fundamental if and only if aA = P−n or, equivalently, a ∈ Π −n UA for a prime element Π of A. (2) Let π be an irreducible smooth representation of G with (π) > 0. If (π) = n/2 ∈ / Z, then π contains a fundamental stratum (I, n, a). Proof. The ﬁrst assertion concerns only the conjugacy class of the stratum, so we can take A = I. The result then follows from 12.8 Gloss. In (2), 12.9 Theorem says that π contains a fundamental stratum; since (π) ∈ / Z, it must be conjugate to one of the form (I, n, a). Deﬁnition. A ramiﬁed simple stratum is a fundamental stratum (A, n, α) in which eA = 2 and n is odd. Proposition 2. If 0 < (π) = n ∈ Z, then π contains a fundamental stratum of the form (M, n, α). Proof. This follows from 12.9 Theorem and the observation that n+1 n ⊂ UI2n+1 ⊂ UI2n ⊂ UM . UM

(13.1.1)

For this reason, there is no need to consider fundamental strata of the form (I, 2n, a), n ∈ Z. More precisely, Corollary. Let π be an irreducible smooth representation of G with (π) > 0. Then π contains a fundamental stratum (A, n, a) such that gcd(n, eA ) = 1. 13.2. Consider a stratum (A, n, α) in which eA = 1. We may write α = −n α0 , for some α0 ∈ A. Let fα (t) ∈ o[t] be the characteristic polynomial of α0 , and let f˜α (t) ∈ k[t] be its reduction modulo p: thus f˜α (t) is the characteristic polynomial of the image of α0 in A/P ∼ = M2 (k), where P = rad A. If we regard the prime element as ﬁxed, the polynomial f˜α (t) depends only on the equivalence class of the stratum (A, n, α), and is indeed an invariant of the G-conjugacy class of the stratum. Observe that the stratum (A, n, α) is fundamental if and only if f˜α (t) = t2 (12.8 Proposition). Deﬁnition. Let (A, n, α) be a fundamental stratum with eA = 1. We say that (A, n, α) is ⎫ ⎧ ˜ unramiﬁed simple⎪ ⎪ ⎬ ⎨ fα (t) is irreducible in k[t], split if f˜α (t) has distinct roots in k, ⎪ ⎪ ⎭ ⎩˜ essentially scalar fα (t) has a repeated root in k× .

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97

A stratum (A, n, α) is called simple if it is either ramiﬁed or unramiﬁed simple. Remark. All of the qualities assigned to fundamental strata (A, n, α) in the preceding two deﬁnitions are qualities of the associated character ψα of UAn . Proposition. (1) A ramiﬁed simple stratum cannot intertwine with any fundamental stratum of the form (M, n, α). (2) Let (M, n, α), (M, n, β) be fundamental strata which intertwine. We then have f˜α (t) = f˜β (t). Proof. (1) follows from 12.8 Lemma 2. In (2), there exists g ∈ G and β ∈ such that g −1 β g ∈ α+P1−n β+P1−n M M . The characteristic polynomial of the element n g −1 β g is the same as that of n β , so, when we reduce it modulo p, we get f˜β (t). On the other hand, since g −1 β g ∈ α+P1−n M , this reduction is also f˜α (t). Thus f˜β = f˜α , as required. 13.3. One of the cases of 13.2 Deﬁnition is easily explained. If π is an irreducible smooth representation of G and if χ is a character of F × , we recall that χπ denotes the representation g → χ(det g)π(g). Theorem. Let π be an irreducible smooth representation of G with (π) > 0. The following are equivalent: (1) The representation π contains an essentially scalar stratum (M, n, α). (2) There is a character χ of F × such that (χπ) < (π). Proof. Suppose that (M, n, α) is an essentially scalar stratum occurring in π. Replacing α by a UM -conjugate, we can assume (mod p1−n M), (13.3.1) α ≡ −n a0 ab for a prime element of F and a ∈ UF , b ∈ o. Let χ be a character of F × , of level n, such that χ(1+x) = ψ(−a−n x), x ∈ pn (cf. 1.8). We then n = ψ−a−n , so the representation χπ contains the stratum have χ ◦ det | UM (M, n, β), with (mod p1−n M). β ≡ −n 00 0b This is not fundamental and so (χπ) < n. For the converse, we change notation and take for π an irreducible smooth n representation of G with (π) < n. Thus π contains the trivial character of UM × −n (13.1.1). Let φ be a character of F , of level n, with φ(1+x) = ψ(a x), x ∈ pn and some a ∈ UF . The representation φπ then contains a stratum of the form (13.3.1).

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Corollary. Let π be an irreducible smooth representation of G such that 0 < (π) (χπ), for every character χ of F × . One and only one of the following holds: (1) π contains a split fundamental stratum. (2) π contains a ramiﬁed simple stratum. (3) π contains an unramiﬁed simple stratum. Proof. Since (π) > 0, the representation π contains a fundamental stratum. By the theorem, this stratum is not essentially scalar. We can assume (13.1 Propositions) it is either ramiﬁed simple, unramiﬁed simple, or split. Proposition 13.2 shows that only one of these possibilities can occur. We will see in the next section that the Corollary partly reﬂects the initial partition of the irreducible representations of G into cuspidal and non-cuspidal ones. 13.4. We need more understanding of the structures underlying simple strata. Deﬁnition. An element α ∈ GZ is called minimal over F if the sub-algebra E = F [α] of A is a ﬁeld and, setting n = −υE (α), one of the following holds: (1) E/F is totally ramiﬁed and n is odd; (2) E/F is unramiﬁed and, for a prime element of F , the coset n α+pE generates the ﬁeld extension kE /k. The hypothesis α ∈ / Z implies that [E:F ] = 2. We also have f (E|F )υE (α) = υF (det α), where f (E|F ) is the residue class degree [kE : k]. The deﬁnition does not really depend on α being an element of G: we could equally take a quadratic ﬁeld extension E/F and apply it to elements α of E F . Exercise. Let E/F be a quadratic ﬁeld extension, let α ∈ E × , and write ∩ F = ∅. n = −υE (x). Show that α is minimal over F if and only if α+p1−n E Lemma. Let α ∈ G be minimal over F ; set E = F [α], n = −υE (α), and choose a prime element of F . Deﬁne (n+1)/2 α if E/F is ramiﬁed, (13.4.1) α0 = if E/F is unramiﬁed. n α We then have oE = o[α0 ]. Proof. In both cases, the ring o[α0 ] contains a prime element of E and a k-basis of oE /oE . There is a close connection between minimal elements and simple strata. In one direction:

13. Classiﬁcation of Fundamental Strata

99

Proposition. Let (A, n, α) be a simple stratum in A. Then: (1) (2) (3) (4)

α is minimal over F ; F [α]× ⊂ KA ; e(F [α]|F ) = eA ; every α ∈ α+P1−n = αUA1 is minimal over F , where P = rad A.

Proof. We ﬁx a prime element of F . Suppose ﬁrst that (A, n, α) is ramiﬁed. Thus n = 2m+1 is odd and the element α0 = 1+m α satisﬁes α0 A = P (13.1). Therefore υF (det α0 ) = 1 and υF (tr α0 ) 1 (cf. (12.2.1)). The minimal polynomial of α0 over F is thus an Eisenstein polynomial, so E = F [α] = F [α0 ] is a ramiﬁed quadratic ﬁeld extension of F . Moreover, υE (α0 ) = 1, υE (α) = −n, and n is odd. Thus α is minimal over F . If (A, n, α) is unramiﬁed, we put α0 = n α ∈ A. The minimal polynomial f (t) of α0 over F remains irreducible on reduction modulo p. In particular, f (t) is irreducible and E = F [α] is an unramiﬁed ﬁeld extension of degree α0 ], as 2. Further, if α ˜ 0 is the reduction of α0 modulo pE , we have kE = k[˜ required. In both cases, oE = o[α0 ] ⊂ A. If L is the chain of A-lattices in V = F 2 , every L ∈ L is an oE -lattice. Moreover, KA contains a prime element E of E which is also a prime element of A: we take E = α0 in the ramiﬁed case, E = in the unramiﬁed one. It follows that L is the chain of all oE -lattices in V , and that E × ⊂ KA . Finally, let α ∈ α+P1−n . The stratum (A, n, α ) is equivalent to (A, n, α), hence simple, and the same argument applies. 13.5. Thus simple strata give rise to minimal elements. The converse also holds: Proposition. Let α ∈ G be minimal over F . There exists a unique chain order A in A such that α ∈ KA . Moreover, F [α]× ⊂ KA and, if n = −υF [α] (α), the triple (A, n, α) is a simple stratum. Proof. Put E = F [α] and n = −υE (α). Let A be the unique chain order in A which is stable under conjugation by E × (as in 12.4). In particular, α ∈ KA . Deﬁne α0 as in (13.4.1). Let B be a chain order with α ∈ KB . We need to show that B = A. We have α0 ∈ KB and υF (det α0 ) 0. Therefore α0 ∈ B and so, by 13.4 Lemma, oE ⊂ B. Exactly as in the proof of 13.4 Proposition, we get E × ⊂ KB and therefore B = A. We now have αA = P−n , where P = rad A. The ﬁnal assertion is immediate, on comparing the deﬁnitions. Remark. Combining the last two propositions, we see that if (A, n, α) is a simple stratum and α ∈ αUA1 , then F [α ]/F is a quadratic ﬁeld extension

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4 Cuspidal Representations

with e(F [α ]|F ) = e(F [α]|F ). However, the ﬁelds F [α], F [α ] need not be F -isomorphic.

14. Strata and the Principal Series We start to exploit the theory of strata in the classiﬁcation of the irreducible representations of G. We show that, once the obfuscatory eﬀect of twisting (13.3) has been taken into account, one can distinguish between cuspidal and non-cuspidal representations of G using only fundamental strata. This, combined with a parallel result in level zero, is the content of the Exhaustion Theorem (14.5), the main result of the section. The subgroups B, N , T , Z of G are as in 5.1. 14.1. We start with one of the cases of 13.3 Corollary. We use throughout the notation (12.1.2). Proposition. Let (π, V ) be an irreducible smooth representation of G, and suppose that π contains a split fundamental stratum (M, n, α). One may choose α ∈ T and, in that case, the Jacquet module (πN , VN ) contains the n ∩ T. character ψα | UM In particular, VN = 0 and π is not cuspidal. Proof. We are at liberty to replace the coset α+P1−n M by a UM -conjugate. We can therefore take (14.1.1) α = −n a0 0b , where is a prime element of F and a, b ∈ o ∩ F × , with a ≡ b (mod p): in particular, α ∈ T . n . It is enough to show that the space V ξ has Let us write ξ = ψα | UM non-zero image in VN . Suppose, for a contradiction, that V ξ ⊂ V (N ). Thus, for each v ∈ V ξ , there is a compact open subgroup N (v) of N such that π(u)v du = 0 N (v)

(8.1). We write

Nj =

1 pj 0 1

.

The representation π is admissible (10.2 Corollary), so the space V ξ is ﬁnitedimensional. Therefore there exists j ∈ Z such that π(u)v du = 0, Nj

14. Strata and the Principal Series

101

for all v ∈ V ξ . We choose j maximal for this property, so there exists v1 ∈ V ξ such that π(u)v1 du = 0. Nj+1 0 ( 0 1 ). t

The element t then intertwines the character ξ, that is, the We set t = characters ξ, ξ , agree on the group n n p p n n ∩ t−1 UM t = 1 + pn+1 pn . Y = UM Lemma. n (1) Any irreducible representation of UM , containing ξ | Y , is of dimension one. n such that φ | Y = ξ | Y . There exists x ∈ N0 (2) Let φ be a character of UM x such that φ = ξ. n+1 , and so (1) follows immediately. In part Proof. The group Y contains UM (2), we have φ = ψδ , where a x (mod p1−n M), δ ≡ −n 0 b

for some x ∈ o. For y ∈ o, we have ax 1y 1 −y 0 1

0 b

0 1

=

a z 0 b

,

where z = x + (b−a)y. We can surely choose y ∈ o so that z = 0, as required. We consider the vector v2 = π(t−1 )v1 . By the deﬁnition of j, π(u)v2 du = π(ut−1 )v1 du Nj

Nj −1

= π(t

π(tut−1 )v1 du

) Nj

−1

= c π(t

π(u)v1 du = 0,

) Nj+1

for some c > 0. n which agree with ξ on Y . By the Let Φ be the set of characters φ of UM

v , for certain vectors vφ ∈ V φ . There exists lemma, we have v2 = φ∈Φ φ φ ∈ Φ such that π(u)vφ du = 0. Nj

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4 Cuspidal Representations

By part (2) of the lemma, φx = ξ, for some x ∈ N0 . Thus v3 = π(x−1 )vφ lies in V ξ and π(u)v3 du = 0, Nj

contrary to hypothesis. 14.2. In the opposite direction, it is easy to spot a fundamental stratum in an induced representation IndG B χ: Proposition. Let χ = χ1 ⊗ χ2 be a character of T and set Σ = IndG B χ. Let ni be the level of χi . (1) If n = max(n1 , n2 ) > 0 and χ1 χ−1 | UFn = 1, then Σ contains a split 2 fundamental stratum. | UFn is trivial, then Σ contains an (2) If n1 = n2 = n = 0 and χ1 χ−1 2 essentially scalar fundamental stratum. (3) If n1 = n2 = 0, then Σ contains the trivial character of UI1 . Proof. In case (1), we choose ai ∈ p−n so that χi (1+x) = ψ(ai x), x ∈ pn . At least one of the ai satisﬁes υF (ai ) = −n, and a1 ≡ a2 (mod p1−n ). Set 1 0 a1 0 , Nn = . a= pn 1 0 a2 The triple (M, n, a) is then a split fundamental stratum. Let f ∈ IndG Bχ n = BNn and be ﬁxed by Nn . We then have uf = ψa (u)f , have support BUM n , so (M, n, a) occurs in Σ. The other cases are similar. u ∈ UM 14.3. Let (π, V ) be an irreducible smooth representation of G with (π) = 0. We suppose that π contains the trivial character of UI1 : this is the second case of 11.5 Theorem and the third of 14.2 Proposition. Write I = UI . Since I/UI1 ∼ = k× × k× , π contains a character φ of I trivial on UI1 . Proposition. Let (π, V ) be an irreducible smooth representation containing a character φ of I trivial on UI1 . The canonical map V → VN is injective on the isotypic space V φ . In particular, (π, V ) is not cuspidal. Proof. We use the notation U 0 , T 0 = 0F U F

Nj =

for j ∈ Z. We also put

t=

for some prime element of F .

1 pj 0 1

0 0 1

,

,

Nj =

1 0 j

p 1

,

14. Strata and the Principal Series

103

We ﬁx a Haar measure µ on G and form the algebra Hφ (G) = eφ ∗ H(G) ∗ eφ , in the notation of 4.4. This is the algebra of functions h ∈ H(G) such that h(j1 gj2 ) = φ(j1 j2 )−1 h(g), for g ∈ G and j1 , j2 ∈ I. (Thus Hφ (G) = H(G, φ−1 ), in the notation of 11.2.) We view the isotypic space V φ as Hφ (G)module, as in 4.4. The proof is based on the following: Lemma. Let f ∈ Hφ (G) have support ItI, such that f (t) = 1. Then f is an invertible element of Hφ (G). We defer consideration of this lemma to 14.4. Accepting it for the moment, we assume for a contradiction that there exists v ∈ V φ , v = 0, with zero image in VN . Thus (8.1 Lemma (1)) there exists j such that π(x)v dx = 0, (14.3.1) Nj

where dx is some Haar measure on N . The representation (π, V ) is admissible (10.2 Corollary), so dim V φ is ﬁnite. We may therefore choose j maximal for the property that there exists v ∈ V φ , v = 0, satisfying (14.3.1). We consider the element w = π(t)v. This has the properties π(x)w dx, 0= Nj+1

π(y)w = φ(y)w, We now put u = π(eφ )w = q −1

y ∈ N1 T 0 N1 .

π(z)w.

z∈N0 /N1

Thus u ∈ V φ and

π(x)u dx = 0. Nj+1

However, u = µ(ItI)−1 π(f )v so, by the lemma, u = 0. We have contradicted the deﬁnition of j, and the proposition is proved. φ|T 0

Remark. In the proposition, one can show that the canonical map V φ → VN is an isomorphism.

14.4. We return to the lemma of 14.3. The restriction φ | T 0 is the inﬂation of a character φ˜ = φ˜1 ⊗ φ˜2 of k× × k× . We prove 14.3 Lemma in the case φ˜1 = φ˜2 . Let h ∈ Hφ (G) have support −1 It I, and h(t−1 ) = 1. Consider the function f ∗h. This has support contained in ItIt−1 I = IN0 I. Suppose z ∈ N0 lies in the support. Thus z intertwines the character φ (11.2 Lemma). However, z ∈ UM , M = M2 (o), and the image z˜ of z in Gk = GL2 (k) therefore intertwines the character φ˜ of the group

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4 Cuspidal Representations

Bk of upper triangular matrices in Gk . However, the character φ˜ induces irreducibly to Gk (6.3 Proposition), so z˜ = 1 and z ∈ N1 . The support of f ∗ h is therefore contained in I and f ∗h(1) = µ(ItI) = h∗f (1), by straightforward calculations. The lemma follows in this case. We turn to the case φ˜1 = φ˜2 . Let φ1 be a character of F × such that φ1 | UF is the inﬂation of φ˜1 . The map h → (φ1 ◦ det) · h gives an algebra isomorphism Hφ (G) → H(G, I), which preserves support of functions. This means we need only treat the case where φ˜ is trivial. This requires a diﬀerent technique, so we deal with it in 17.3 below. In eﬀect, we have reduced 14.3 Proposition to the case where (π, V ) has an Iwahori-ﬁxed vector. Irreducible representations (π, V ) such that V I = 0 have subtle and interesting features: we discuss them in more detail in §17. Here we note only one particularly signiﬁcant instance, that of the Steinberg representation StG introduced in 9.10: Example. If (π, V ) = StG , then dim V I = 1 and (π) = 0. I For, if we set (Σ, X) = IndG = 2. The B 1T , then (7.3.3) implies dim X 1-dimensional G-subspace 1G of X has only a one-dimensional space of Iﬁxed vectors, so dim V I = 1 also.

14.5. We can now characterize the irreducible cuspidal representations of G in terms of strata: Exhaustion theorem. Let (π, V ) be an irreducible smooth representation of G, which satisﬁes (π) (χπ), for every character χ of F × . The following are equivalent: (1) The representation π is cuspidal. (2) Either (a) (π) = 0 and π contains a representation of UM ∼ = GL2 (o) inﬂated from an irreducible cuspidal representation of GL2 (k), or (b) (π) > 0 and π contains a simple stratum. Proof. Suppose ﬁrst that (π) = 0. The result then follows from 11.5 Theorem and 14.3 Proposition. We therefore assume (π) > 0. If π does not contain a simple stratum, then it must contain a split fundamental stratum (13.3 Corollary). It is then not cuspidal (14.1 Proposition) and we have shown that (1) ⇒ (2). Conversely, let us assume that π is not cuspidal. We identity π with a Gsubspace of a representation Σ = IndG B χ, for some character χ = χ1 ⊗ χ2 of T (9.1 Proposition). Suppose ﬁrst that Σ is irreducible. In particular, π = Σ. If some χi has level 1, 14.2 Proposition says that Σ contains either a split or an essentially scalar fundamental stratum. The second possibility is excluded

15. Classiﬁcation of Cuspidal Representations

105

by hypothesis and 13.3 Theorem. Thus Σ = π contains a split fundamental stratum and so cannot contain a simple stratum. If both χi have level zero, then Σ = π contains the trivial character of UI1 loc. cit., and so (π) = 0, contrary to hypothesis. We therefore assume that Σ is not irreducible. Thus π is either φ ◦ det or φ · StG , for some character φ of F × (9.11). Each of these representations has normalized level l, where l is the level of φ. By the minimality hypothesis, l = 0 and π contains the trivial character of UI1 . By 11.5 Theorem, π cannot satisfy condition (2). This completes the proof of the theorem.

15. Classiﬁcation of Cuspidal Representations In this section, we reﬁne the Exhaustion Theorem (14.5) into a classiﬁcation of the irreducible cuspidal representations of G = GL2 (F ). The arguments rely on some quite remarkable intertwining properties of various characters deﬁned by simple strata. While these are quite easy to state and use, their proofs require special techniques. We have therefore deferred them until the next section, working out their consequences in this one. Throughout, ψ denotes a ﬁxed character of F of level one. 15.1. In this paragraph, we ﬁx a simple stratum (A, n, α) in A, with n 1. The algebra E = F [α] is then a ﬁeld such that [E:F ] = 2, E × ⊂ KA , and α is minimal over F (13.4 Proposition). The function ψα = ψA,α deﬁnes a [n/2]+1 which is trivial on UAn+1 (12.5). character of UA Intertwining Theorem. Let (A, n, α) be a simple stratum, and put E = F [α]. Let g ∈ G. The following are equivalent: [n/2]+1

; (1) g intertwines the character ψα of UA [n/2]+1 (2) g normalizes the character ψα of UA ; [(n+1)/2] (3) g ∈ E × UA . Observe, in (3), that E × ⊂ KA (13.4 Proposition), so E × UA subgroup of KA .

[(n+1)/2]

is a

15.2. We will also need: Conjugacy Theorem. For i = 1, 2, let (A, n, αi ) be a simple stratum, n 1. [n/2]+1 intertwine in G if and only if they are conjugate The characters ψαi of UA by an element of UA . We defer the proofs of these two theorems to §16. We remark that the second condition in the Conjugacy Theorem is equivalent to the cosets [(n+1)/2] being UA -conjugate: the argument is identical to αi +P−[n/2] = αi UA 12.7. This, however, does not imply that the elements αi are conjugate, nor even that the ﬁelds F [αi ] are isomorphic.

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4 Cuspidal Representations

15.3. We reap the consequences for the representation theory of G. Let (A, n, α) be a simple stratum in A, with n 1 and E = F [α], as in 15.1. We set [(n+1)/2] . (15.3.1) J α = E × UA Thus Jα ⊂ KA is open in G; it contains and is compact modulo Z ∼ = F ×. Theorem. With the preceding notation, let Λ be an irreducible representation [n/2]+1 of Jα which contains the character ψα of UA . Then: [n/2]+1

(1) The restriction of Λ to UA (2) The representation

is a multiple of ψα .

πΛ = c-IndG Jα Λ

(15.3.2)

is irreducible and cuspidal. [n/2]+1

is a direct sum of irreducible Proof. The restriction of Λ to the group UA representations, any two of which are conjugate under Jα . Among these irreducible components is the character ψα , which is normalized by Jα (15.1), whence follows (1). Therefore, if g ∈ G intertwines Λ, it must also intertwine ψα . Thus (15.1) g ∈ Jα . Assertion (2) is now given by 11.4 Theorem. It will be convenient to have a special notation for this class of representations. Deﬁnition. Let (A, n, α), n 1, be a simple stratum. Let C(ψα , A) denote the set of equivalence classes of irreducible representations Λ of the group [(n+1)/2] [n/2]+1 such that Λ | UA is a multiple of ψα . Jα = F [α]× UA 15.4. We have a strong uniqueness property: Theorem. For i = 1, 2, let (Ai , ni , αi ) be a simple stratum in A, ni 1, and let Λi ∈ C(ψαi , Ai ). Suppose that the representations πΛi = c-IndG Jα Λ i , i

i = 1, 2,

are equivalent. Then n1 = n2 and there exists g ∈ G such that A2 = g −1 A1 g,

Jα2 = g −1 Jα1 g,

Λ2 = Λg1 .

If A1 = A2 , we may choose g ∈ UA1 . Proof. We identify πΛ1 = πΛ2 = π, say. The representation π contains each simple stratum (Ai , ni , αi ). These strata are either both ramiﬁed or both unramiﬁed (13.3 Corollary). Both Ai are conjugate to I in the ﬁrst case, to M in the second (notation of (12.1.2)). In other words, we can assume A1 = A2 = A, say. Further, ni /eA = (π) (12.9 Theorem), so n1 = n2 = n,

15. Classiﬁcation of Cuspidal Representations

107

[n/2]+1

say. The characters ψαi of UA intertwine in G (11.1), and so are UA conjugate (15.2), say ψα2 = ψαg 1 , g ∈ UA . The G-normalizers Jαi of these characters ψαi are therefore conjugate under g. Consider the representation Λ3 = Λg1 of Jα2 . The restriction of Λ3 to [n/2]+1 UA is a multiple of ψα2 , and Λ3 intertwines with Λ2 . If this intertwining is realized by an element h of G, then h also intertwines ψα2 and lies in Jα2 . It therefore ﬁxes Λ3 , whence Λ3 ∼ = Λ2 , as required. 15.5. It will be convenient to introduce a new term: Deﬁnition. A cuspidal type in G is a triple (A, J, Λ), where A is a chain order in A, J is a subgroup of KA and Λ is an irreducible smooth representation of J, of one of the following kinds: (1) A ∼ = M, J = ZUA , and Λ | UA is the inﬂation of an irreducible cuspidal representation of the group UA /UA1 ∼ = GL2 (k); (2) there is a simple stratum (A, n, α), n 1, such that J = Jα and Λ ∈ C(ψα , A); (3) there is a triple (A, J, Λ0 ) satisfying (1) or (2), and a character χ of F × , such that Λ ∼ = Λ0 ⊗ χ ◦ det. Note that the class of cuspidal types is deﬁned independently of the choice of the character ψ, and is stable under conjugation by G. There is an overlap between case (3) of the deﬁnition and the others. This will cause no problems but, for further comments, see 15.9 below. The component A in this notation is, strictly speaking, redundant, since A is in all cases the unique chain order such that KA ⊃ J. It is more convenient, however, to retain it. If (A, J, Λ) is a cuspidal type in G, the representation c-IndG J Λ is irreducible and cuspidal (15.3 Theorem). We come to the main result. Induction Theorem. Let π be an irreducible cuspidal representation of G = GL2 (F ). There exists a cuspidal type (A, J, Λ) in G such that π ∼ = c-IndG J Λ. The representation π determines (A, J, Λ) uniquely, up to G-conjugacy. Proof. Let π be an irreducible cuspidal representation of G. In case (3) of the deﬁnition above, we have G ∼ c-IndG J (Λ0 ⊗ χ ◦ det) = χ · c-IndJ Λ0 ,

so it is enough to treat the case where π satisﬁes (π) (χπ), for all characters χ of F × . If (π) = 0, the result is given by the Exhaustion Theorem (14.5) and 11.5 Theorem. We assume (π) > 0. By the Exhaustion Theorem, there is a

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4 Cuspidal Representations

simple stratum (A, n, α), n 1, such that π contains the character ψα of UAn . [n/2]+1 The representation π therefore contains a character ξ of UA such that ξ | UAn = ψα . The character ξ is of the form ψβ , for some β ≡ α (mod P1−n ), where P = rad A. By 13.4, 13.5 Propositions, the stratum (A, n, β) is also simple. The representation π then contains some irreducible smooth represen[n/2]+1 contains ψβ . This restriction is therefore tation Λ of Jβ such that Λ | UA a multiple of ψβ (15.3 Theorem), so the triple (A, Jβ , Λ) is a cuspidal type occurring in π. Since the representation πΛ = c-IndG Jβ Λ is irreducible, we conclude that π ∼ = πΛ , as desired. If (A , J , Λ ) is another cuspidal type occurring in π, then (π) > 0 (12.9) and 13.3 Corollary shows that Λ is of the second kind in the deﬁnition. The uniqueness statement therefore follows from 15.4 Theorem. Consequently: Corollary (Classiﬁcation Theorem).The map (A, J, Λ) −→ πΛ = c-IndG J Λ induces a bijection between the set of conjugacy classes of cuspidal types in G and the set of equivalence classes of irreducible cuspidal representations of G. Remark. Thus an irreducible cuspidal representation π of G contains a cuspidal type (A, J, Λ). If (π) = 0, this type is of the ﬁrst kind in 15.5 Deﬁnition. If 0 < (π) (χπ), for all characters χ of F × , then it is of the second kind. Otherwise, it is of the third kind. 15.6. Corollary 15.5 reduces the study of cuspidal representations of G to that of cuspidal types in G. We therefore need to investigate the structure of cuspidal types. In the deﬁnition of these, it is only the second case of which there is anything to say. We therefore take a simple stratum (A, n, α), n 1, E = F [α], and describe the representations Λ ∈ C(ψα , A). We need some intermediate groups: [n/2]+1

Hα1 = UE1 UA

,

[(n+1)/2]

Jα1 = Jα ∩ UA1 = UE1 UA

.

(15.6.1)

Observe that Jα1 = Hα1 if and only if n is odd. Proposition 1. Suppose that n is odd. (1) Every Λ ∈ C(ψα , A) has dimension 1, and (2) two characters Λ1 , Λ2 ∈ C(ψα , A) intertwine in G if and only if Λ1 = Λ2 . = UE Proof. We have E × ∩ UA [n/2]+1 with ψα on UE , the map [n/2]+1

[n/2]+1

Λ : xu −→ φ(x)ψα (u),

. If φ is a character of E × agreeing

x ∈ E × , u ∈ UA

[n/2]+1

,

15. Classiﬁcation of Cuspidal Representations

109

deﬁnes a character of Jα (since ψα is invariant under conjugation by E × ). Any irreducible representation of Jα containing ψα is of this form. This proves (1). In (2), any g ∈ G which intertwines Λ1 with Λ2 must intertwine ψα itself, and hence lie in Jα . It therefore ﬁxes each Λi and the assertion follows. We turn to the case where n is even. In particular, E/F is unramiﬁed and A∼ = M. In parallel to the ﬁrst case, ψα admits extension to a character of the group Hα1 . We have to consider representations of the intermediate group Jα1 . Lemma. Suppose that n is even, and let θ be a character of Hα1 extending ψα . There is a unique irreducible representation ηθ of Jα1 such that ηθ | Hα1 contains θ. Moreover, (1) dim ηθ = q, and (2) ηθ | Hα1 is a multiple of θ. We give the proof later, in 16.4. Using the same notation, we prove: Proposition 2. (1) An element g ∈ G intertwines ηθ if and only if g ∈ Jα . (2) The representation ηθ admits extension to an irreducible representation of Jα , and any such extension lies in C(ψα , A). (3) A representation Λ ∈ C(ψα , A) satisﬁes Λ | Jα1 ∼ = ηθ , for a uniquely 1 determined character θ of Hα . (4) Two representations Λ1 , Λ2 ∈ C(ψα , A) intertwine in G if and only if they are equivalent. Proof. The intertwining statements follow the standard course, so we do not repeat the details. In (2), the representation ηθ surely admits extension to a representation ρ of F × Jα1 . Since ηθ is stable under conjugation by Jα , so is ρ. Since n is even, the ﬁeld extension E/F is unramiﬁed, so Jα /F × Jα1 is cyclic, of order q+1. Thus ρ extends to Jα . 15.7. We note a useful consequence of these discussions: Proposition. Let (A, J, Λ) be a cuspidal type in G. Set J 0 = J ∩ UA and λ = Λ | J 0. (1) The representation λ is irreducible. (2) An element g ∈ G intertwines λ if and only if g ∈ J. (3) Let (π, V ) be an irreducible smooth representation of G which contains λ. Then π is cuspidal and λ occurs in π with multiplicity one. Proof. The ﬁrst assertion follows immediately from the constructions in (15.6). The second is parallel to earlier arguments. For the third, we consider

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4 Cuspidal Representations

the natural representation of J on the isotypic space V λ . This is a direct sum of irreducible representations of J extending λ and agreeing with ωπ on Z. If we choose one such component Λ, we have π ∼ = c-Ind Λ. Any other component must intertwine with, hence be equivalent to, Λ. However, HomJ (Λ, π) = EndG (V ) is one-dimensional. That is, Λ occurs in π with mul tiplicity one, and V λ = V Λ . 15.8. In a cuspidal type (A, J, Λ), the representation Λ has a singularly straightforward structure, but the group J may be diﬃcult to specify without supplementary information. It is sometimes more convenient, therefore, to use a variant of the construction in which the group is standard but the representation may be less explicit. Deﬁnition. A cuspidal inducing datum in G is a pair (A, Ξ), where A is a chain order in A and Ξ is an irreducible smooth representation of KA of the A form Ξ = IndK J Λ, for some cuspidal type (A, J, Λ). It is straightforward to give a direct deﬁnition of cuspidal inducing datum, parallel to 15.5. By transitivity of induction, we have: Proposition 1. (1) If (A, Ξ) is a cuspidal inducing datum in G, the representation πΞ = c-IndG KA Ξ is an irreducible cuspidal representation of G. (2) The map (A, Ξ) → πΞ induces a bijection between the set of G-conjugacy classes of cuspidal inducing data in G and the set equivalence classes of irreducible cuspidal representations of G. If (A, Ξ) is a cuspidal inducing datum, we use the notation: A (Ξ) = min {n 0 : UAn+1 ⊂ Ker Ξ} = eA (πΞ ).

(15.8.1)

An analogue of 15.7 Proposition also holds here: Proposition 2. Let (A, Ξ) be a cuspidal inducing datum in G. Set ξ = Ξ | UA . (1) The representation ξ is irreducible, and occurs in πΞ with multiplicity one. (2) An element g ∈ G intertwines ξ if and only if g ∈ KA . Proof. Let (A, J, Λ) be a cuspidal type which induces Ξ. If we set λ = Λ | A J ∩ UA , then ξ = IndU J∩UA λ. All assertions now follow from 15.7 Proposition. Exercise. Let A be a chain order in A and let Θ be an irreducible smooth representation of KA . Let n be the least integer 0 such that UAn+1 ⊂ Ker Θ. Suppose that n 1 and that there is a simple stratum (A, n, α) such that Θ | UAn contains ψα . Show that (A, Θ) is a cuspidal inducing datum in G.

16. Intertwining of Simple Strata

111

15.9. Looking back to the deﬁnition (15.5) of cuspidal type, we see that there is a degree of overlap between the third kind and the other two. That between the ﬁrst and third is insigniﬁcant: if (A, J, Λ) is a cuspidal type of the ﬁrst kind, then so is (A, J, χ ◦ det ⊗Λ) if and only if χ has level zero. For the other case, let (A, n, α), n 1 be a simple stratum and let χ be a character of F × . If χ is of level zero, the map Λ −→ χΛ = Λ ⊗ χ ◦ det is a permutation of the set C(ψα , A). Suppose, on the other hand, that χ has level l 1. If l > n/eA , then (A, J, χ ◦ det ⊗Λ) is unequivocally of the third kind, so assume that eA l n. The character χ ◦ det | UA1 thus has level eA l n. If we take c ∈ p−l such [n/2]+1 that χ(1+x) = ψ(cx) for x ∈ p[l/2]+1 , then χ ◦ det | UA = ψc . The triple (A, n, α+c) is a simple stratum. With this notation, we have: Proposition. The map Λ → χΛ is a bijection C(ψα , A) → C(ψα+c , A).

16. Intertwining of Simple Strata We prove the results quoted in §15 concerning intertwining properties of various characters deﬁned by simple strata. 16.1. We start with the Intertwining Theorem of 15.1. Throughout, we write P = rad A. The implication (2) ⇒ (1) is trivial. We show next that (3) ⇒ (2). The [(n+1)/2] [n/2]+1 is contained in KA and so normalizes UA . If we take group E × UA u ∈ E × , y ∈ P[(n+1)/2] and x ∈ P[n/2]+1 , we get ψα u(1+y)(1+x)(1+y)−1 u−1 = ψA αu (1+y)(1+x)(1+y)−1 − 1 u−1 = ψA u−1 αu (1+y)(1+x)(1+y)−1 − 1 = ψA α (1+y)(1+x)(1+y)−1 − 1 ; We have (1+y)(1+x)(1+y)−1 ≡ 1+x (mod Pn+1 ), so the last expression reduces to ψα (1+x), as required. We next show that (1) ⇒ (2). We proceed via the following very useful lemma: Lemma. Let (A, n, α), (A, n, β) be simple strata with n 1. Let g ∈ G and suppose that g intertwines ψα with ψβ on UAn : then g ∈ KA and the characters ψα | UAn , ψβ | UAn are conjugate under g. Proof. By hypothesis and 12.7 Proposition, we have (α+P1−n ) ∩ g −1 (β+P1−n )g = ∅.

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4 Cuspidal Representations

Let γ ∈ (α+P1−n ) ∩ g −1 (β+P1−n )g. As γ ∈ α+P1−n = αUA1 , it is minimal over F and F [γ]× ⊂ KA (13.4). If L is the chain of A-lattices in V = F 2 , then L is the chain of all oF [γ] -lattices in V . On the other hand, gγg −1 ∈ βUA1 so g −1 L is the chain of all oF [γ] -lattices in V . That is, g −1 L = L and so g ∈ KA . Finally, the cosets αUA1 and g −1 βUA1 g = g −1 βgUA1 intersect, and so they are equal. [n/2]+1

, it surely inReturning to the main proof, if g ∈ G intertwines ψα on UA n tertwines ψα | UA . By the lemma, it lies in KA . In particular, g normalizes the [n/2]+1 [n/2]+1 group UA . Since it intertwines ψα | UA , it must actually normalize this character. Thus (1) ⇒ (2). 16.2. We have to show that (2) ⇒ (3). Let g ∈ G normalize the character ψα [n/2]+1 of UA ; in particular, g ∈ KA (16.1 Lemma). Moreover, conjugation by g [(n+1)/2] ﬁxes the coset αUA = α+P−[n/2] . × We have KA = E UA ; we may as well, therefore, take g ∈ UA . The intertwining condition then translates into g −1 αg ≡ α (mod P−[n/2] ) or, equivalently, (16.2.1) αgα−1 ≡ g (mod P[(n+1)/2] ). The following lemma, applied with k = [(n+1)/2], then completes the proof of the Intertwining Theorem. Lemma. Let g ∈ A, k ∈ Z, k 1; then αgα−1 ≡ g (mod Pk ) if and only if g ∈ oE +Pk . Proof. The “if” implication is clear. We prove the converse by induction on k. We ﬁrst assume k 2 and, by inductive hypothesis, g ∈ oE +Pk−1 . We may as k−1 g0 , well take g ∈ Pk−1 . If E is a prime element of E, we can write g = E −1 ≡ g0 (mod P). We are thus reduced to the where g0 ∈ A satisﬁes αg0 α case k = 1. We deal with this case by direct computation. Suppose ﬁrst that E/F is unramiﬁed. Conjugating, we may take A = M. We have α = −n α0 , where ˜ 0 of α0 in is a prime element of F and α0 ∈ M. By deﬁnition, the image α M/M ∼ = M2 (k) generates a quadratic ﬁeld extension l of k. Moreover, l is the image of oE in M2 (k). The centralizer of l in M2 (k) is l itself (5.3), and the result follows in this case. Suppose, on the other hand, that E/F is totally ramiﬁed; we conjugate to achieve A = I. The element α is of the form uΠ −n , for some u ∈ UI and some prime element Π of I. (We recall also that n is odd in this case.) We have I/PI = k ⊕ k, and conjugation by a prime element interchanges the factors. The result again follows on observing that the image of oE in k ⊕ k consists of the elements (x, x).

16. Intertwining of Simple Strata

113

16.3. We now prove the Conjugacy Theorem of 15.2. We are given an element [n/2]+1 . An argument identical g ∈ G which intertwines the characters ψαi of UA to that of 12.7 Proposition gives (α1 + P−[n/2] ) ∩ g −1 (α2 + P−[n/2] )g = ∅. In particular, (α1 + P1−n ) ∩ g −1 (α2 + P1−n )g = ∅. (16.1) Lemma yields g ∈ KA ; since KA = F [α2 ]× UA , we may as well take g ∈ UA . The cosets α1 +P−[n/2] = α1 UA

[(n+1)/2]

g

−1

−[n/2]

(α2 +P

)g = g

−1

,

[(n+1)/2] α2 g UA ,

intersect. They are therefore equal, as required.

16.4. We have to prove 15.6 Lemma. The proof has two components, the ﬁrst of which is an application of the theory of the preceding paragraphs. To simplify the notation, we set n = 2m, so that Jα1 = UE1 UAm and Hα1 = 1 m+1 UE UA . We deﬁne m+1 V = Jα1 /Hα1 ∼ . = UAm /UEm UAm+1 ∼ = Pm /pm E +P

Thus V is a k-vector space of dimension 2 and, in particular, an elementary abelian p-group, where p is the characteristic of k. For x, y ∈ Jα1 , we consider the quantity θ[x, y], where [x, y] denotes the commutator xyx−1 y −1 . Using standard commutator identities, the fact that θ is ﬁxed under conjugation by Jα1 implies θ[xx , y] = θ[x, y]θ[x , y] and symmetrically. Moreover, θ[x, y] = 1 if either x or y lies in Hα1 . In all, we have a bi-additive pairing hα : V × V −→ C× , induced by (x, y) → θ[x, y]. Further, hα is alternating, in that hα (x, x) = 1 for all x. Explicitly, if x, y ∈ Pm , we have hα (1+x, 1+y) = ψα [1+x, 1+y] = ψA α(xy−yx) . The key point is: Lemma 1. The pairing hα on V is non-degenerate. The group Jα1 /Ker θ has centre Hα1 /Ker θ.

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4 Cuspidal Representations

Proof. This reduces to the following assertion: if x ∈ Pm satisﬁes ψA (α(xy−yx)) = 1,

y ∈ Pm ,

m+1 . As ψA (α(xy−yx)) = ψA ((αx−xα)y), this is equivalent then x ∈ pm E +P 1−m , or αxα−1 ≡ x (mod Pm+1 ). Lemma 16.2 now implies to αx−xα ∈ P x ∈ Hα1 , as required. If we view θ as a character of Hα1 /Ker θ, it is faithful. If x lies in the centre of Jα1 /Ker θ, then hα (x, V) = 1, and x ∈ Hα1 /Ker θ, as required for the second assertion.

We recall a standard group-theoretic argument: Lemma 2. Let G be a ﬁnite group, with cyclic centre N , such that V = G/N is an elementary abelian p-group. Let χ be a faithful character of N . The pairing hχ : V × V → C× induced by (x, y) −→ χ[x, y], is nondegenerate. There is a unique irreducible representation ζ of G such that ζ | N contains χ. Moreover: (1) (2) (3) (4)

ζ | N is a multiple of χ; dim ζ = |V |1/2 ; |V |1/2 ; IndG N χ=ζ if H is a subgroup of G, containing N , such that (G:H) = |V |1/2 , and such that hχ is null on H/N , then ζ = IndG H φ, for any character φ of H such that φ | N = χ.

Proof. As V is an elementary abelian p-group, the values hχ (V, V ) lie in the group µC (p) of p-th roots of unity in C. Composing with some group isomorphism µC (p) ∼ = Fp , the form hχ becomes an Fp -bilinear form V × V → Fp . It is alternating; it is nondegenerate because, if x ∈ G satisﬁes χ[x, y] = 1 for all y ∈ G, then [x, y] = 1 and x is central in G. In particular, dimFp V = 2d, say, is even. There is a vector subspace U of V , of dimension d, such that hχ (u, u ) = 0, for all u, u ∈ U . Since hχ is nondegenerate, (a) U is its own orthogonal complement in V , and (b) any linear map U → Fp is of the form u → hχ (u, v), for a uniquely determined element v ∈ V /U . Returning to the multiplicative context, let H be the inverse image of U in G. The triviality of hχ on U then implies H is abelian. In particular, χ admits extension to a character χ of H. By property (b), any two such extensions are G-conjugate, and the G-normalizer of χ is precisely H itself. We deduce that d 1/2 . If η is an irreducible η = IndG H χ is irreducible. Its dimension is p = |V |

17. Representations with Iwahori-Fixed Vector

115

representation of G containing χ, then η | H is a sum of characters, all extending χ. All such characters are G-conjugate, so η | H contains χ and η ∼ = η. All other assertions follow easily. In the context of 15.6, we apply Lemma 2 with G = Jα1 /K, where K = Ker θ. The subgroup N = Hα1 /K ∼ = θ(Hα1 ) is certainly central in G. The nondegeneracy of hα (Lemma 1) implies that N is the centre of G. Lemma 15.6 now follows directly from Lemma 2.

17. Representations with Iwahori-Fixed Vector Let I = UI be the standard Iwahori subgroup of G = GL2 (F ). The class of irreducible representations (π, V ) of G for which V I = 0 is particularly subtle and interesting. We study these in order to ﬁnish the proof of 14.3 Lemma, but we also need an analysis of the coeﬃcients of the Steinberg representation for use in Chapter VI. Throughout the section, we use the following notation: j 1 0 U 0 , Nj = 1 p , Nj = pj 1 , w = 01 10 , T 0 = 0F U 0 1

F

for j ∈ Z. In particular, we have I = N1 T 0 N0 (with the factors in any order). 17.1. We ﬁx a prime element of F . Let W be the set (indeed group) of all matrices of the form a 0 0 a or , b b 0

0

for a, b ∈ Z. Proposition. The elements of W form a set of coset representatives for I\G/I. Proof. We start from the Cartan decomposition a 0 K, G= K b 0

ab

a where K = GL2 (o) (7.2.2). We decompose each coset K 0 0b K into (I, I)double cosets. The presence of a central makes no diﬀerence, so we may −a factor as well treat the coset Ca = K 0 10 K, for an integer a 0. If a = 0, the coset C0 is just K and we know already that K = I ∪ IwI (7.3.2). We therefore assume that a 1. We show that Ca is the disjoint union of the cosets −a −a 1 0 0 1 I, I 0 I, I I. (17.1.1) I 0 I, I −a −a 0

1

0

1

0

0

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4 Cuspidal Representations

All of these cosets are certainly contained in Ca . Let µ be the Haar measure on G for which µ(I) = 1. Thus µ(K) = q+1 and µ(Ca ) = (q+1)2 q a−1 . We have

I∩

−a 0 0 1 −a

0

−a −1 I 0 = N−a T 0 Na+1 , 0

1

0

1

−a −1 0 I 0 = N0 T 0 Na+1 .

1

This group has index q a in I, so −a µ(I 0 10 I) = q a . Similarly, µ(I µ(I µ(I

1

0 I) −a 0 −a 0 I) 1 0 0 1 I) −a 0

= qa , = q a−1 , = q a+1 .

Comparing measures, we see that the only possible equality in the list (17.1.1) is between the ﬁrst two, given by the diagonal matrices. If they were the same, we’d have an equation −a 1 0 y, x 0 = −a 0

1

0

with x, y ∈ I. Examining only the (1, 1)-entry in these matrices shows this is impossible. The measures of the cosets (17.1.1) add up to that of Ca , while any coset IgI is open of positive measure. We deduce that Ca is the disjoint union of the four cosets (17.1.1), as required. We remark that the group W is an example of an aﬃne Weyl group. It has a normal subgroup W0 = {x ∈ W : det x = 1}, 0 1 ) is the and it is the semi-direct product W = Π W0 , where Π = ( 0 standard prime element of I.

17.2. Using the same Haar measure µ, we consider some special elements of the Hecke algebra H(G, I) = eI ∗ H(G) ∗ eI of compactly supported functions on G which are invariant under translation by I on each side. For g ∈ G, we let [g] ∈ H(G, I) denote the characteristic function of IgI. In particular, [Π] is the characteristic function of IΠI = IΠ = ΠI.

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117

Lemma 1. For any g ∈ G, we have [Π] ∗ [g] = [Πg] and [g] ∗ [Π] = [gΠ]. In particular, [Π] is invertible in H(G, I), with inverse [Π −1 ]. Proof. Since Π normalizes I, this reduces to an easy calculation.

Lemma 2. The function [w] satisﬁes the relation [w] ∗ [w] + (1−q)[w] − qeI = 0. Proof. The support of [w] ∗ [w] is contained in IwIwI ⊂ GL2 (o) = I ∪ IwI, so [w] ∗ [w] = a[w]+beI , for some constants a, b. We ﬁrst compute [w] ∗ [w](1) = [w](x)[w](x−1 ) dµ(x) = µ(IwI) = q, G

xwI. Next, [w](x)[w](x−1 w) dµ(x) = [w] ∗ [w](w) =

since IwI =

x∈N0 /N1

G

[w](wyw).

y∈N0 /N1

The term y ∈ N1 contributes nothing, since [w](w2 ) = [w](1) = 0. For y ∈ N0 N1 , we use the identity −1 −1 1 0 = −a 1 w 1 a a1 0

a

0

1

to get [w] ∗ [w](w) = q−1. Therefore [w] ∗ [w] = (q−1)[w] + qeI , as required. 17.3. We now complete the proof of 14.3 Lemma by observing that, in our present notation, the characteristic function of ItI (as in 14.3) is [t] = [w]∗[Π], which is an invertible element of H(G, I). This proves the Lemma and the Proposition of 14.3, and ﬁlls the last gap in the proof of the Exhaustion Theorem (14.5). 17.4. We turn to a diﬀerent aspect of the matter. Let (π, V ) be an irreducible smooth representation of G. One says that π is square-integrable (modulo the centre Z of G) if |ωπ | = 1 and, for a Haar measure dg˙ on G/Z, we have: | ˇ v , π(g)v|2 dg˙ < ∞, (17.4.1) G/Z

for every v ∈ V , vˇ ∈ Vˇ . Exactly in parallel with 10.1 Proposition (2), (17.4.1) holds for all vectors vˇ ⊗ v if it holds for one such vector vˇ ⊗ v = 0. One says that an irreducible smooth representation π of G is essentially square-integrable, or is in the discrete series of G, if it is of the form φπ0 , for a character φ of F × and a square-integrable representation π0 of G.

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4 Cuspidal Representations

17.5. If π is cuspidal, then certainly it is essentially square-integrable; it is square-integrable if and only if |ωπ | = 1. Thus it is the non-cuspidal representations which are of interest here: Theorem. (1) The Steinberg representation of G is square-integrable. (2) Let π be an irreducible smooth representation of G, and suppose that π is not cuspidal. It is square-integrable if and only if it is of the form π∼ = φ · StG , for a character φ of F × satisfying |φ| = 1. 17.6. To prove part (1) of 17.5 Theorem, we need to produce one non-trivial coeﬃcient of StG which is square-integrable modulo Z. We write StG = (π, V ), −1 and identify it with the unique irreducible G-subspace of (Σ, W ) = IndG B δB (cf. (9.10.4). (9.10.5)). Thus V is the kernel of the G-homomorphism

W −→ C,

f −→

f (g) d¯ g, B\G

where G acts trivially on C and d¯ g is a semi-invariant measure on B\G. Equivalently, V = {f ∈ W : f (k) dk = 0}, K

where K = GL2 (o) and dk is some Haar measure on K. As in 14.4, dim V I = 1, and so V I is spanned by the function θ ∈ W deﬁned by: θ(gj) = θ(g), θ(1) = 1,

g ∈ G, j ∈ I θ(w) = −q −1 .

ˇ W ˇ ) = IndG 1T ; we let τ ∈ W ˇ be the characteristic function of We have (Σ, B BI. The function τ (k)θ(kg) dk = θ(jg) dj f (g) = τ, π(g)θ = K

I

is then a non-trivial coeﬃcient of π, ﬁxed on either side by I. We henceforward work with the Haar measure µ on G for which µ(I) = 1. The algebra H(G, I) acts on V I in the natural way; we let it act on Vˇ I ⊗V I via the second factor. We identify Vˇ I ⊗V I with a subspace of C(π). This subspace has dimension 1, and is spanned by f . The action of H(G, I) is given by ˇ φ · f (g) = f ∗ φ(g),

φ ∈ H(G, I),

and there exists αφ ∈ C such that φ · f = αφ f,

φ ∈ H(G, I).

(17.6.1)

17. Representations with Iwahori-Fixed Vector

119

Writing f (g) = τ, π(g)θ as above, we have φ · f (g) = τ, π(g)π(φ)θ, so also π(φ) θ = αφ θ,

φ ∈ H(G, I).

(17.6.2)

The map φ → αφ is an algebra homomorphism H(G, I) → C. 17.7. Take Π as in 17.1. Proposition. Let g ∈ G and, as before, let µ be the Haar measure on G such that µ(I) = 1 and [g] the characteristic function of IgI. We then have α[g] = [g] · f (1) = µ(IgI) f (g). In particular, α[w] = −1,

α[Π] = −1.

Proof. The ﬁrst equality is a matter of deﬁnition. For the second, f (x) [g](x) dµ(x) = f (g) µ(IgI), [g] · f (1) = G

as required. The ﬁrst of the special cases is given by the original deﬁnition of f . In the second, we get 0 )−1 f (w) = −1, α[Π] = f (Π) = δB ( 10 as desired. The volume µ(IgI) depends only on the coset ZIgI so, at least formally, we have |f (g)|2 dg˙ = |α[g] |2 µ(IgI)−1 , (17.7.1) G/Z

g∈ZI\G/I

where dg˙ is the Haar measure on G/Z for which IZ/Z has measure 1. To prove part (1) of 17.5 Theorem, we have to check that this series is convergent. 17.8. To do this, we use a systematic description of the algebra H(G, I). Since Π 2 = 1G ∈ Z, a set of representatives for the space ZI\G/I is given by the matrices Π m x, with m ∈ {0, 1} and x ∈ W0 . Besides the standard permutation matrix w ∈ W0 , we shall also need the element 0 −1 −1 . w = ΠwΠ = 0 Lemma 1. The set S = {w, w } generates the group W0 .

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4 Cuspidal Representations

Proof. Simple calculations give −1 w w = 0

and so on: −a 0

0 a

= (w w)a ,

Conjugating by Π, we get a 0 = (ww )a , −a 0

This proves Lemma 1.

0

a

0

ww w =

,

−a

0

0

−1

0

= w(w w)a ,

0

−(a+1)

(a+1)

0

= w (ww )a ,

,

a 0.

a 0.

(17.8.1)

(17.8.2)

Given x ∈ W0 , there exist elements w1 , w2 , . . . wr ∈ S such that x = w1 w2 . . . wr . The length of x, denoted (x), is the least integer r 0 for which there is such an expression. There is one element of length 0 (namely 1), and two elements of length 1 (namely w and w ). The expressions (17.8.1), (17.8.2) are clearly minimal; it follows that W0 has exactly two elements of length a, for every integer a 1. Lemma 2. Let µ be the Haar measure on G for which µ(I) = 1. If x ∈ W0 , let [x] denote the characteristic function of IxI. (1) We have µ(IgI) = µ(IΠgI) = q (g) , g ∈ W0 . (2) Let g ∈ W0 , let x ∈ S = {w, w }, and suppose that (xg) = (g)+1. Then [xg] = [x] ∗ [g]. (3) For g ∈ W0 , we have α[g] = (−1)(g) . Proof. The list above, together with the calculations in 17.1, give part (1). Likewise (2), while (3) follows from 17.7 Lemma. 17.9. We now complete the proof of 17.5 Theorem (1), starting from (17.7.1): |f (g)|2 dg˙ = |α[g] |2 µ(IgI)−1 G/Z

g∈ZI\G/I

=2

q −(g) .

g∈W0

Given an integer b 1, there are exactly 2 elements of W0 of length b, so this series converges. Remark. The same argument shows that, for the coeﬃcient f above, the integral |f (g)|1+ dg˙ G/Z

converges for any > 0. It follows that it also converges for any f ∈ C(π).

17. Representations with Iwahori-Fixed Vector

121

17.10. We now prove 17.5 Theorem (2). The trivial (one-dimensional) representation of G is certainly not square-integrable, so it is enough to show: Proposition. Let χ be a character of T such that (π, V ) = IndG B χ is irreducible. The representation π is then not essentially square-integrable. Proof. Twisting by an unramiﬁed character of F × , we can assume that ωπ = χ | Z has absolute value 1. If π were essentially square-integrable it would be square-integrable. We have to show it is not. r = 1 + pr M2 (o). We choose r suﬃciently If r 1 is an integer, let Kr = UM large to ensure that χ is trivial on Kr ∩ T . We deﬁne a function φ ∈ V by supp φ = BKr , φ(bk) = χ(b),

b ∈ B, k ∈ Kr .

−1 −1 . We similarly take φ ∈ Vˇ Kr to have support We have (ˇ π , Vˇ ) ∼ = IndG B δB χ BKr and φ (1) = 1. We consider the coeﬃcient ¯ f : g −→ φ , π(g)φ = φ (h)φ(hg) dh B\G = φ (k)φ(kg) dk, K

¯ on B\G or a Haar measure dk on K. We show for a semi-invariant measure dh that |f (g)|2 dg˙ (17.10.1) G/Z

diverges, where dg˙ is a Haar measure on G/Z. Let χ = χ1 ⊗ χ2 , let be a prime element of F , and deﬁne t ∈ T as follows: ⎧ 0 ⎪ ⎨ if |χ1 ()| q −1/2 , 0 1 −1 t= ⎪ ⎩ 0 if |χ1 ()| < q −1/2 . 0

1

We take n > r, and consider the contribution to the integral from the set ˙ n ) = cq n (where Cn = ZKr tn Kr . There is a constant c > 0 such that µ(C dg˙ = dµ(g)). ˙ As usual, we set 1 0 Nk = pk 1 , k ∈ Z. In the ﬁrst case, the support of the function k → φ (k)φ(ktn ), k ∈ K, contains (K ∩ B)N0 , so there is a constant c1 > 0 such that |f (g)| > c1 q −n/2 ,

g ∈ Cn .

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4 Cuspidal Representations

In the second case, the support of this function contains (K ∩ B)Nn+r , and again |f (g)| > c2 q −n/2 , g ∈ Cn ,

for a constant c2 > 0. In either case, therefore, there is a constant c3 > 0 such that |f (g)|2 dg˙ > c3 , n > r, Cn

and the integral (17.10.1) therefore diverges. This completes the proof of 17.5 Theorem (2). Exercises. (1) Calculate the formal degree (see (10a.2) Remark) of the Steinberg representation. (2) An irreducible representation (π, V ) is called spherical if V K = 0, K = GL2 (o). Show that π is spherical if and only if it is a composition factor of IndG B (χ1 ⊗ χ2 ) with both χi unramiﬁed and π is not a twist of the Steinberg representation. Show that, if π is spherical, then dim V K = 1. (3) Alternative version: Let K = GL2 (o); show that the algebra H(G, K) is commutative. Deduce that any simple H(G, K)-module has dimension one and, if (π, V ) is spherical, then dim V K = 1. Further reading. The discussion of hereditary (or chain) orders and fundamental strata has a close analogue for GLn (F ): see [69] for the theory of hereditary orders and [11] for fundamental strata. The concept of fundamental stratum, and with it 12.9 Theorem, has been generalized to arbitrary connected reductive groups in [66]. This chapter follows exactly the course of the classiﬁcation of the irreducible cuspidal representations of GLn (F ) [19], minus several levels of complexity. It is eﬀectively Kutzko’s original treatment [50], [51] ameliorated by hindsight. Carayol’s account [21] of GLp (F ) (p prime) is in much the same spirit. It provides a more general, but easily accessible, example. The same sort of approach is eﬀective in constructing cuspidal representations of much more general groups: see, for example, [73] or [80]. The importance of representations with Iwahori-ﬁxed vector was ﬁrst recognized in [7]. A survey of the general case is given in [24]. The signiﬁcance of these ideas for investigating the structure of induction functors is discussed, for GLn (F ), in [19] and for general groups in [20].

5 Parametrization of Tame Cuspidals

18. 19. 20. 21. 22.

Admissible pairs Construction of representations The parametrization theorem Tame intertwining properties A certain group extension

The cuspidal types, and hence the cuspidal representations of G, constructed in §15 are visibly related to (multiplicative) characters of quadratic ﬁeld extensions of F . When the characteristic p of the ﬁnite ﬁeld k = o/p is odd, this relation can be made precise and informative: the connection is the subject of this chapter. When p = 2, only certain types are amenable to such a convenient description. However, it will be useful, and costs no extra eﬀort, to carry this case along. We therefore impose no restriction on the prime number p.

18. Admissible Pairs We start with some purely ﬁeld-theoretic constructions. Throughout this section, ψ denotes a character of F of level one. 18.1. Let E/F be a ﬁnite ﬁeld extension. We recall that E/F is tamely ramiﬁed if p e(E|F ). Equivalently, E/F is tamely ramiﬁed if and only if TrE/F (oE ) = o. In particular, every quadratic ﬁeld extension E/F is tamely ramiﬁed if p = 2 while, in the case p = 2, a quadratic extension E/F is tamely ramiﬁed if and only if it is unramiﬁed. We gather some elementary properties: Lemma. Let E/F be a tamely ramiﬁed ﬁeld extension, and set e = e(E|F ).

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5 Parametrization of Tame Cuspidals

1+[r/e] (1) For r ∈ Z, we have TrE/F (p1+r = p1+r ∩ F. E )=p E (2) For m 1, the norm map NE/F induces an isomorphism ≈

UEem /UEem+1 → UFm /UFm+1 satisfying NE/F (1+x) ≡ 1 + TrE/F (x) (mod pm+1 ),

x ∈ pem E .

(3) The norm NE/F induces a map UE /UE1 → UF /UF1 which is surjective if E/F is unramiﬁed, and has kernel and cokernel of order gcd (e, q−1) if E/F is totally ramiﬁed. For E/F as in the lemma, we set ψE = ψ ◦ TrE/F . Proposition. (1) The character ψE has level one. (2) Let χ be a character of F × of level m 1, and let χE = χ ◦ NE/F . The character χE has level em. If c ∈ p−m satisﬁes χ(1+x) = ψ(cx), x ∈ p[m/2]+1 , then χE (1+y) = ψE (cy),

[em/2]+1

y ∈ pE

.

Proof. Part (1) and the ﬁrst assertion of (2) follow from part (2) of the lemma. [em/2]+1 , then If x ∈ pE NE/F (1+x) ≡ 1 + TrE/F (x) (mod pm+1 ), and the last assertion follows.

18.2. We consider a pair (E/F, χ), where E/F is a tamely ramiﬁed quadratic ﬁeld extension and χ is a character of E × . Deﬁnition. The pair (E/F, χ) is called admissible if (1) χ does not factor through the norm map NE/F : E × → F × and, (2) if χ | UE1 does factor through NE/F , then E/F is unramiﬁed. Admissible pairs (E/F, χ), (E /F, χ ) are said to be F -isomorphic if there is an F -isomorphism j : E → E such that χ = χ ◦ j. In the case E = E , this amounts to χ = χσ , for some σ ∈ Gal(E/F ). We write P2 (F ) for the set of F -isomorphism classes of admissible pairs (E/F, χ). If (E/F, χ) is an admissible pair, and if φ is a character of F × , the pair (E/F, χ ⊗ φE ) is also admissible, where φE = φ ◦ NE/F . Let (E/F, χ) be an admissible pair, and let n be the level of χ. We say that (E/F, χ) is minimal if χ | UEn does not factor through NE/F . Clearly, any admissible pair (E/F, χ) is isomorphic to one of the form (E/F, χ ⊗ φE ), for a character φ of F × and a minimal pair (E/F, χ ).

19. Construction of Representations

125

Proposition. Let E/F be a tamely ramiﬁed quadratic ﬁeld extension, and let satisfy χ(1+x) = ψE (αx), χ be a character of E × of level m 1. Let α ∈ p−m E . Then (E/F, χ) is a minimal (admissible) pair if and only if the x ∈ pm E element α is minimal over F . Proof. Proposition 18.1 shows that χ | UEm factors through NE/F if and only ). if there exists c ∈ F such that c ≡ α (mod p1−m E On the other hand, the deﬁnition of minimal element amounts to the following: if α ∈ E and υE (α) = −m, then α is not minimal over F if and ) (cf. 13.4 Exercise). only if there exists c ∈ F such that α ≡ c (mod p1−m E The result now follows. Exercise. (1) Let (E1 /F, χ1 ), (E2 /F, χ2 ) be admissible pairs. Choose an F -embedding ji : Ei → A, i = 1, 2. Show that the pairs (Ei /F, χi ) are F -isomorphic if and only if there exists g ∈ G such that j2 (E2 ) = gj1 (E1 )g −1 and, for x ∈ E1× , χ1 (x) = χ2 (j2−1 (gj1 (x)g −1 )). (2) Let E1 /F , E2 /F be tamely ramiﬁed quadratic ﬁeld extensions. Show, as consequences of 18.1 Lemma, that NEi /F (Ei× ) is a subgroup of F × of index 2, and that E1 is F -isomorphic to E2 if and only if NE1 /F (E1× ) = NE2 /F (E2× ). Remark. Exercise (2) is a simple instance of local class ﬁeld theory, for which see §29 below.

19. Construction of Representations In this section, we attach to an admissible pair (E/F, χ) an irreducible cuspidal representation πχ of G = GL2 (F ). 19.1. We start with the special case of an admissible pair (E/F, χ) in which χ has level zero. Thus, by deﬁnition, E/F is unramiﬁed. Lemma. Let E/F be an unramiﬁed quadratic extension, let χ be a character of E × of level zero, and let σ ∈ Gal(E/F ), σ = 1. The following are equivalent: (1) the pair (E/F, χ) is admissible; (2) χ = χσ ; (3) χ | UE = χσ | UE . Proof. Since E/F is unramiﬁed, we have E × = F × UE , so surely (2) ⇔ (3). Since E/F is cyclic, the kernel of NE/F consists of all elements xσ /x, x ∈ E × .

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5 Parametrization of Tame Cuspidals

Thus χ factors through NE/F if and only if χ = χσ . The second condition in the deﬁnition of admissibility is empty in this case, so (1) ⇔ (2). Returning to the admissible pair (E/F, χ) of level zero, we write kE = oE /pE : thus kE /k is a quadratic ﬁeld extension. We choose an F -embedding E → A, and let A be the unique chain order with E × ⊂ KA (cf. 12.4). Conjugating by an element of G, we can take A = M = M2 (o). This gives an embedding of oE in M and hence a k-embedding of kE in M2 (k). × ˜ of kE . Condition The character χ | UE is the inﬂation of a character χ × (cf. 6.4). (3) in the lemma is equivalent to χ ˜ being a regular character of kE ˜ of GL2 (k) As in 6.4, χ ˜ gives rise to an irreducible cuspidal representation λ ˜ to a representation ˜ = πχ˜ in the notation of §6). Let λ be the inﬂation of λ (λ of UM = GL2 (o). The restriction of λ to UF is a multiple of χ | UF (as follows from (6.4.1)); we therefore extend λ to a representation Λ of KM = F × UM by deeming that Λ | F × be a multiple of χ. The triple (M, KM , Λ) is then a cuspidal type. We set πχ = c-IndG KM Λ. Thus πχ is an irreducible cuspidal representation of G such that (π) = 0. × , χ) ˜ Isomorphic pairs (E/F, χ) (of level zero) give rise to conjugate pairs (kE in GL2 (k), so the equivalence class of πχ depends only on the F -isomorphism class of (E/F, χ). Write P2 (F )0 for the set of isomorphism classes of admissible pairs (E/F, χ) in which χ has level zero. Likewise, let A02 (F )0 denote the set of equivalence classes of irreducible cuspidal representations π of G such that

(π) = 0. Proposition. The map (E/F, χ) → πχ induces a bijection ≈ A02 (F )0 . P2 (F )0 −→

(19.1.1)

Further, if (E/F, χ) ∈ P2 (F )0 , then: (1) if φ is a character of F × of level zero, then πχφE = φπχ ; (2) if π = πχ , then ωπ = χ | F × ; (3) the pair (E/F, χ) ˇ is admissible and π ˇχ = πχˇ . Proof. The analysis of cuspidal representations of GL2 (k) in 6.4, together with the lemma, shows that any cuspidal type of level zero arises from an admissible pair (E/F, χ) ∈ P2 (F )0 . The Exhaustion Theorem (14.5) then implies that the map (19.1.1) is surjective. To prove injectivity, suppose we have pairs (Ei /F, χi ) such that the representations πχi are equivalent. The extensions Ei /F are unramiﬁed and so F -isomorphic: we may as well take E1 = E2 = E, say. The central character

19. Construction of Representations

127

relation gives χ1 | F × = χ2 | F × . Corollary 15.5 implies that the cuspidal representations πχ˜i of GL2 (k) are equivalent (notation of 6.4). The charac× are then Galois-conjugate (6.4 Theorem (2)). It follows that the ters χ ˜i of kE characters χi | UE are Galois-conjugate and then that the pairs (E/F, χi ) are F -isomorphic. Thus (19.1.1) is injective and hence bijective. Properties (1) and (2) are immediate from the construction, while (3) is given by the construction and 11.4 Remark 3. Exercise. Find an example of a representation π ∈ A02 (F )0 and a ramiﬁed character φ of F × such that φπ ∼ = π. 19.2. We now ﬁx a character ψ ∈ F, of level one. Let (E/F, χ) be a minimal admissible pair such that χ has level n 1. We set ψE = ψ ◦ TrE/F , ψA = ψ ◦ trA . Next, we choose an element α ∈ p−n E such that χ(1+x) = ψE (αx), x ∈ [n/2]+1 pE . We choose an F -embedding of E in A = M2 (F ) and we let A be the unique chain order in A such that E × ⊂ KA (12.4). Then eA = e(E|F ) and the triple (A, n, α) is a simple stratum (18.2, 13.5). Attached to the simple stratum (A, n, α), we have the subgroups Jα , Jα1 , 1 Hα , as in §15. The next step is to deﬁne an irreducible representation Λ ∈ C(ψα , A) (notation of 15.5). 19.3. Suppose in this paragraph that n = 2m+1 is odd. The desired representation Λ is then the character of Jα = E × UAm+1 given by Λ | UAm+1 = ψα ,

Λ | E × = χ.

(19.3.1)

Since trA | E = TrE/F and E ∩ UAm+1 = UEm+1 , these two conditions are consistent. The triple (A, Jα , Λ) is a cuspidal type in G, and so πχ = c-IndG Jα Λ is an irreducible cuspidal representation of G containing the fundamental stratum (A, n, α). Thus

(πχ ) = n/e(E|F ),

ωπ χ = χ | F × .

(19.3.2)

19.4. In this paragraph, we assume that (E/F, χ) is a minimal pair in which χ has even level n = 2m > 0. In particular, E/F is unramiﬁed. We deﬁne a character θ of Hα1 = UE1 UAm+1 by θ(ux) = χ(u) ψα (x),

x ∈ UAm+1 , u ∈ UE1 .

(19.4.1)

These conditions are consistent, as before. We let η = ηθ be the unique irreducible representation of Jα1 = UE1 UAm which contains θ (15.6 Lemma).

128

5 Parametrization of Tame Cuspidals

The cyclic group µE of roots of unity, of order prime to p in E, acts on Jα1 by conjugation, ﬁxing the representation η (15.6 Proposition 2). The subgroup µF acts trivially. Proposition. There is a unique irreducible representation η˜ of µE /µF Jα1 such that η˜ | Jα1 ∼ = η and tr η˜(ζu) = −θ(u), for u ∈ Hα1 and every ζ ∈ µE /µF , ζ = 1. We prove this later, in §22. We need the following consequence: Corollary. There is a unique irreducible representation Λ of Jα such that (1) Λ | Jα1 ∼ = η; (2) Λ | F × is a multiple of χ | F × ; (3) for every ζ ∈ µE µF , we have tr Λ(ζ) = −χ(ζ). Proof. The stated conditions certainly determine Λ uniquely. We have to show it exists. We identify µE /µF with E × /F × UE1 . We take the representation η˜ given by the proposition, and view it as a representation of µE /µF Jα1 /Ker θ. We inﬂate it to a representation ν of the group E × Jα1 /Ker θ (in which E × acts on Jα1 /Ker θ via the isomorphism E × /UE1 F × ∼ = µE /µF ). We next deﬁne a character χ ˜ of E × Jα1 /Ker θ by deeming it to be trivial on Jα1 /Ker θ and to ˜ ⊗ ν of E × Jα1 /Ker θ. agree with χ on E × . We form the representation Λ˜ = χ × There is a surjective group homomorphism E Jα1 /Ker θ → Jα /Ker θ given by (x, j) → xj (mod Ker θ). The kernel of this map is the group of elements (x−1 , x Ker θ), x ∈ UE1 . The representation Λ˜ is trivial on this kernel, so Λ˜ is the inﬂation of an irreducible representation Λ1 of Jα /Ker θ. The inﬂation of Λ1 to Jα is the representation Λ demanded by the Corollary. The representation Λ of the corollary lies in C(ψα , A). We deﬁne: πχ = c-IndG Jα Λ.

(19.4.2)

Thus πχ is an irreducible cuspidal representation of G satisfying

(πχ ) = n,

ωπχ = χ | F × .

(19.4.3)

19.5. We have to check that the construction of πχ is independent of choices: Proposition. Let (E/F, χ) be a minimal pair in which χ has positive level. The representation πχ depends, up to equivalence, only on the isomorphism

20. The Parametrization Theorem

129

class of the pair (E/F, χ). In particular, it is independent of the choices of ψ, α and of the embedding E → A. Moreover, if φ is a character of F × such that (E/F, χφE ) is also minimal, then πχφE = φπχ . Proof. Changing the choices of ψ and of α (but ﬁxing the embedding) changes neither the group Jα nor the representation Λ. Any two F -embeddings of E in A are G-conjugate, and so give rise to conjugate cuspidal types (A, Jα , Λ). Moreover, if we have minimal pairs (Ei /F, χi ) which are F -isomorphic, and ji : Ei → F is an F -embedding, the isomorphism (E1 /F, χ1 ) → (E2 /F, χ2 ) can be realized by a G-conjugation taking j1 (E1 ) to j2 (E2 ) which matches the characters (cf. 18.2 Exercise). In the ﬁnal assertion, the character χφE gives rise to the cuspidal type (A, Jα , Λ ⊗ φ ◦ det) (cf. 15.9). 19.6. Let (E/F, χ) be an admissible pair. As in 18.2, there is a character φ of F × and a character χ of E × such that (E/F, χ ) is minimal and χ = χ φE . We deﬁne (19.6.1) πχ = φπχ . The result is independent of the choice of decomposition χ = χ φE , by the ﬁnal assertion of 19.5 Proposition. Immediately:

(πχ ) = n/e(E|F ),

ωπ χ = χ | F × ,

(19.6.2)

where n is the level of χ. In all cases, the equivalence class of the representation πχ depends only on the isomorphism class of the admissible pair (E/F, χ). Writing A02 (F ) for the set of equivalence classes of irreducible cuspidal representations of G = GL2 (F ), we have a map P2 (F ) −→ A02 (F ), (E/F, χ) −→ πχ ,

(19.6.3)

deﬁned independently of all choices.

20. The Parametrization Theorem From now on, we denote by A02 (F ) the set of equivalence classes of irreducible cuspidal representations of G = GL2 (F ). 20.1. Let π be an irreducible cuspidal representation of G = GL2 (F ). We say that π is unramiﬁed if there exists an unramiﬁed character φ = 1 of F × such that φπ ∼ = π. 0 We denote by Anr 2 (F ) the set of unramiﬁed classes in A2 (F ). A representation π ∈ A02 (F ) Anr 2 (F ) will be called totally ramiﬁed.

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5 Parametrization of Tame Cuspidals

20.2. We come to the main result of the section: Tame Parametrization Theorem. The map (E/F, χ) → πχ of (19.6.3) induces a bijection ≈ A02 (F ) P2 (F ) −→ ≈ Anr P2 (F ) −→ 2 (F )

if p = 2, or if p = 2.

(20.2.1)

If (E/F, χ) ∈ P2 (F ), then: (1) (2) (3) (4)

if χ has level (χ), then (πχ ) = (χ)/e(E|F ); ω πχ = χ | F × ; ˇχ ; the pair (E/F, χ) ˇ is admissible and πχˇ = π × if φ is a character of F , then πχφE = φπχ .

Properties (1), (2) and (4) have already been observed. For (3), write πχ = c-IndG J Λ, for the cuspidal type (A, J, Λ) constructed from (E/F, χ). The pair (E/F, χ) ˇ gives rise to the type (A, J, Λ∨ ), and so (3) follows from 11.4 Remark 3. We therefore have only to show that the map (20.2.1) is a bijection. Remark. The properties (1)–(4) do not determine the bijection (20.2.1) uniquely. It is canonical, in that it does not depend on any of the choices made in its construction, but the lack of a simple, external characterization is a severe drawback. It does, however, serve to provide a usefully explicit approximation to the Langlands correspondence of Chapter VIII (which is speciﬁed by a short list of properties). 20.3. Let π ∈ A02 (F ). As a prelude to proving the theorem, we demonstrate that useful information can be gleaned from the group of characters φ of F × such that φπ ∼ = π. Note ﬁrst that, comparing central characters, the relation ∼ φπ = π implies φ2 = 1. Lemma. Let π be an irreducible cuspidal representation of G, containing a cuspidal inducing datum (A, Ξ). The representation π is unramiﬁed if and only if A ∼ = M. Proof. Let φ be the unramiﬁed character of F × of order 2. If A ∼ = M, then Ξ and det KA = (F × )2 UF ⊂ Ker φ; consequently Ξ ⊗ (φ ◦ det) ∼ = G ∼ φπ = c-IndG KA (Ξ ⊗ (φ ◦ det)) = c-IndKA (Ξ) = π.

Conversely, suppose that A ∼ = I (notation of (12.1.2)). In this case, π = c-IndG Λ, for a character Λ of a group J on which φ ◦ det is not trivial. The J relation φπ ∼ π would imply that the characters Λ, Λ ⊗ (φ ◦ det) intertwine = in G, contrary to 15.6 Proposition 1. Proposition. Suppose p = 2, and let π ∈ A02 (F ) be totally ramiﬁed.

21. Tame Intertwining Properties

131

(1) There exists a unique character φ of F × , φ = 1, such that φπ ∼ = π. The character φ is ramiﬁed, of level 0, and of order 2. (2) Let (A, n, α) be a simple stratum, with n 1, and suppose that π = θπ0 , for a character θ of F × and a representation π0 containing the character [n/2]+1 . The ﬁeld E = F [α] then satisﬁes NE/F (E × ) = Ker φ. ψα | UA Proof. Nothing is changed if we replace π by a twist, so we can assume

(π) (ξπ), for all characters ξ of F × . The Exhaustion Theorem and the lemma together imply that there is a ramiﬁed simple stratum (A, n, α) such [n/2]+1 . The integer n = 2m+1 is odd; that π contains the character ψα of UA m+1 putting E = F [α], we have Jα = E × UA , and the representation π contains a cuspidal type (A, Jα , Λ). Since E/F is totally tamely ramiﬁed, det E × = NE/F (E × ) ⊃ UF1 . On the other hand, det UAm+1 ⊂ det UA1 = UF1 . Therefore det Jα = NE/F (E × ),

(20.3.1)

which is a subgroup of F × of index 2 (18.2 Exercise 2). If φ is the nontrivial character of F × vanishing on NE/F (E × ), then Λ ⊗ φ ◦ det = Λ, whence π = c-Ind Λ = c-Ind Λ ⊗ φ ◦ det = φπ. Certainly, φ is ramiﬁed, of level zero, and of order 2. This also proves part (2). To prove uniqueness, let ξ be a character of F × such that ξπ ∼ = π. Since 2 ξ = 1 and p is odd, the restriction ξ | UF1 is trivial and so ξ has level zero. The representation π thus contains the two characters Λ, Λ ⊗ ξ ◦ det of Jα . These extend ψα on UAm+1 and so are equal (15.6 Proposition 1). Thus ξ vanishes on NE/F (E × ), and ξ is therefore either trivial or equal to φ.

21. Tame Intertwining Properties We now prove the Tame Parametrization Theorem. 21.1. The ﬁrst, and main, step in the proof is to show: Proposition. The map (E/F, χ) → πχ is injective on isomorphism classes of minimal pairs. Proof. Suppose we have minimal pairs (Ei /F, χi ), i = 1, 2, such that πχ1 ∼ = πχ2 = π, say. If π is unramiﬁed, then so are both extensions Ei /F (20.3 Lemma). Otherwise, there is a non-trivial, ramiﬁed character φ of F × such that φπ ∼ = π, and Ei /F satisﬁes NEi /F (Ei× ) = Ker φ (20.3 Proposition). In both cases therefore, E1 ∼ = E2 (cf. 18.2 Exercise (2)), so we can take E1 = E2 = E, say. Further, the level of χi is n = e(E|F ) (π). We choose an F -embedding of E in A, and let A be the chain order nor[(n+1)/2] . The pair malized by (the image of) E × . We form the group J = E × UA

132

5 Parametrization of Tame Cuspidals [n/2]+1

(E/F, χi ) gives a cuspidal type (A, J, Λi ). The restriction of Λi to UA is . These two cuspidal types intertwine in a multiple of ψαi , for some αi ∈ p−n E [n/2]+1 G and, in particular, the characters ψα1 , ψα2 of UA are UA -conjugate (15.2). We prove, under these hypotheses: Lemma. There exists u ∈ UA such that uα2 u−1 ∈ E and uα2 u−1 ≡ α1 −[n/2] (mod pE ). Before proving this, we complete the main argument. The lemma shows that, after applying a suitable conjugation, we can assume that the αi both lie [n/2]+1 are the same. The representations in E and that the characters ψαi of UA Λi intertwine in G, and so are equivalent (15.6). If n is odd, the character χi is Λi | E × ; if n is even, χi is given by the formula of 19.4 Corollary. In either case, we get χ1 = χ2 , as required. [(n+1)/2]

21.2. We prove the lemma of 21.1. By hypothesis, the cosets αi UA are UA -conjugate. In particular, the cosets αi UA1 are UA -conjugate. Under these circumstances: Lemma 1. There exists σ ∈ Gal(E/F ) such that α2 ≡ α1σ (mod UE1 ). Proof. Suppose ﬁrst that E/F is unramiﬁed, and pick a prime element of F . Thus αi ≡ −n ζi (mod UE1 ), for some root of unity ζi ∈ µE µF . The cosets ζi UA1 are still UA -conjugate. The images ζ˜i of ζi in A/P ∼ = M2 (k) are therefore conjugate under GL2 (k). We have k[ζ˜1 ] = k[ζ˜2 ] ∼ = Fq2 , so the ζ˜i are Galois-conjugate. It follows that the ζi are Galois-conjugate. Suppose now that E/F is totally ramiﬁed: thus p = 2 and UE = UF UE1 . Thus there exists y ∈ UF such that α2 ≡ yα1 (mod UE1 ). If the cosets αi UA1 are to be UA -conjugate, the only possibilities (comparing norms) are y ≡ ±1 (mod UE1 ). In this case, if σ is the non-trivial F -automorphism of E, we have α1σ ≡ −α1 (mod UE1 ), and the lemma again follows. The next point to observe is that, if we write x → j(x) for our chosen embedding of E in A, and if σ ∈ Gal(E/F ), the embedding x → j(xσ ) likewise carries E × into KA . It is therefore UA -conjugate to j. We conclude: Lemma 2. There exists u ∈ UA such that uEu−1 = E and uα2 u−1 ≡ α1 (mod UE1 ). We have therefore reduced to the case α1 ≡ α2 (mod UA1 ) or, equivalently, α1 ≡ α2 (mod UE1 ). It is now enough to prove: [(n+1)/2]

Lemma 3. Suppose that α1 ≡ α2 (mod UE1 ), and that the cosets α1 UA [(n+1)/2] [(n+1)/2] α2 UA are UA -conjugate. Then α1 ≡ α2 (mod UE ).

,

21. Tame Intertwining Properties

133

Proof. By hypothesis, there exists u ∈ UA such that uα2 u−1 ≡ α1 (mod P1−n ), P = rad A. By 16.2 Lemma, u ∈ UE UA1 , so we may as well take u = 1+x ∈ UA1 . We now proceed inductively, using: Lemma 4. Let m 1 be an integer. Suppose that α1 ≡ α2 (mod pm−n ), and E suppose there exists x ∈ Pm such that (1+x)α2 (1+x)−1 ≡ α1 (mod Pm+1−n ). m+1 and α1 ≡ α2 (mod pm+1−n ). Then x ∈ pm E +P E Proof. We write α2 = α1 +c, where c ∈ pm−n and expand E (1+x)α2 (1+x)−1 ≡ α1 +c − α1 x + xα1

(mod Pm+1−n ).

That is, α1 x−xα1 ≡ c (mod Pm+1−n ). Consider the character y → ψE (cy), . We have y ∈ pn−m E ψE (cy) = ψA (cy) = ψA (α1 x−xα1 )y = ψA (yα1 −α1 y)x = 1, since y commutes with α1 . This implies c ∈ pm+1−n , since ψE has level one. E m+1−n ). Also, α x−xα ∈ P , and 16.2 Lemma That is, α2 ≡ α1 (mod pm+1−n 1 1 E m m+1 , as required. implies x ∈ pE +P This proves Lemma 3, and also 21.1 Lemma. 21.3. If (E/F, χ) is a minimal pair of positive level, the representation πχ constructed above contains a simple stratum. It therefore satisﬁes (πχ )

(φπχ ), for any character φ of F × (13.3). The converse also holds: Proposition. Let π ∈ A02 (F ) satisfy 0 < (π) (φπ) for all characters φ of F × . Suppose, in the case p = 2, that π is unramiﬁed. There exists a minimal pair (E/F, χ) ∈ P2 (F ) such that π ∼ = πχ . Proof. The representation π contains a cuspidal type (A, J, Λ) attached to a [n/2]+1 simple stratum (A, n, α). We have J = Jα and Λ | UA is a multiple of × [(n+1)/2] ψα . Setting E = F [α], we have Jα = E UA . If n is odd, we put χ = Λ | E × to get the desired minimal pair (E/F, χ). If n is even, we let θ be the unique character of Hα1 occurring in Λ | Hα1 . Lemma. There exists a unique character χ of E × such that χ | UE1 = θ | UE1 ,

χ | F × = ωπ ,

χ(ζ) = −tr Λ(ζ),

for every ζ ∈ µE µF . The pair (E/F, χ) is a minimal admissible pair. Proof. Let χ be a character of E × agreeing with θ on UE1 and ωπ on F × . The pair (E/F, χ ) is then admissible and minimal. As in 19.4 Corollary, there is a unique representation Λ ∈ C(ψα , A) such that tr Λ (ζ) = −χ (ζ), ζ ∈ µE µF . There is a unique character φ of the group Jα /F × Jα1 ∼ = µE /µF such that Λ ∼ = φ ⊗ Λ . We view φ as a character of E × via the isomorphism E × /F × UE1 ∼ = µE /µF ; the desired character is then χ = φχ . The remaining assertions are immediate. This completes the proof of the proposition.

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5 Parametrization of Tame Cuspidals

21.4. We now ﬁnish the proof of the Tame Parametrization Theorem (20.2). We take π ∈ A02 (F ) (p = 2) or Anr 2 (F ) (if p = 2). We choose a character φ of F × to minimize the level of ρ = φ−1 π. Thus (ρ) (θρ) for all characters θ of F × . By 21.3 or 19.1, there exists (E/F, ξ) ∈ P2 (F ) such that ρ = πξ . Setting χ = ξφE , we get π = πχ and so (20.2.1) is surjective. Suppose now we have pairs (Ei /F, χi ) ∈ P2 (F ), i = 1, 2, with πχ1 ∼ = πχ2 . Twisting by a character of F × , we can assume that (E1 /F, χ1 ) is minimal. The representation πχi then has minimal level relative to twisting, so (E2 /F, χ2 ) is minimal (19.6). Thus (21.1, 19.1) (E1 /F, χ1 ) is isomorphic to (E2 /F, χ2 ).

22. A Certain Group Extension We have to prove 19.4 Proposition. This is part of a larger episode from the representation theory of ﬁnite groups, but we focus as closely as possible on the very special case to hand. 22.1. We ﬁx a prime number p. We consider a ﬁnite p-group G, with cyclic centre Z, such that V = G/Z is an elementary abelian p-group. Let θ be a faithful character of Z. The alternating pairing h : V × V −→ C× , induced by (x, y) → θ[x, y], is then nondegenerate. There exists a unique irreducible representation η of G such that η | Z contains θ (cf. 16.4). In addition, we are given a ﬁnite cyclic group A of automorphisms of G, of order relatively prime to p. We assume that A acts trivially on Z. It therefore ﬁxes (the equivalence class of) the representation η. Since A is cyclic, the representation η admits extension to an irreducible representation of the semi-direct product AG = A G. Lemma. Let η0 be some irreducible representation of AG such that η0 | G ∼ = η, and let χ range over the irreducible characters of AG/G ∼ = A. The representations χ ⊗ η0 are then distinct and χ ⊗ η0 . IndAG G η = χ

Proof. The determinant of the representation χ ⊗ η0 is χdim η det η0 . Since dim η is relatively prime to |A|, these determinants are distinct, whence so are the representations χ⊗η0 . They all occur in IndAG G η, and the second assertion follows on comparing dimensions. 22.2. We are only concerned with a very special case, so we impose some correspondingly restrictive hypotheses:

22. A Certain Group Extension

135

Hypotheses. (a) |V | = q 2 , where q is some power of p; (b) |A| = q+1; (c) if a ∈ A, a = 1, then a has only the trivial ﬁxed point in V . Observe that condition (c) implies V to be simple as Fp [A]-module: otherwise, V would have an A-subspace of cardinality q on which a generator of A would act as a cyclic automorphism with an orbit of q+1 elements. Lemma. Under these hypotheses, there is a unique irreducible representation η1 of AG such that η1 | G ∼ = η and tr η1 (a) = −1, for all a ∈ A, a = 1. Proof. Let ηi , 1 i q+1, be the distinct extensions of η to AG. Consider the induced representation ξ = IndAG AZ 1A ⊗ θ, where 1A ⊗ θ denotes the character (a, z) → θ(z) of AZ ∼ = A × Z. This takes the form q+1 mi ηi , ξ= i=1

for integers mi 0. We have ξ | G ∼ = IndG Z θ = qη, so q+1

mi = q.

(22.2.1)

i=1

Consider also the restriction ξ | AZ =

g IndAZ g −1 AZg∩AZ (1A ⊗ θ) .

g∈AZ\AG/AZ

The coset space AZ\AG/AZ is just the orbit space V /A. Hypothesis (c) implies g −1 AZg ∩ AZ = Z unless g represents the trivial orbit (of 1G ). This gives us (22.2.2) ξ | AZ = 1A ⊗ θ + (q−1) RegA ⊗ θ, where RegA denotes the regular representation of A. Thus m2i . q = 1A ⊗ θ, ξAZ = ξ, ξAG = i

Combining this with (22.2.1), we see that all the integers mi are equal to 1, except for one which is zero. We re-number so that m1 = 0, mi = 1 for i 2. Thus ξ = RegA ⊗ η1 − η1 , or η1 = IndAG G η − ξ. Let a ∈ A, a = 1. From (22.2.2), we get tr ξ(a) = 1, so tr η1 (a) = −1. Thus η1 is the representation demanded by the lemma.

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5 Parametrization of Tame Cuspidals

22.3. It will be useful later to know more about the conjugacy class structure of the group AG. With the hypotheses of 22.2: Lemma. Let a ∈ A, a = 1. (1) Let g ∈ G; the element ag is then G-conjugate to one of the form az, z ∈ Z. (2) For z ∈ Z, the G-conjugacy class and the AG-conjugacy class of az are the same, and this class has q 2 elements. (3) For z1 , z2 ∈ Z, the elements az1 , az2 are AG-conjugate if and only if z1 = z 2 . Proof. We ﬁrst observe that the commutator map V −→ V, v −→ ava−1 v −1 is bijective. Assertion (1) follows immediately. In (2), the group A centralizes az, so the AG-conjugacy class is the G-conjugacy class. As g ranges over AG/AZ = G/Z, the conjugates gazg −1 = az(a−1 gag −1 ) are distinct, and there are q 2 of them. This proves (2). The commutator a−1 gag −1 lies in Z if and only if g ∈ Z, so (3) holds. 22.4. To prove 19.4 Proposition, we have to show that the situation of 19.4 satisﬁes the hypotheses of 22.2. The group G is Jα1 /Ker θ; its centre Hα1 /Ker θ, of which θ is a faithful character (16.4). Thus V = Jα1 /Hα1 ∼ = A/oE +P, which 2 has order q . The group A is µE /µF , which has order q+1 and acts on A/oE +P by conjugation. We can identify A/P with M2 (k), and the image of oE is the residue ﬁeld kE . We choose σ ∈ GL2 (k) which normalizes kE and acts on it (by conjugation) as the non-trivial element of Gal(kE /k). We then have × , a root M2 (k) = kE ⊕ σkE . Under the natural conjugation action of µE ∼ = kE of unity ζ ﬁxes the factor kE and acts on σkE as right multiplication by ζ q /ζ. Thus all the hypotheses of 22.2 are satisﬁed. Further reading. The constructions of this chapter are more general than they appear. Fixing n 2 and assuming, for simplicity, that n ≡ 0 (mod p), one can use the same deﬁnition for admissible pairs of degree n. The idea originates with Howe [45]; that paper describes a method of constructing, from an admissible pair (E/F, χ) of degree n, an irreducible cuspidal representation πχ of GLn (F ), and shows that the map (E/F, χ) → πχ is injective. Moy [65] subsequently proved, by an indirect method, that the map is surjective. A more direct proof, based on [19] and closer in spirit to the methods here, is given in [17]. More surprisingly, Yu [89] shows how to construct cuspidal representations of quite general groups in this way, under the hypothesis that the residual characteristic p is much larger than the rank of the group.

6 Functional Equation

23. 24. 25. 26. 27.

Functional equation for GL(1) Functional equation for GL(2) Cuspidal local constants Functional equation for non-cuspidal representations Converse theorem

In this chapter, we take an irreducible smooth representation π of G = GL2 (F ) and attach to it a pair of invariants L(π, s), ε(π, s, ψ). Here, s is a complex variable and ψ is a non-trivial character of F . The L-function L(π, s) is an elementary function of the form f (q −s )−1 , where f (t) is a complex polynomial of degree at most two. The local constant ε(π, s, ψ) is of the form cq −ms , for a non-zero constant c and an integer m. This theory generalizes a classical one for characters of F × . Indeed, for non-cuspidal representations of G, the L-function and local constant are expressed directly in terms of the corresponding objects for characters of F × . We therefore start the chapter with an account of the one-dimensional theory. For cuspidal representations π of G, the picture is rather diﬀerent. One invariably has L(π, s) = 1, and the local constant ε(π, s, ψ) assumes greater importance. This generalizes the one-dimensional theory in a diﬀerent way. If χ is a ramiﬁed character of F × , the local constant ε(χ, s, ψ) is given in terms of a classical, or abelian, local Gauss sum. In the two-dimensional case, the local constant ε(π, s, ψ) of a cuspidal representation π can be expressed in terms of a “non-abelian Gauss sum” attached to a cuspidal inducing datum occurring in π: the parallel is very close. These Gauss sums provide us with a very eﬀective computational tool. The main consequence in the present chapter is the Converse Theorem. This asserts that an irreducible smooth representation π of G is determined,

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6 Functional Equation

up to equivalence, by the pair of functions χ → L(χπ, s), χ → ε(χπ, s, ψ), where χ ranges over the characters of F × . There is, in fact, an absolute dichotomy: if π is not cuspidal, it is determined by the L-functions L(χπ, s) while, if π is cuspidal, the L-functions give no information whatsoever, and the representation π is determined by the local constants ε(χπ, s, ψ).

23. Functional Equation for GL(1) We start with a detailed review of the theory of the local functional equation for characters of F × = GL1 (F ). As well as providing an illustrative introduction to the corresponding theory for GL2 (F ), many of the results and calculations have direct relevance for later parts of the chapter. 23.1. Recall that, if a ∈ F and ψ ∈ F, then aψ denotes the character x → ψ(ax). Further, if ψ = 1, the map a → aψ gives an isomorphism F ∼ = F (1.7). We ﬁx ψ ∈ F , ψ = 1, and a Haar measure µ on F . For Φ ∈ Cc∞ (F ), we ˆ of Φ (relative to µ and ψ) by deﬁne the Fourier transform Φ ˆ Φ(y)ψ(xy) dµ(y), x ∈ F. Φ(x) = F

The integrand is locally constant and the integral reduces to a ﬁnite sum. Proposition. ˆ lies in Cc∞ (F ). (1) For Φ ∈ Cc∞ (F ), the function Φ (2) There is a positive real number c = c(ψ, µ) such that ˆ ˆ Φ(x) = c Φ(−x),

Φ ∈ Cc∞ (F ), x ∈ F.

(23.1.1)

(3) For given ψ, there is a unique Haar measure µψ for which c(ψ, µψ ) = 1. This measure satisﬁes µψ (o) = q l/2 , where l is the level of ψ. 1 (4) For a ∈ F × , we have µaψ = a 2 µψ . Proof. Let l be the level of ψ, and let Φj be the characteristic function of pj . For a ∈ F , the character aψ | pj is trivial if and only if a ∈ pl−j . The support ˆj is therefore pl−j and of Φ ˆj (x) = µ(o) q −j , Φ

x ∈ pl−j .

Assertions (1) and (2) therefore hold for all functions Φj , with c = µ(o)2 q −l . Let Φ ∈ Cc∞ (F ), a ∈ F , and let Ψ denote the function x → Φ(x−a). We then have ˆ ˆ Φ(y−a)ψ(xy) dµ(y) = aψ(x)Φ(x). Ψ (x) = F

23. Functional Equation for GL(1)

139

ˆ does. The function aψ is locally constant, so Ψˆ lies in Cc∞ (F ) provided Φ Calculating the Fourier transform again, we get ˆ ˆˆ Ψˆ (x) = Φ(a+x). If parts (1) and (2) of the Lemma hold for Φ, they also therefore hold for Ψ with the same value of c. However, they do hold for the functions Φj , and the functions x → Φj (x−a), a ∈ F , j ∈ Z, span Cc∞ (F ) (7.4). The ﬁrst two parts of the proposition therefore hold for all Φ ∈ Cc∞ (F ), with c = µ(o)2 q −l , as above. For b > 0, we have c(ψ, bµ) = b2 c(ψ, µ); put another way, to achieve c(ψ, µ) = 1, we must have µ(o)2 q −l = 1, so part (3) follows. Part (4) is an immediate consequence. The measure µψ is called the self-dual Haar measure on F , relative to ψ. Using µψ to compute the Fourier transform: ˆ Φ(y) ψ(xy) dµψ (y), Φ(x) = F

(23.1.1) gives the Fourier inversion formula ˆ ˆ Φ(x) = Φ(−x), ∗

Φ ∈ Cc∞ (F ), x ∈ F.

(23.1.2)

×

23.2. Let µ be a Haar measure on F . Let χ be a character of F × , let Φ ∈ Cc∞ (F ), and choose a prime element of F . For m ∈ Z, let em denote the characteristic function of the set m UF = m p pm+1 . For Φ ∈ Cc∞ (F ), the function Φm = em Φ lies in Cc∞ (F × ) ⊂ Cc∞ (F ). It is identically zero for m 0. We may therefore set Φ(x)χ(x) dµ∗ (x), m ∈ Z, zm = zm (Φ, χ) = m UF

and deﬁne a formal Laurent series Z(Φ, χ, X) ∈ C((X)) by Z(Φ, χ, X) = zm X m . m∈Z

Clearly, Φ → Z(Φ, χ, X) is a linear map Cc∞ (F ) → C((X)). We denote its image Z(χ) = Z(χ, X) = {Z(Φ, χ, X) : Φ ∈ Cc∞ (F )}. For a ∈ F × and Φ ∈ Cc∞ (F ), we denote by aΦ the function x → Φ(a−1 x). We have (23.2.1) Z(aΦ, χ, X) = χ(a) X υF (a) Z(Φ, χ, X). Thus Z(χ) is a module over the ring C[X, X −1 ] of Laurent polynomials in X. This ring has useful properties: particularly, it is a principal ideal domain and its unit group consists of the monomials aX b , a ∈ C× , b ∈ Z.

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6 Functional Equation

Proposition. Let χ be a character of F × ; then Z(χ, X) = Pχ (X)−1 C[X, X −1 ],

where Pχ (X) =

1−χ()X

if χ is unramiﬁed,

1

otherwise.

Proof. Suppose ﬁrst that Φ(0) = 0. Thus Φ | F × lies in Cc∞ (F × ) and Z(Φ, χ, X) has only ﬁnitely many non-zero coeﬃcients. That is, Z(Φ, χ, X) ∈ C[X, X −1 ]. Further, if Φ is the characteristic function of a suﬃciently small neighbourhood of 1, then Z(Φ, χ, X) is a positive constant. Thus 1 ∈ Z(χ) and {Z(Φ, χ, X) : Φ ∈ Cc∞ (F × )} = C[X, X −1 ]. Let Φ0 be the characteristic function of o; we have m m χ( ) X χ(x) dµ∗ (x). Z(Φ0 , χ, X) = m0

(23.2.2)

UF

The inner integral is µ∗ (UF ) if χ is unramiﬁed, and is zero otherwise. That is, (1−χ()X)−1 if χ is unramiﬁed, µ∗ (UF )−1 Z(Φ0 , χ, X) = 0 otherwise. Since Cc∞ (F ) is spanned by Φ0 and Cc∞ (F × ), the result follows.

23.3. We take a character ψ ∈ F, ψ = 1. For Φ ∈ Cc∞ (F ), let Φˆ denote the Fourier transform of Φ, calculated using the Haar measure on F which is self-dual relative to ψ. Theorem. Let χ be a character of F × . There is a unique rational function c(χ, ψ, X) ∈ C(X) such that 1 ˆ χ, ) = c(χ, ψ, X) Z(Φ, χ, X), Z(Φ, ˇ qX

for all Φ ∈ Cc∞ (F ). Proof. Consider the space Λ of linear maps λ : Cc∞ (F ) → C(X) satisfying λ(aΦ) = χ(a) X υF (a) λ(Φ),

Φ ∈ Cc∞ (F ), a ∈ F × .

Surely Λ is a C(X)-vector space. It contains the map λ0 : Φ −→ Z(Φ, χ, X),

(23.3.1)

23. Functional Equation for GL(1)

141

which is non-zero (23.2 Proposition). For Φ ∈ Cc∞ (F ) and a ∈ F × , a simple calculation yields ˆ = a a−1 Φ. aΦ It follows that the map ˆ χ, λ1 : Φ −→ Z(Φ, ˇ 1/qX) also lies in Λ. The theorem is thus a consequence of: Lemma. The space Λ has dimension one over C(X). Proof. Choose n 0 such that UFn ⊂ Ker χ. Let Φk denote the characteristic function of UFk , k 1. We consider the map Λ −→ C(X), λ −→ λ(Φn ). We show this is injective, and the lemma will follow. Suppose that λ(Φn ) = 0. The deﬁning property (23.3.1) gives λ(aΦk ) = λ(Φk ), for k n, a ∈ UFn and λ ∈ Λ, so k n.

λ(Φk ) = q n−k λ(Φn ) = 0,

Any Φ ∈ Cc∞ (F × ) is a ﬁnite linear combination of F × -translates of functions Φk , k n. So, if λ(Φn ) = 0 then λ(Φ) = 0 for all Φ ∈ Cc∞ (F × ). For Φ ∈ Cc∞ (F ), therefore, the value λ(Φ) depends only on Φ(0). Thus λ(aΦ) = λ(Φ), for all a ∈ F × , and (23.3.1) implies that λ(Φ) = 0, whence λ = 0. This completes the proof of the theorem. We introduce a more traditional notation. We let s be a complex variable, and set ζ(Φ, χ, s) = Z(Φ, χ, q −s ), L(χ, s) = Pχ (q −s )−1 , γ(χ, s, ψ) = c(χ, ψ, q In particular,

ζ(Φ, χ, s) =

−s

(23.3.2)

).

Φ(x)χ(x)xs dµ∗ (x)

(23.3.3)

F×

in the following sense. The calculations in the proof of 23.2 Proposition (especially (23.2.2)) show that this integral converges, absolutely and uniformly in vertical strips, in some half-plane Re s > s0 . It there represents a rational function in q −s and so admits analytic continuation to a meromorphic function on the whole s-plane. The two languages are equivalent, and the relation between them is transparent. We use whichever seems more convenient at the time.

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6 Functional Equation

23.4. In the classical language, (1 − χ()q −s )−1 L(χ, s) = 1

if χ is unramiﬁed, otherwise.

(23.4.1)

The L-function L(χ, s) thus says nothing about χ when χ is ramiﬁed. If, however, χ is unramiﬁed, then it is determined completely by its L-function. Corollary 1. Let χ1 , χ2 be unramiﬁed characters of F × . The following are equivalent: (1) the meromorphic functions L(χ1 , s), L(χ2 , s) have a pole in common; (2) the meromorphic functions L(χ1 , s), L(χ2 , s) have the same sets of poles; (3) χ1 = χ2 . The structure of the function γ(χ, s, ψ) is of particular importance. We deﬁne a rational function ε(χ, s, ψ) ∈ C(q −s ) by ε(χ, s, ψ) = γ(χ, s, ψ)

L(χ, s) . L(χ, ˇ 1−s)

Corollary 2. The function ε(χ, s, ψ) satisﬁes the functional equation ε(χ, s, ψ) ε(χ, ˇ 1−s, ψ) = χ(−1).

(23.4.2)

It is of the form ε(χ, s, ψ) = q ( 2 −s)n(χ,ψ) ε(χ, 12 , ψ), 1

for some n(χ, ψ) ∈ Z. Proof. The functional equation in 23.3 Theorem reads ˆ χ, ζ(Φ, ˇ 1−s) = γ(χ, s, ψ) ζ(Φ, χ, s).

(23.4.3)

Applying this twice, we get ˆ ˆ χ, s) = γ(χ, ζ(Φ, ˇ 1−s, ψ) γ(χ, s, ψ) ζ(Φ, χ, s). ˆˆ Fourier inversion (23.1.2) gives ζ(Φ, χ, s) = χ(−1)ζ(Φ, χ, s), whence γ(χ, ˇ 1−s, ψ) γ(χ, s, ψ) = ε(χ, ˇ 1−s, ψ) ε(χ, s, ψ) = χ(−1), and the relation (23.4.2) follows. We can re-write (23.4.3) in the form ˆ χ, ζ(Φ, χ, s) ζ(Φ, ˇ 1−s) = ε(χ, s, ψ) , L(χ, ˇ 1−s) L(χ, s)

(23.4.4)

23. Functional Equation for GL(1)

143

the fraction on either side lying in C[q s , q −s ]. As in 23.2 Proposition, we may choose Φ such that ζ(Φ, χ, s) = L(χ, s). We deduce that ε(χ, s, ψ) ∈ C[q s , q −s ]. Likewise, we have ε(χ, ˇ 1−s, ψ) ∈ C[q s , q −s ]. The functional equation (23.4.2) implies that ε(χ, s, ψ) is an invertible element of the ring C[q s , q −s ], and hence equal to aq ms , for some a ∈ C× , m ∈ Z. This can surely be written in the required form. Remark. The relation (23.4.4) is generally referred to as “Tate’s (local) functional equation”. The function ε(χ, s, ψ) is the Tate local constant of χ (relative to ψ). 23.5. It will be essential to have a table of values for the local constant ε(χ, s, ψ). The dependence on ψ is transparent (cf. 23.1 Proposition (4)): Lemma 1. Let a ∈ F × ; we then have 1

ε(χ, s, aψ) = χ(a) as− 2 ε(χ, s, ψ). We therefore only calculate ε(χ, s, ψ) for characters ψ of level one. The self-dual Haar measure for ψ then satisﬁes dx = q 1/2 . (23.5.1) o

For the purposes of calculation, it will be convenient to choose the Haar measure d∗ x on F × such that d∗ x = 1. UF

Proposition. Suppose that χ is unramiﬁed and that ψ has level one. If is a prime element of F , then ε(χ, s, ψ) = q s− 2 χ()−1 . 1

Proof. If Φ denotes the characteristic function of o, we get ζ(Φ, χ, s) = L(χ, s), ˆ as in the proof of 23.2 Proposition. On the other side, Φ(x) = q 1/2 Φ(−1 x), so −1 1/2 ˆ Φ(−1 x) χ(x)−1 xs d∗ x ζ(Φ, χ , s) = q F× = q 1/2−s χ()−1 Φ(x) χ(x)−1 xs d∗ x =q

1/2−s

χ()

−1

F× −1

L(χ

, s).

It follows that ε(χ, s, ψ) = as required.

ˆ χ−1 , 1−s) 1 ζ(Φ, = q s− 2 χ()−1 , L(χ−1 , 1−s)

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6 Functional Equation

Theorem. Suppose that χ has level n 0 and is not unramiﬁed. Let ψ ∈ F have level one. Then 1 ε(χ, s, ψ) = q n( 2 −s) χ(αx)−1 ψ(αx) q (n+1)/2 , n+1 x∈UF /UF

for any α ∈ F × such that υF (α) = −n. Proof. Let Φ be the characteristic function of UFn+1 , so that ζ(Φ, χ, s) = µ∗ (UFn+1 ). ˆ has support p−n and The Fourier transform Φ 1 ˆ Φ(y) = q 2 −n−1 ψ(y),

Thus ˆ χ−1 , s) = q 2 −n−1 ζ(Φ, 1

y ∈ p−n .

p−n {0}

= q 2 −n−1 1

ψ(y)χ(y)−1 ys d∗ y

zm q −ms ,

m−n

where zm =

ψ(y)χ(y)−1 d∗ y.

pm pm+1

ˆ χ−1 , s) must reduce to a monomial in We know (23.4 Corollary 2) that ζ(Φ, s q , so only one of the coeﬃcients zm is non-zero. The ﬁrst one is z−n = ψ(αu)χ−1 (αu) d∗ u, UF

for any α ∈ F with υF (α) = −n. The integrand is constant on cosets of UFn+1 and so the integral reduces to ψ(αu)χ−1 (αu) d∗ u = µ∗ (UFn+1 ) χ(αx)−1 ψ(αx). UF

n+1 x∈UF /UF

The result will follow, therefore, when we show: Lemma 2. The coeﬃcients zm , m 1−n, are all zero. Proof. We take β ∈ F , with υF (β) = m > −n, and consider the coeﬃcient zm = ψ(βu)χ(βu)−1 d∗ u. (23.5.2) UF

23. Functional Equation for GL(1)

Taking v ∈ UFn , we have ψ(βu)χ(βu)−1 d∗ u = UF

145

ψ(βuv)χ(βuv)−1 d∗ u.

UF

Setting v = 1+y, the ﬁrst factor reduces to ψ(βu)ψ(βuy) = ψ(βu), since βuy ∈ pm+n ⊂ p ⊂ Ker ψ. Thus −1 ∗ −1 ψ(βu)χ(βu) d u = χ(v) ψ(βu)χ(βu)−1 d∗ u. UF

UF

We may certainly choose v so that χ(v) = 1, so the integral (23.5.2) vanishes, as required. This completes the proof of the theorem. Remark. A variation on the same technique shows that ψ(βu)χ(βu)−1 d∗ u = 0 UF

for any β ∈ F × such that υF (β) = −n. Exercise. Let χ be a ramiﬁed character of F × and ψ ∈ F, ψ = 1. Let d∗ x = x−1 dx, where dx is the self-dual Haar measure on F relative to ψ. Let c ∈ F × . Show that ε(χ, 1, ψ) if υF (c) = −n(χ, ψ), −1 ∗ χ(cx) ψ(cx) d x = 0 otherwise. UF 23.6. The trigonometric sum appearing in 23.5 Theorem is of particular interest. Formally, let χ be a ramiﬁed character of F × of level n 0, and let ψ ∈ F have level one. Let c ∈ F satisfy υF (c) = −n. The sum χ(cx)ψ(cx) ˇ (23.6.1) τ (χ, ψ) = n+1 x∈UF /UF

is called the Gauss sum of χ (relative to ψ). The deﬁnition (23.6.1) is independent of the choice of coset representatives x, and of the element c. In these terms, 23.5 Theorem reads: ε(χ, s, ψ) = q n( 2 −s) τ (χ, ψ)/q (n+1)/2 . 1

(23.6.2)

The functional equation for the local constant (23.4.2) yields the classical relation (23.6.3) τ (χ, ψ) τ (χ, ˇ ψ) = χ(−1) q n+1 . One can usefully simplify the deﬁning expression (23.6.1) in most cases:

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6 Functional Equation

Proposition. Suppose that χ has level n 1. Let c ∈ F satisfy x ∈ p[n/2]+1 .

χ(1+x) = ψ(cx), Then τ (χ, ψ) = q [(n+1)/2]

χ(cy)ψ(cy), ˇ

(23.6.4)

y [(n+1)/2]

where y ranges over UF

[n/2]+1

/UF

. [n/2]+1

Proof. In the deﬁning sum (23.6.1), write x = y(1+z), with y ∈ UF /UF and z ∈ p[n/2]+1 /pn+1 . The typical term is then χ(cy(1+z))ψ(cy(1+z)) ˇ = χ(cy)ψ(cy) ˇ ψ(c(y−1)z). Summing: τ (χ, ψ) =

χ(cy)ψ(cy) ˇ

y

ψ(c(y−1)z).

z

The map z → ψ(c(y−1)z) is a character of p[n/2]+1 /pn+1 , since c(y−1) ∈ p−n . It is the trivial character if and only if y ≡ 1 (mod pn−[n/2] ), that is, if and [(n+1)/2] [(n+1)/2] . The inner sum therefore vanishes unless y ∈ UF , only if y ∈ UF in which case it takes the value (p[n/2]+1 : pn+1 ) = q [(n+1)/2] , as required. Remark. In an equally valid convention, what we have called τ (χ, ψ) would be denoted τ (χ, ˇ ψ). Our version is slightly more convenient for present purposes. 23.7. The Gauss sum τ (χ, ψ), in the case where χ is of level zero, relates to a particularly well-known species of classical Gauss sum. For the moment, let k be a ﬁnite ﬁeld; let θ be a character of k× and η a non-trivial character of k. We can form the classical Gauss sum ˇ gk (θ, η) = θ(x)η(x). x∈k×

In the case where χ is a ramiﬁed character of F × of level zero, the restriction ˜ of k× . Likewise, as ψ is of level one, the χ | UF is the inﬂation of a character χ restriction ψ | o is the inﬂation of a non-trivial character ψ˜ of k. Immediately: ˜ ˜ ψ). τ (χ, ψ) = gk (χ,

(23.7.1)

(Observe that, when χ is unramiﬁed, this does not give the correct formula for the local constant.)

24. Functional Equation for GL(2)

147

23.8. The formula of 23.6 Proposition has an important application: Stability theorem. Let θ, χ be characters of F × , of level l 0, n 1 respectively. Suppose that 2l < n. Let ψ ∈ F, ψ = 1, and let c ∈ F satisfy χ(1+x) = ψ(cx), x ∈ p[n/2]+1 . Then ε(θχ, s, ψ) = θ(c)−1 ε(χ, s, ψ). Proof. By 23.5 Lemma 1, it is enough to treat the case where ψ has level one. [(n+1)/2] . Applying 23.6 The character θχ has level n, and agrees with χ on UF Proposition, we get τ (θχ, ψ) = q [(n+1)/2] θˇχ(cy)ψ(cy), ˇ y [(n+1)/2] [n/2]+1 UF /UF .

where the sum is taken over u ∈ Since θ is trivial on [(n+1)/2] UF , this reduces to θ(c)−1 τ (χ, ψ), as required.

24. Functional Equation for GL(2) We turn to the functional equation of Godement and Jacquet, which generalizes the ideas of §23 to representations of G = GL2 (F ). As in the classical case, there are two steps. The ﬁrst leads to an L-function, and the second is a functional equation giving rise to a local constant. In this section, we state the results in general, but prove them only for cuspidal representations. We calculate the local constants of cuspidal representations in §25. We deal with the non-cuspidal representations separately, in §26. We use the description of cuspidal representations given in §15. One can proceed quite well without this level of detail but there is no point neglecting it, since it will be needed for the computation of local constants. We state the results here in a classical language of analytic functions. The method of proof we adopt for cuspidal representations is, however, algebraic and developed from that of §23. 24.1. Let M = M2 (o) ⊂ M2 (F ) = A. In parallel to the one-dimensional case, the space Cc∞ (A) is spanned by the characteristic functions of the sets a+pj M, a ∈ A, j ∈ Z. We ﬁx ψ ∈ F, ψ = 1, and we set ψA = ψ ◦ trA . We deﬁne the Fourier ˆ of Φ ∈ C ∞ (A) by transform Φ c ˆ Φ(y)ψA (xy) dµ(y), Φ ∈ Cc∞ (A), Φ(x) = A

relative to a Haar measure µ on A. Then Φˆ ∈ Cc∞ (A) and, exactly as in 23.1, there is a unique Haar measure µA ψ for which the Fourier inversion formula ˆ ˆ Φ(x) = Φ(−x),

Φ ∈ Cc∞ (A),

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6 Functional Equation

holds. This is the self dual Haar measure on A, relative to ψ. As in 23.1, we get the relations 2l µA ψ (M) = q , (24.1.1) 2 A µA aψ = a µψ , where a ∈ F × and l is the level of ψ. 24.2. Let (π, V ) be an irreducible smooth representation of G. Let C(π) be the space of coeﬃcients of π (10.1). As in the proof of 10.1 Proposition, the space C(π) carries a smooth representation of G × G and the canonical map Vˇ ⊗ V → C(π) is a (G × G)-isomorphism. In particular, C(π) is irreducible over G × G. We consider integrals of the form Φ(x) f (x) det xs dµ∗ (x), (24.2.1) ζ(Φ, f, s) = G

Cc∞ (A)

and f ∈ C(π), where s is a complex variable and µ∗ is a Haar for Φ ∈ measure on G. We usually abbreviate d∗ x = dµ∗ (x). Theorem 1. Let (π, V ) be an irreducible smooth representation of G. (1) There exists s0 ∈ R such that the integral (24.2.1) converges, absolutely and uniformly in vertical strips in the region Re s > s0 , for all Φ and f . The integral represents a rational function in q −s . (2) Deﬁne Z(π) = Z(π, X) = {ζ(Φ, f, s+ 12 ) : Φ ∈ Cc∞ (A), f ∈ C(π)}. There is a unique polynomial Pπ (X) ∈ C[X], satisfying Pπ (0) = 1, and Z(π) = Pπ (q −s )−1 C[q s , q −s ]. One sets L(π, s) = Pπ (q −s )−1 .

(24.2.2)

This deﬁnition is, we note, independent of the choice of Haar measure µ∗ . ˆ the Next, we choose ψ ∈ F, ψ = 1. For Φ ∈ Cc∞ (A), we denote by Φ Fourier transform of Φ using the Haar measure on A which is self-dual with respect to ψ. If (π, V ) is an irreducible smooth representation of G and f ∈ C(π), the π ). Indeed, the map f → fˇ gives a linear function fˇ : g → f (g −1 ) lies in C(ˇ ∼ isomorphism C(π) = C(ˇ π ). Theorem 2. Let (π, V ) be an irreducible smooth representation of G. There is a unique rational function γ(π, s, ψ) ∈ C(q −s ) such that ˆ fˇ, 3 −s) = γ(π, s, ψ) ζ(Φ, f, 1 +s), ζ(Φ, 2 2 for all Φ ∈ Cc∞ (A), f ∈ C(π).

(24.2.3)

24. Functional Equation for GL(2)

149

Before starting the proofs of these theorems, we deduce: Corollary. Deﬁne ε(π, s, ψ) = γ(π, s, ψ)

L(π, s) . L(ˇ π , 1−s)

The function ε(π, s, ψ) satisﬁes the functional equation ε(π, s, ψ) ε(ˇ π , 1−s, ψ) = ωπ (−1).

(24.2.4)

Moreover, there exist a ∈ C× and b ∈ Z such that ε(π, s, ψ) = aq bs . ˆ fˇ, 1−s) in place of (Φ, f, s), reads Proof. The equation (24.2.3), applied to (Φ, ˆ ˆ f, 1 +s) = γ(ˇ ˆ fˇ, 3 −s), ζ(Φ, π , 1−s, ψ) ζ(Φ, 2 2 so

ˆ ˆ f, 1 +s) = γ(ˇ ζ(Φ, π , 1−s, ψ) γ(π, s, ψ) ζ(Φ, f, 12 +s). 2

ˆˆ The Fourier inversion formula (24.1) gives ζ(Φ, f, s) = ωπ (−1)ζ(Φ, f, s), whence follows the relation (24.2.4). By deﬁnition, we can ﬁnd Φi ∈ Cc∞ (A), fi ∈ C(π), 1 i r, such that r

ζ(Φi , fi , s+ 12 ) = L(π, s).

i=1

The equation (24.2.3) gives us L(ˇ π , 1−s)−1

r

ζ(Φˆi , fˇi , 32 −s) = ε(π, s, ψ).

i=1

By deﬁnition, the left hand side lies in C[q s , q −s ], so ε(π, s, ψ) ∈ C[q s , q −s ]. Likewise, ε(ˇ π , 1−s, ψ) lies in C[q s , q −s ]. The relation (24.2.4) now implies that ε(π, s, ψ) is an invertible element of the ring C[q s , q −s ], and hence of the desired form. 24.3. The relation (24.2.3) is called the “Godement-Jacquet functional equation” for π. The quantity ε(π, s, ψ) is the Godement-Jacquet local constant of π. We may write 1 (24.3.1) ε(π, s, ψ) = q n(π,ψ)( 2 −s) ε(π, 12 , ψ), for an integer n(π, ψ). The choice of Haar measure µ∗ on G has no eﬀect on the deﬁnitions of the L-function L(π, s) and γ(π, s, ψ). The choice of Haar measure µ on A, used for computing Fourier transforms, is dictated by the character ψ. Changing ψ has no eﬀect on the L-function, but both γ(π, s, ψ), ε(π, s, ψ) do genuinely depend on ψ. Following the deﬁnitions through, one ﬁnds:

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6 Functional Equation

Proposition. Let (π, V ) be an irreducible smooth representation of G. Let ψ ∈ F, ψ = 1, and let a ∈ F × . Then: ε(π, s, aψ) = ωπ (a) a2s−1 ε(π, s, ψ), γ(π, s, aψ) = ωπ (a) a2s−1 γ(π, s, ψ). 24.4. We make some general remarks concerning the integrals (24.2.1), and describe them in terms of power series, much as in §23. The space Cc∞ (A) carries a smooth representation of the locally proﬁnite group G × G by (g, h)Φ : x −→ Φ(g −1 xh),

g, h ∈ G, Φ ∈ Cc∞ (A), x ∈ A.

This extends to an action of H(G × G) on Cc∞ (A), as in 4.2. Identifying the algebra H(G × G) with H(G) ⊗ H(G) (cf. 3.2), the action is given by (φ1 ⊗ φ2 ) : Φ −→ φ1 ∗ Φ ∗ φˇ2 ,

Φ ∈ Cc∞ (A), φi ∈ H(G),

where φˇ2 is the function g → φ2 (g −1 ). Explicitly, φ1 (g) Φ(g −1 xh) φ2 (h) d∗ g d∗ h. φ1 ∗ Φ ∗ φˇ2 (x) = G×G

Similarly for C(π). These actions interact predictably with the Fourier transform. A simple calculation yields: ˆ (g, h)Φ = det gh−1 2 (h, g)Φ,

g, h ∈ G, Φ ∈ Cc∞ (A).

(24.4.1)

For an integer m, let Gm = {x ∈ G : υF (det x) = m}, and let em denote the characteristic function of Gm in A. Deﬁne Φm = em Φ,

Φ ∈ Cc∞ (A).

(24.4.2)

Lemma 1. Let Φ ∈ Cc∞ (A) and m ∈ Z. The function Φm lies in Cc∞ (G) for all m, and is identically zero for m 0. Proof. The function Φm is locally constant on G, so we have to show that its support is compact. Let K0 = GL2 (o) and let be a prime element of F . Consider the set of pairs of integers (a, b), such that a b and a supp(Φ) ∩ K0 0 0b K0 = ∅.

24. Functional Equation for GL(2)

151

The integers a, b are individually bounded below by this condition. If we further impose the condition a+b = m, there are only ﬁnitely many such pairs (a, b). Thus supp Φm meets only ﬁnitely many K0 -double cosets contained in Gm , as required. Let (π, V ) be an irreducible smooth representation of G. For m ∈ Z, Φ ∈ Cc∞ (A), and f ∈ C(π), we deﬁne ∗ zm (Φ, f ) = Φm (x)f (x) d x = Φ(x)f (x) d∗ x. (24.4.3) G

Gm

The integrand here lies in Cc∞ (G), and is zero for m 0 (Lemma 1), so we can deﬁne a formal Laurent series Z(Φ, f, X) ∈ C((X)) by Z(Φ, f, X) = zm (Φ, f )X m . m∈Z

We put Z(π) = {Z(Φ, f, q −1/2 X) : Φ ∈ Cc∞ (A), f ∈ C(π)}. Lemma 2. (1) The space Z(π) is a C[X, X −1 ]-module, containing C[X, X −1 ]. (2) If f0 ∈ C(π), f0 = 0, the set of series {Z(Φ, f0 , q −1/2 X) : Φ ∈ Cc∞ (A)}, generates Z(π) as C[X, X −1 ]-module. Proof. For Φ ∈ Cc∞ (A), f ∈ C(π) and g, h ∈ G, we have −1

Z((g, h)Φ, (g, h)f, X) = X υF (det gh

)

Z(Φ, f, X).

(24.4.4)

It follows that Z(π) is a C[X, X −1 ]-module. Suppose f (1) = 0 and let K be a compact open subgroup of G such that K ×K ﬁxes f . If Φ is the characteristic function of K, then Z(Φ, f, X) is a positive constant. We deduce that Z(π) contains C[X, X −1 ]. As noted in 24.2, the space C(π) is irreducible over G × G. Thus every f ∈ C(π) is a ﬁnite linear combination of functions (g, h)f0 , for various g, h ∈ G. Assertion (2) now follows from (24.4.4). 24.5. In this paragraph, we prove 24.2 Theorem 1 under the assumption that π is cuspidal. More precisely: Proposition. Let (π, V ) be an irreducible cuspidal representation of G. Then Z(π) = C[X, X −1 ].

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6 Functional Equation

Proof. As in 15.8, there is a cuspidal inducing datum (A, Ξ) which occurs in π. We write ξ = Ξ | UA . The representation ξ is irreducible and V ξ = V Ξ (15.8 Proposition 2). We let eξ ∈ H(G) be the idempotent attached to ξ, as in 4.4. ˇ We choose elements v ∈ V ξ and vˇ ∈ Vˇ ξ so that the coeﬃcient v , π(g)v f0 = γvˇ⊗v : g −→ ˇ satisﬁes f0 (1) = 1. We have eˇξ ∗ f0 ∗ eˇξ = f0 (and eˇξ = eξˇ), so f0 has support contained in KA (15.8 Proposition 2, 11.1 Proposition 2). Also, since every g in the support of eξ satisﬁes υF (det g) = 0, a simple calculation yields Z(Φ, f0 , X) = Z(eξ ∗ Φ ∗ eξ , f0 , X),

Φ ∈ Cc∞ (A).

(24.5.1)

Lemma. Take f0 = γvˇ⊗v ∈ C(π) as above. (1) Let Φ be the characteristic function of A; then Z(Φ, f0 , X) = 0. (2) Let Φ ∈ Cc∞ (A) and suppose that eξ ∗ Φ ∗ eξ = Φ. Then Φ ∈ Cc∞ (G) and the support of Φ is contained in KA . (3) We have Z(Φ, f0 , X) ∈ C[X, X −1 ], for all Φ ∈ Cc∞ (A). Proof. In (1), the function Φ is invariant under translation, on both sides, by UA . It follows that eξ ∗ Φ ∗ eξ = 0, whence, by (24.5.1), Z(Φ, f0 , X) = 0. In (2), the relation eξ ∗ Φ ∗ eξ = Φ implies, via 11.1 Proposition 2, that supp Φ ∩ G ⊂ KA . We next show that supp Φ ⊂ G. Suppose, on the contrary, that supp Φ contains a singular matrix x. It therefore contains an open neighbourhood N of x. The open set N contains a diagonalizable matrix x with eigenvalues a1 , a2 ∈ F × , such that a1 = a2 . (The matrix x is G-conjugate to a triangular matrix, and the assertion is obvious for triangular matrices.) Thus x ∈ G KA but x ∈ supp Φ ∩ G. This is impossible, so supp Φ ⊂ G, as asserted. It follows that supp Φ is contained in a union of cosets Π r UA , r ∈ Z, where Π is a prime element of A. Consider the set of integers r such that supp Φ ∩ Π r UA is non-empty. This set is bounded below, since supp Φ is compact. Since Φ(0) = 0, it is also bounded above. This proves (2), and (3) follows from (24.5.1). The proposition follows from part (3) of the lemma and 24.4 Lemma 2. For integrals of the form ζ(Φ, f0 , s), the convergence statement of 24.2 Theorem 1(1) is implied by the fact that supp f0 ⊂ KA . The general case follows from (24.4.4). Replacing X by q −s , all assertions of 24.2 Theorem 1 have been proved in this case. Moreover: Corollary. Let (π, V ) be an irreducible cuspidal representation of G. The L-function of π is then trivial: L(π, s) = 1.

24. Functional Equation for GL(2)

153

24.6. We now prove 24.2 Theorem 2 in the case where π is cuspidal, starting with: Proposition. Let (π, V ) be an irreducible cuspidal representation of G. There is a unique rational function c(π, X, ψ) ∈ C(X) such that ˆ fˇ, 21 ) = c(π, X, ψ) Z(Φ, f, X), Z(Φ, q X

(24.6.1)

for all Φ ∈ Cc∞ (A) and all f ∈ C(π). Proof. We deﬁne an action of the group G × G on the ﬁeld C(X) of rational functions by −1 (g, h)f (X) = X υF (det gh ) f (X). We consider the space L = HomG×G Cc∞ (A) ⊗C C(π), C(X) . This is a vector space over the ﬁeld C(X), with the key property: Lemma. The space L has C(X)-dimension one. We give the proof in the next paragraph, after completing the proof of the proposition. Consider the pairings Cc∞ (A) × C(π) → C(X) given by λ : (Φ, f ) −→ Z(Φ, f, X), ˆ fˇ, 1/q 2 X). λ : (Φ, f ) −→ Z(Φ, The ﬁrst of these belongs to L by (24.4.4). Combining (24.4.4) with (24.4.1), we see that λ ∈ L also. The existence of the rational function c(π, X, ψ) now follows from the lemma. In (24.6.1), we put X = q −s to obtain ˆ fˇ, 2−s) = c(π, q −s , ψ) ζ(Φ, f, s). ζ(Φ, Setting c(π, q − 2 −s , ψ) = γ(π, s, ψ), 1

(24.6.2)

Theorem 2 of 24.2 follows in this cuspidal case. Here we have γ(π, s, ψ) = ε(π, s, ψ) = c(π, q − 2 −s , ψ), 1

by 24.5 Corollary. Moreover, c(π, X, ψ) is a monomial in X.

(24.6.3)

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6 Functional Equation

24.7. We prove the lemma of 24.6. Let h : Cc∞ (A) ⊗ C(π) −→ C(X) be a G × G-equivariant pairing. Let f0 ∈ C(π) be the coeﬃcient introduced in 24.5. Since f0 spans C(π) as G × G-space, the pairing h is determined by its restriction to the subspace Cc∞ (A) ⊗ f0 . Since every g in the support of the idempotent eξ has υF (det g) = 0, the property eˇξ ∗ f0 ∗ eˇξ = f0 implies h(Φ ⊗ f0 ) = h(eξ ∗ Φ ∗ eξ ⊗ f0 ), for every Φ ∈ Cc∞ (A). It follows (24.5 Lemma) that h is determined by its restriction to Cc∞ (G) ⊗ C(π). Next, HomG×G (Cc∞ (G) ⊗ C(π), C(X)) ∼ = HomG×G (C(π), Cc∞ (G)∗ ⊗ C(X)), where Cc∞ (G)∗ denotes the abstract linear dual of Cc∞ (G). The image of any G × G-map C(π) → Cc∞ (G)∗ ⊗ C(X) lies in the subspace of smooth vectors. Since C(X) is smooth over G × G, this subspace is Cc∞ (G)∨ ⊗ C(X), so HomG×G (Cc∞ (G) ⊗ C(π), C(X)) ∼ = HomG×G (C(π), Cc∞ (G)∨ ⊗ C(X)). We describe the G × G-space Cc∞ (G)∨ . Let H denote the diagonal subgroup {(g, g) : g ∈ G} of G×G. For φ ∈ Cc∞ (G), we deﬁne a function fφ : G×G → C by fφ (g1 , g2 ) = φ(g1−1 g2 ). The map φ → fφ is then a G × G-isomorphism (1H ). Thus Cc∞ (G) → c-IndG×G H (1H ). Cc∞ (G)∨ ∼ = IndG×G H There is a canonical G × G-injection (1H ) ⊗ C(X) −→ IndG×G (C(X)) IndG×G H H given by mapping the tensor f ⊗ φ(X) to the function y → f (y)φ(X), y ∈ G × G. This induces HomG×G (C(π), Cc∞ (G)∨ ⊗ C(X)) → HomH (C(π), C(X)). Since H acts trivially on C(X), this is the same as HomG (C(π), C(X)), where G acts on C(π) ∼ = Vˇ ⊗ V in the natural (diagonal) way, and trivially on C(X). We have HomG (C(π), C(X)) ∼ = HomG (Vˇ ⊗ V , C) ⊗ C(X) ∼ = EndG (V ) ⊗ C(X) ∼ = C(X), and the result follows.

25. Cuspidal Local Constants

155

24.8. It is sometimes useful to assemble the series Z(Φ, f, X), or the integrals ζ(Φ, f, s), into a more compact form: we will need this in the next section. Let (π, V ) be a irreducible smooth representation of G. We shall assume π is cuspidal, although this is not strictly necessary. For m ∈ Z and Φ ∈ Cc∞ (A), deﬁne Φm as in (24.4.2). Then (24.4 Lemma 1) Φm ∈ H(G) and we can form zm (Φ, π) = Φm (x)π(x) d∗ x = π(Φm ). G

If K is a compact open subgroup of G, such that K × K ﬁxes Φ, this operator annihilates V (K) (2.3) and is, in eﬀect, an endomorphism of the ﬁnitedimensional space V K . We deﬁne Z(Φ, π, X) = zm (Φ, π)X m = π(Φm )X m . m∈Z

m∈Z

This is a formal Laurent series with coeﬃcients in EndC (V ): indeed, if K × K ﬁxes Φ, the coeﬃcients lie in EndC (V K ). If we take v ∈ V , vˇ ∈ Vˇ and form the coeﬃcient function f = γvˇ⊗v ∈ C(π), then

Z(Φ, f, X) = vˇ, Z(Φ, π, X)v . Choosing a basis of V K and the dual basis of Vˇ K , one may think of Z(Φ, π, X) as a ﬁnite matrix whose entries are various series Z(Φ, f, X). We can also form a zeta-function ζ(Φ, π, s) = Z(Φ, π, q −s ). This can be expressed as an integral, ζ(Φ, π, s) = Φ(x)π(x) det xs d∗ x, G

convergent in a half-plane. ˇ π As an operator on Vˇ , ζ(Φ, π ˇ , s) has a transpose ζ(Φ, ˇ , s) ∈ EndC (V ), deﬁned in 2.10 by:

ˇ π ζ(Φ, π ˇ , s)ˇ v , v = vˇ, ζ(Φ, ˇ , s)v . In these terms, the functional equation (24.2.3) reads: ˇ Φ, ˆ π ζ( ˇ , 32 −s) = ε(π, s, ψ) ζ(Φ, π, s+ 12 ).

(24.8.1)

25. Cuspidal Local Constants We give an explicit formula for the local constant ε(π, s, ψ), when π is an irreducible cuspidal representation of G = GL2 (F ). Throughout this section, ψ denotes a character of F of level one. We put ψA = ψ ◦ trA .

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6 Functional Equation

The discussion centres on a non-abelian Gauss sum, generalizing the abelian Gauss sum appearing in (23.6.1). Indeed, the whole discussion closely parallels that in 23.5, and leads to a Stability Theorem generalizing 23.8. 25.1. Let (A, Ξ) be a cuspidal inducing datum in G (15.8), say Ξ : KA → AutC (W ). Let n = A (Ξ) (15.8.1). ˇ ) by We deﬁne an element T (Ξ, ψ) of EndC (W ˇ T (Ξ, ψ) = (25.1.1) Ξ(cx) ψA (cx), n+1 x∈UA /UA

where c ∈ KA satisﬁes cA = P−n , P = rad A. Lemma. (1) The deﬁnition (25.1.1) is independent of the choices of coset representatives x ∈ UA /UAn+1 and of c ∈ KA such that cA = P−n . (2) There exists τ (Ξ, ψ) ∈ C such that T (Ξ, ψ) = τ (Ξ, ψ) 1W ˇ .

(25.1.2)

ˇ ˇ Proof. For y ∈ UAn+1 , we have Ξ(cxy) = Ξ(cx), by deﬁnition. Also, ψA (cxy) = −n ψA (cx)ψA (cx(y−1)). Since cx ∈ P , (y−1) ∈ Pn+1 , we have ψA (cx(y−1)) = 1 (12.5). This proves the ﬁrst assertion in (1) and the second follows immediately. For g ∈ KA , we have −1 ˇ ˇ −1 = ˇ Ξ(g) T (Ξ, ψ) Ξ(g) gxg −1 ) ψA (cx). Ξ(gcg n+1 x∈UA /UA

However, ψA (cx) = ψA (gcxg −1 ); the set {gxg −1 : x ∈ UA /UAn+1 } is a set of coset representatives for UA /UAn+1 , and gcg −1 ∈ KA generates P−n . It follows from (1) that ˇ ˇ −1 = T (Ξ, ψ), Ξ(g) T (Ξ, ψ) Ξ(g) whence T (Ξ, ψ) is a scalar operator, as required for (2). The complex number τ (Ξ, ψ) is the non-abelian Gauss sum of Ξ. One can equally express it in the form τ (Ξ, ψ) =

for the same c as before.

1 dim Ξ

n+1 x∈UA /UA

ˇ tr(Ξ(cx)) ψA (cx),

(25.1.3)

25. Cuspidal Local Constants

157

25.2. We use the Gauss sum to calculate the local constant ε(π, s, ψ) of an irreducible cuspidal representation π of G. We deﬁne (π) as in 12.6. Theorem. Let (π, V ) be an irreducible cuspidal representation of G, and let ψ be a character of F of level one. Let (A, Ξ) be a cuspidal inducing datum which occurs in π. Let n = A (Ξ) = eA (π). Writing P = rad A, we have ε(π, s, ψ) = (P−n : A)( 2 −s)/2 1

= q 2 (π)( 2 −s) 1

τ (Ξ, ψ) 1

(A : Pn+1 ) 2 τ (Ξ, ψ) 1 . (A : Pn+1 ) 2

(25.2.1)

Proof. Fix a Haar measure µ∗ on G and write d∗ g = dµ∗ (g). In this argument, it will be more convenient to use the operator-valued integrals ζ(Φ, π, s) of 24.8. We temporarily write K = UAn+1 , K = KA . We choose Φ = eK ; this gives ζ(Φ, π, s) = π(eK ). Let ξ = Ξ | UA : we know (15.8 Proposition 2) that ξ is irreducible and V Ξ = V ξ . Consequently, if v ∈ V ξ , then Ξ(g) v if g ∈ K, π(eξ ) π(g) π(eξ ) v = 0 otherwise, while π(eξ ) π(g) π(eξ ) annihilates the K-complement of V ξ in V . A direct calculation yields π(eξ ) ζ(Φ, π, s) π(eξ ) = ζ(eξ ∗ Φ ∗ eξ , π, s) = π(eξ ). We abbreviate Ψ = eξ ∗ Φ ∗ eξ = eξ . We then have Ψˆ = eˇξ ∗ Φˆ ∗ eˇξ ; since eˇξ = eξˇ, the support of Ψˆ is contained in K (24.5 Lemma). The transpose of the functional equation (24.8.1) gives ˇ (ˇ eξ ) = ζ(Ψˆ , π ˇ , 2−s) ε(π, s− 12 , ψ) π Ψˆ (x) π ˇ (x) det x2−s d∗ x = K = eˇξ ∗ Ψˆ ∗ eˇξ (x) π ˇ (x) det x2−s d∗ x K = π (x)ˇ π (ˇ eξ ) det x2−s d∗ x. Ψˆ (x) π ˇ (ˇ eξ )ˇ K

ˇ As an operator on Vˇ ξ , this reduces to

ˇ (ˇ eξ ) = ε(π, s− 12 , ψ) π

K

= K

ˇ Ψˆ (x) Ξ(x) det x2−s d∗ x ˆ ˇ Φ(x) Ξ(x) det x2−s d∗ x.

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6 Functional Equation

ˆ The self-dual Haar measure µ on A satisﬁes µ(A) = (A : P)1/2 . We calculate Φ. ˆ is P−n and The support of Φ ˆ Φ(x) = µ∗ (K)−1 µ(Pn+1 ) ψA (x), So: K

x ∈ P−n .

ˆ ˇ Φ(x) Ξ(x) det x2−s d∗ x

µ(Pn+1 ) ˇ ψA (x) Ξ(x) det x2−s d∗ x. µ∗ (K) P−n ∩K The range of integration is eﬀectively r−n Π r UA , where Π is a prime element of A. We know that ε(π, s, ψ) is a monomial in q −s , so only one of the “shells” Π r UA contributes. =

Lemma. We have

ΠrU

ˇ ψA (x) Ξ(x) det x2−s d∗ x = 0 A

for all r 1−n. ˇ Proof. As an operator on Vˇ , the integral in question stabilizes Vˇ ξ and annihilates all other UA -isotypic subspaces of Vˇ . We abbreviate c = Π r , for some r 1−n. The factor det x2−s in the integral is a non-zero constant, so we have to consider ˇ ˇ ψA (x) Ξ(x) d∗ x = ψA (xu) Ξ(xu) d∗ x, cUA

cUA

for any u ∈ UA . Let u ∈ UAn . Writing u = 1+y, we have ψA (xu) = ψA (x), since xy ∈ P ⊂ Ker ψA . Thus ˇ ˇ ˇ ψA (x) Ξ(x) d∗ x = ψA (x) Ξ(x) d∗ x · Ξ(u). cUA

The operator

cUA

ˇ ψA (x) Ξ(x) d∗ x

cUA ˇ ˇ therefore annihilates the subspace Vˇ ξ (UAn ) (notation of 2.3). Since Vˇ ξ has no n UA -ﬁxed points, this operator is zero, as required.

Returning to the theorem, the shell r = −n must give a non-zero integral, and the result follows.

25. Cuspidal Local Constants

159

The proof gives the relation ε(π, s, ψ) dim Ξ =

µ(Pn+1 ) µ∗ (UAn+1 )

ˇ tr Ξ(x) ψA (x) det x 2 −s d∗ x. 3

Π −n U

A

(25.2.2) As a further consequence, we get the following functional equation for the Gauss sum, ˇ ψ) = ωΞ (−1) (A : Pn+1 ), τ (Ξ, ψ) τ (Ξ, where ωΞ is the central character of Ξ. Remark. The formula (25.2.1) says that, when π is cuspidal and ψ has level one, the integer n(π, ψ) of (24.3.1) satisﬁes n(π, ψ) = 2(π).

(25.2.3)

25.3. For an argument in §27, we need to generalize this calculation. Keeping ψ of level one as above, one deﬁnes T (Ξ, aψ), a ∈ F × , by the same formula (25.1.1), except that the element c ∈ KA has to satisfy cA = a−1 P−n . Clearly, its eigenvalue satisﬁes τ (Ξ, aψ) = ωΞ (a) τ (Ξ, ψ), and so, comparing with 24.3, ε(π, s, aψ) = (a−1 P−n : A)( 2 −s)/2 1

τ (Ξ, aψ) 1

(A : Pn+1 ) 2

.

(25.3.1)

In this more general context, the analogue of the 25.2 Lemma says: ˇ (25.3.2) Ξ(x) aψA (x) det x2−s d∗ x = 0, r 1−n−eA υF (a). Π r UA

25.4. When the cuspidal inducing datum (A, Ξ) is of level zero, the Gauss sum τ (Ξ, ψ) can be given a more familiar form by using the classical Gauss sum gk (θ, η) of 23.7. As Ξ has level zero, we can take A = M2 (o); the representation ξ = Ξ | UA is the inﬂation of an irreducible cuspidal representation ξ˜ of UA /UA1 = GL2 (k). We use the description of such representations given in §6. Let l/k be the quadratic ﬁeld extension of k. There is a regular character θ of l× such that ξ˜ ∼ = πθ , in the notation of that section. One can calculate τ (Ξ, ψ) explicitly, using (25.1.3) and the character table (6.4.1). The character ψ | o is the inﬂation of a character ψ˜ of k; we set ψ˜l = ψ˜ ◦ Trl/k . In the deﬁnition of T (Ξ, ψ), we take c = 1. An entertaining calculation gives: τ (Ξ, ψ) = −q gl (θ, ψ˜l ).

(25.4.1)

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6 Functional Equation

25.5. It will be helpful to translate the formalism of Gauss sums of cuspidal inducing data back into the language of cuspidal types. Lemma. Let (A, Ξ) be a cuspidal inducing datum, and set n = A (Ξ), P = rad A. Let (A, J, Λ) be a cuspidal type which induces Ξ. The Gauss sum τ (Ξ, ψ) is then the unique eigenvalue of the scalar operator Λ∨ (cx)ψA (cx), (25.5.1) n+1 x∈J∩UA /UA

for any c ∈ J such that cA = P−n . ˇ ψ) Proof. For notational convenience, we work with Ξˇ in place of Ξ. Thus τ (Ξ, is the unique eigenvalue of the scalar operator Ξ(cx)ψA (cx), n+1 x∈UA /UA

for any c ∈ KA generating P−n . Let Ξ act on the space W ; the isotypic space W Λ is then irreducible over J and J-equivalent to Λ. Let eΛ denote the J-projection W → W Λ . For g ∈ KA , we have Λ(g) if g ∈ J, eΛ Ξ(g) eΛ = 0 otherwise. Choosing c ∈ J, we get the result immediately.

We can simplify further: [n/2]+1

is a multiple of ψc . The Proposition. Choose c ∈ J such that Λ | UA Gauss sum τ (Ξ, ψ) is the unique eigenvalue of the scalar operator ∨ A : P[(n+1)/2] Λ (cy)ψA (cy), (25.5.2) y [(n+1)/2]

where y ranges over UA

[n/2]+1

/UA

. [(n+1)/2]

Proof. In the sum (25.5.1), we write x = y(1+z), y ∈ J 0 /UA , z ∈ P[n/2]+1 /Pn+1 . The sum over z vanishes unless z → ψ(−cz + cyz) is the [(n+1)/2] . In that case, trivial character of P[n/2]+1 /Pn+1 , that is, unless y ∈ UA [n/2]+1 n+1 [(n+1)/2] :P ) = (A : P ), as required. it takes the value (P We will rely heavily on this result for comparisons of local constants. We therefore re-formulate it explicitly in these terms.

25. Cuspidal Local Constants

161

Corollary. Let (A, J, Λ) be a cuspidal type in G. Let n be the least integer 0 [n/2]+1 is a multiple of such that UAn+1 ⊂ Ker Λ. Choose c ∈ J such that Λ | UA G ψc . If π = c-IndJ Λ, then tr Λ∨ (cx)ψA (cx), ε(π, 12 , ψ) = q a x

where x ranges over

[(n+1)/2] [n/2]+1 UA /UA

q a dim Λ =

and

1

if n is odd, −1/2

(A : P)

if n is even.

25.6. We have to prove an analogue of the Stability Theorem (23.8). To this end, we introduce another species of trigonometric sum. Let A be a chain order with radical P and e = eA . Let χ be a character of F × of level l 1. We view χ ◦ det as character of KA , when it has level el. We can form a Gauss sum χ(det ˇ cx)ψA (cx), τA (χ, ψ) = el+1 x∈UA /UA

for any c ∈ F × such that co = p−l . An easier version of the proof of 25.5 Proposition yields: Lemma. Suppose that c ∈ F satisﬁes χ(1+x) = ψ(cx), x ∈ p[l/2]+1 . Then [el/2]+1 = ψc and χ ◦ det | UA χ(det ˇ cy)ψA (cy), τA (χ, ψ) = A : P[(el+1)/2] y

where y ranges over

[(el+1)/2] [el/2]+1 UA /UA .

A direct calculation now yields: τA (χ, ψ) τ (χ, ψ)2 = , q l+1 (A : Pel+1 )1/2

(25.6.1)

where τ (χ, ψ) is deﬁned in (23.6.1). We can deﬁne, purely formally at the moment, ε(χ ◦ det, s, ψ) = (P−el : A)( 2 −s)/2 τA (χ, ψ)/(A : Pel+1 )1/2 , 1

(25.6.2)

for some A and ψ of level one. We note that this is independent of the choice of A, by (25.6.1). We use 24.3 to formally extend the deﬁnition (25.6.2) to arbitrary characters of F by setting ε(χ ◦ det, s, aψ) = χ(a)2 a2s−1 ε(χ ◦ det, s, ψ),

a ∈ F ×.

(25.6.3)

Remark. Theorem 26.1 will show that the deﬁnitions (25.6.2), (25.6.3) are correct: see particularly the propositions of 26.6, 26.7.

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6 Functional Equation

25.7. We prove: Stability Theorem. Let π be an irreducible cuspidal representation of G and let χ be a character of F × , of level m, such that m > 2(π). Let µ ∈ F, µ = 1. Let c ∈ F × satisfy χ(1+x) = µ(cx), x ∈ p[m/2]+1 . Then: ε(χπ, s, µ) = ωπ (c)−1 ε(χ ◦ det, s, µ). Proof. By (25.6.3), 24.3, it is enough to treat the case where µ has level one. The representation π contains a cuspidal type (A, J, Λ), so χπ contains the type (A, J, χΛ), where χΛ : g → χ(det g)Λ(g). Put e = eA . As representation of J, Λ has level e(π) and χΛ has level em. The hypothesis on levels implies [(em+1)/2] , and the result follows straightaway, from Λ is trivial on the group UA 25.5 Corollary.

26. Functional Equation for Non-Cuspidal Representations In this section, we prove Theorems 1 and 2 of 24.2 for non-cuspidal irreducible representations of G: we use the classical language of zeta-integrals and analytic functions, since it is marginally easier here. The subgroups B, N , T , Z of G are as in 5.1. 26.1. We also compute the associated L-functions and local constants: Theorem. Let χ = χ1 ⊗ χ2 be a character of the group T , and let π be a G-composition factor of ιG B χ. For any ψ ∈ F , ψ = 1, we have L(π, s) = L(χ1 , s) L(χ2 , s), ε(π, s, ψ) = ε(χ1 , s, ψ) ε(χ2 , s, ψ),

(26.1.1)

except when π ∼ = φ · StG , for an unramiﬁed character φ of F × . In this exceptional case, we have L(π, s) = L(φ, s+ 12 ),

ε(π, s, ψ) = −ε(φ, s, ψ).

(26.1.2)

The L-functions and local constants of characters of F × have been worked out in §23. 26.2. Let χ = χ1 ⊗ χ2 be a character of T . We set (π, V ) = ιG Bχ = −1/2 IndG χ ⊗ δ . While π is not necessarily irreducible, we can still consider B B the space C(π) of coeﬃcients of π. If (σ, W ) is a G-subspace of (π, V ), we can identify C(σ) with a G × G-subspace of C(π): it is the linear span of coeﬃˇ ∈ Vˇ . Similarly, if Y = V /W and τ denotes the natural cients γxˇ⊗w , w ∈ W , x

26. Functional Equation for Non-Cuspidal Representations

163

representation of G on Y , then C(τ ) is the span of the coeﬃcients γxˇ⊗v , v ∈ V and x ˇ ∈ Vˇ such that ˇ x, W = 0. We choose a Haar measure µG on G and abbreviate dµG (g) = dg. We form zeta-integrals ζ(Φ, f, s) = Φ(g)f (g) det gs dg, Φ ∈ Cc∞ (A), f ∈ C(π), G

as before, and we write Z(π) = Z(π, q −s ) = {ζ(Φ, f, s+ 12 ) : Φ ∈ Cc∞ (A), f ∈ C(π)}. We ﬁrst prove an analogue of 24.2 Theorem 1 for the representation π. Proposition. Let χ = χ1 ⊗ χ2 be a character of T , and put (π, V ) = ιG B χ. (1) There exists s0 ∈ R, depending only on χ, such that ζ(Φ, f, s) converges, absolutely and uniformly in vertical strips, in the region Re s > s0 . (2) The integral ζ(Φ, f, s) represents a rational function of q −s and Z(π, q −s ) = Z(χ1 , q −s )Z(χ2 , q −s ). Proof. We transfer the zeta-integrals on G to the corresponding objects on T ∼ = F × × F ×. Let D be the algebra of diagonal matrices in A; thus T = D× and the space Cc∞ (D) is canonically isomorphic to Cc∞ (F ) ⊗ Cc∞ (F ) (3.2). We choose a Haar measure dn on the group N of upper triangular unipotent matrices in G. Lemma. Let Φ ∈ Cc∞ (A). There exists a unique function ΦT ∈ Cc∞ (D) such that Φ(tn) dn, t = t01 t02 ∈ T. ΦT (t) = t1 N

The map Φ → ΦT is a linear surjection Cc∞ (A) → Cc∞ (D). Proof. The integral deﬁning ΦT converges absolutely to a function on T , and the map Φ → ΦT is certainly linear. The space Cc∞ (A) is spanned by functions of the form φij (aij ), (26.2.1) Φ = (φij ) : (aij ) −→ i,j

Cc∞ (F ).

with “coordinates” φij ∈ obvious isomorphism N ∼ = F)

For such a function Φ, we have (using the

ΦT (t) = φ11 (t1 ) φ22 (t2 ) φ21 (0)

φ12 (n) dn, F

Thus ΦT ∈ Cc∞ (D), and so the same applies to arbitrary Φ ∈ Cc∞ (A). Surjectivity is now clear.

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6 Functional Equation

We next describe the coeﬃcients of π. Let θ ∈ V , τ ∈ Vˇ ; thus θ, τ are functions G → C satisfying θ(ntg) = δB (t)−1/2 χ(t) θ(g),

τ (ntg) = δB (t)−1/2 χ(t)−1 τ (g),

for n ∈ N , t ∈ T , g ∈ G. The pair (τ, θ) gives a coeﬃcient τ (x) θ(xg) dx, ˙ f (g) = τ, π(g)θ = B\G

for a positive semi-invariant measure dx˙ on B\G. If we put K = GL2 (o), there is a Haar measure dk on K such that f (g) = τ (k) θ(kg) dk K

(cf. 7.6 Corollary). We expand formally: ζ(Φ, f, s) = Φ(g) τ (k) θ(kg) det gs dk dg G K = Φ(k −1 g) τ (k) θ(g) det gs dg dk K G = Φ(k −1 bk ) τ (k) θ(bk ) det bs db dk dk, K

K

B

for a left Haar measure db on B (7.6.2). There exists an open subgroup K1 of K which ﬁxes θ and τ under right translation and satisﬁes Φ(k−1 gk ) = Φ(g),

g ∈ G, k, k ∈ K1 .

The last expression for ζ(Φ, f, s) reduces to Φij (b) θ(bki ) τ (kj ) det bs db, ζ(Φ, f, s) = µ(K1 )2 i,j

(26.2.2)

B

where ki , kj range independently over K/K1 and Φij : x → Φ(ki−1 xkj ). We write db = dt dn for Haar measures dt on T , dn on N . A typical term in the sum in (26.2.2) then reduces to θ(ki ) τ (kj ) Φij (tn) χ(t) δB (t)−1/2 det bs dtdn T N s− 12 = θ(ki ) τ (kj ) Φij dt. T (t) χ(t) det t T

The right hand side has the required convergence properties (23.3, concluding remarks). This argument also shows that ζ(Φ, f, s+ 12 ) ∈ Z(χ1 )Z(χ2 ), and so Z(π) ⊂ Z(χ1 )Z(χ2 ).

26. Functional Equation for Non-Cuspidal Representations

165

To prove the opposite containment, we take φi ∈ Cc∞ (F ), i = 1, 2, such that ζ(φi , χi , s) = L(χi , s). We can assume that φi (uxv) = χi (uv)−1 φi (x),

u, v ∈ UF .

We choose an integer r 0, at least as large as the level of each χi , and form , where the group H = (K ∩ B)Nr+1

Nj = p1j 01 . We then have H = (H ∩ N )(H ∩ T )(H ∩ N ), and the factors may be taken in any order. We deﬁne a character χ ˜ of H to be trivial on the unipotent factors H ∩ N , H ∩ N , and to agree with χ on K ∩ T . We choose Φ ∈ Cc∞ (A) such that ΦT = φ1 ⊗ φ2 ; we may pick Φ to satisfy ˜ 1 h2 )−1 Φ(x), Φ(h1 xh2 ) = χ(h

x ∈ A, hi ∈ H.

We take θ, τ to have support BH and to be ﬁxed by H ∩ N , normalized so that θ(1) = τ (1) = 1. We return to the expression Φ(k −1 bk ) det bs τ (k) θ(bk ) db dk dk . ζ(Φ, f, s) = K

K

B

On following through the procedure above, this reduces to Φ(b) θ(b) det bs db = c L(χ1 , s− 12 ) L(χ2 , s− 12 ), ζ(Φ, f, s) = c B

for a constant c > 0. Therefore Z(π) = Z(χ1 ) Z(χ2 ), as desired. 26.3. We now deal with the functional equation for the (possibly reducible) representation π = ιG B χ. We have to prove: Proposition. Let χ = χ1 ⊗ χ2 be a character of T and set (π, V ) = ιG B χ. Let −s ψ ∈ F , ψ = 1. There is a unique rational function γ(π, s, ψ) ∈ C(q ) such that ˆ fˇ, 3 −s) = γ(π, s, ψ) ζ(Φ, f, s+ 1 ), ζ(Φ, 2 2 for all Φ ∈ Cc∞ (A) and all f ∈ C(π). Moreover, γ(π, s, ψ) = γ(χ1 , s, ψ) γ(χ2 , s, ψ).

(26.3.1)

Proof. We ﬁrst need to work out the relation between the map Φ → ΦT and the various Fourier transforms. Let µF be the self-dual Haar measure on F with respect to the character ψ. If we identify A = M2 (F ) with F ⊕ F ⊕ F ⊕ F in the obvious way, then µF ⊗ µF ⊗ µF ⊗ µF is the self-dual Haar measure

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6 Functional Equation

on A relative to the character ψA = ψ ◦ trA . The restriction ψD = ψA | D is a non-trivial character of D, and the self-dual measure on D, relative to ψD , is µF ⊗ µF . We use these measures to compute Fourier transform in Cc∞ (A), Cc∞ (D). We can view µF as a Haar measure on N via the obvious isomorphism N∼ = F , and we deﬁne the map Φ → ΦT of 26.2 using this measure. T . Lemma. For Φ ∈ Cc∞ (A), we have Φˆ T = Φ Proof. It is enough to treat a function Φ = (φij ) as in 26.2. We can compute the Fourier transform term by term: ˆ ˆ ˆ = φ11 φ21 . Φ φˆ12 φˆ22 As in Lemma 26.2, ΦT (δ) = φ11 (δ1 ) φ22 (δ2 ) φ21 (0)

φ12 (x) dµF (x), F

δ=

δ1

0 0 δ2

∈ D.

Likewise, ˆT (δ) = φˆ11 (δ1 ) φˆ22 (δ2 ) φˆ12 (0) Φ

φˆ21 (x) dµF (x)

F

= φˆ11 (δ1 ) φˆ22 (δ2 ) φ21 (0)

φ12 (x) dµF (x) F

T (δ), =Φ as required. We now take f as in 26.2: τ (x) θ(xg) dx, ˙

f (g) = B\G

so that fˇ(g) = f (g −1 ) =

τ (kg) θ(k) dk. K

Using the same procedure as before, we get ˆ fˇ, s) = ˆ τ (kg) θ(k) det gs dkdg ζ(Φ, Φ(g) G K ˆ −1 bk ) τ (bk ) θ(k) det bs dkdk db. = Φ(k B

K

K

26. Functional Equation for Non-Cuspidal Representations

167

Choosing the same open subgroup K1 and coset representatives ki as before, this reduces to 1 −1 ˆ fˇ, s) = µ(K1 ) θ(ki ) τ (kj ) det ts− 2 dt, Φˆji ζ(Φ, T (t) χ(t) i,j

T

ij , so the lemma gives ˆ −1 xki ). We certainly have Φ ˆji = Φ where Φˆji : x → Φ(k j ˆji = Φij , and we deduce that Φ T

T

ˆ fˇ, 3 −s) = γ(χ1 , s, ψ) γ(χ2 , s, ψ) ζ(Φ, f, s+ 1 ), ζ(Φ, 2 2 as required. 26.4. This completes the proofs of 24.2 Theorems 1 and 2 for irreducible representations of the form ιG B χ. We can also write down the L-functions and local constants: Proposition. Let χ = χ1 ⊗ χ2 be a character of T such that π = ιG B χ is irreducible. Then: L(π, s) = L(χ1 , s) L(χ2 , s), ε(π, s, ψ) = ε(χ1 , s, ψ) ε(χ2 , s, ψ),

(26.4.1)

for any ψ ∈ F, ψ = 1. Proof. The L-function relation reﬂects the equality Z(π) = Z(χ1 )Z(χ2 ) (26.2) and the ε-relation comes from the corresponding relation between the γ’s (26.3.1). 26.5. Let χ be a character of T , set Σ = ιG B χ, and let π be a G-composition factor of Σ. We prove the theorems of 24.2 for π. First, we have C(π) ⊂ C(Σ), so the convergence statements hold for π and we have Z(π) ⊂ Z(Σ). On the other hand, if f (1) = 0 and Φ is the characteristic function of a suﬃciently small compact open subgroup of G, then ζ(Φ, f, s) is a non-zero constant. Therefore (26.5.1) C q −s , q s ⊂ Z(π) ⊂ Z(Σ), and, in the obvious notation (cf. 24.2), Pπ (t) divides PΣ (t). ˇ whence Likewise, for f ∈ C(π), we have fˇ ∈ C(ˇ π ) ⊂ C(Σ), ˆ fˇ, 3 −s) = γ(Σ, s, ψ) ζ(Φ, f, s+ 1 ), ζ(Φ, 2 2 so (24.2.3) holds with γ(π, s, ψ) = γ(Σ, s, ψ). The expression γ(π, s, ψ) = ε(π, s, ψ)L(π, s)/L(ˇ π , 1−s) holds, for a monomial ε(π, s, ψ), as in 24.2 Corollary. This completes the proofs of Theorems 1 and 2 of 24.2. We display one of the conclusions of the preceding arguments:

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6 Functional Equation

Corollary. Let χ = χ1 ⊗ χ2 be a character of T and let π be a G-composition factor of Σ = ιG B χ. We have γ(π, s, ψ) = γ(ιG B χ, s, ψ) = γ(χ1 , s, ψ)γ(χ2 , s, ψ), and Pπ (t) divides PΣ (t) = Pχ1 (t)Pχ2 (t). 26.6. We compute the L-function L(π, s) in the cases where π is a composition ±1/2 , for a character factor of a reducible representation Σ = ιG B χ. Thus χ = φδB × φ of F (9.11). One case is completely straightforward: ±1/2

Proposition. Let π be a composition factor of a representation ιG B φδB and suppose that the character φ of F × is not unramiﬁed. Then L(π, s) = 1,

ε(π, s, ψ) = ε(φ, s− 12 , ψ) ε(φ, s+ 12 , ψ). ±1/2

Proof. Writing Σ = ιG B φδB C[q

−s

,

, we have

, q ] ⊂ Z(π) ⊂ Z(Σ) = C[q −s , q s ], s

so L(π, s) = 1. By 26.3 Proposition, we have ε(π, s, ψ) = γ(π, s, ψ) = γ(φ, s+ 12 , ψ) γ(φ, s− 12 , ψ) = ε(φ, s+ 12 , ψ) ε(φ, s− 12 , ψ),

as claimed.

±1/2

, 26.7. It remains to treat the composition factors of representations ιG B φδB where φ is unramiﬁed. These factors are the one dimensional representation φ ◦ det of G and the twist φ · StG of the Steinberg representation. The ﬁrst of them can be dealt with by a direct computation: Proposition. Let φ be an unramiﬁed character of F × and put π = φ ◦ det. Then: L(π, s) = L(φ, s− 12 ) L(φ, s+ 12 ), ε(π, s, ψ) = ε(φ, s− 12 , ψ) ε(φ, s+ 12 , ψ). In particular, if ψ has level one, then ε(φ ◦ det, s, ψ) = φ()−2 q 2s−1 , for any prime element of F . Proof. Introducing a translation on s, we can assume φ is trivial. The constant function g → 1 is a coeﬃcient of 1G . Let Φ denote the characteristic function of M2 (o). Using the Cartan decomposition (7.2.2), one ﬁnds that Φ(g) det gs dµG (g) = µG (K) (1−q −s )−1 (1−q 1−s )−1 . G

The polynomial Pπ (t) therefore has degree at least 2, and the ﬁrst assertion follows from 26.5 Corollary. Likewise the second.

26. Functional Equation for Non-Cuspidal Representations

169 ±1/2

26.8. We take π = StG and let Σ denote one of the representations ιG . B δB Writing 1F for the trivial character of F × and ζF (s) = (1−q −s )−1 , we get γ(Σ, s, ψ) = γ(1F , s+ 12 , ψ) γ(1F , s− 12 , ψ) = q 2s−1 ζF ( 12 −s)ζF ( 32 −s)/ζF (s− 12 )ζF (s+ 12 ) 1

= −q s− 2 ζF ( 32 −s)/ζF (s+ 12 ) = γ(π, s, ψ). Since π ∼ ˇ (9.10.5), we have γ(π, s, ψ) = ε(π, s, ψ)L(π, s)/L(π, 1−s), so there =π are only the possibilities L(π, s) = ζF (s− 12 )ζF (s+ 12 ) or L(π, s) = ζF (s+ 12 ). A similar remark applies to φ · StG , where φ is unramiﬁed. We prove: Proposition. Let φ be an unramiﬁed character of F × and let π = φ · StG ; then L(π, s) = L(φ, s+ 12 ) and ε(π, s, ψ) = −ε(φ, s, ψ). Proof. The formula for ε follows from that for the L-function and the relation γ(π, s, ψ) = γ(Σ, s, ψ). To determine the L-function, we again need only treat the case where φ is trivial. We take f ∈ C(π) and consider the zeta-integral Φ(g)f (g) det gs dg. (26.8.1) ζ(Φ, f, s) = G

The key point is: Lemma. The integral (26.8.1) converges absolutely for Re s > Φ ∈ Cc∞ (A).

1 2

and any

Proof. We ﬁrst choose Haar measures d× x on Z = F × and dg˙ on G/Z so ˙ Since we are only concerned with absolute convergence, we that dg = d× x dg. may as well take Φ to be the characteristic function of M2 (o) and s real. We consider the integral |f (g)| Φ(gx) det gxs d× x dg. ˙ G/Z

F×

The inner integral, call it Js (g), depends only on ZKgK, where K = GL2 (o), so we may take

a g = 0 10 , for an integer a 0 and a prime element of F . For this element g, we get Js (g) = Φ(gx) det gxs d× x = c1 q −as (1−q −2s )−1 , F×

where c1 is a positive constant depending on choice of measure. For a = 0, the coset KgK has measure q a−1 (q+1)µ(K). The function Js (g) is therefore square-integrable over G/Z, provided Re s > 12 . The coeﬃcient f (g) is squareintegrable (17.5 Theorem), so the lemma is proved.

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6 Functional Equation

It follows that the function ζ(Φ, f, s+ 12 ) is holomorphic in the region Re s > 0. The same therefore applies to L(π, s) and the proposition is proved. We have also ﬁnished the proof of 26.1 Theorem. Exercise. Let A be a chain order in A and let Ξ be an irreducible smooth representation of KA of level n 0. One can deﬁne the Gauss sum τ (Ξ, ψ) exactly as in 25.1. The representation Ξ is called nondegenerate if n = 0, or, if n 1, Ξ | UAn contains a character ψc with c ∈ KA . (1) Show that τ (Ξ, ψ) = 0 if and only if Ξ is nondegenerate. (2) Let π be an irreducible smooth representation of G such that (a) π contains a nondegenerate representation Ξ of KA ; (b) L(π, s) = 1. Show that ε(π, s, ψ) is given by the formula (25.2.1). Exercise/Remark. Show that an irreducible representation π of V is spherical (Exercises, §17) if and only if the polynomial Pπ (X) has degree 2.

27. Converse Theorem We prove a fundamental theorem, on which everything depends. It is the analogue, for representations of GL2 (F ), of the famous theorem of Hecke on classical modular forms which can claim to be the starting point of the entire theory. 27.1. We recall that, if χ is a character of F × and π is an irreducible smooth representation of G, then χπ denotes the representation g → χ(det g)π(g). The result to be proved is: Converse Theorem. Let ψ ∈ F, ψ = 1. Let π1 , π2 be irreducible smooth representations of G = GL2 (F ). Suppose that L(χπ1 , s) = L(χπ2 , s)

and ε(χπ1 , s, ψ) = ε(χπ2 , s, ψ),

for all characters χ of F × . We then have π1 ∼ = π2 . 27.2. We can separate the proof into two parts, using the following: Proposition. An irreducible smooth representation π of G is cuspidal if and only if L(φπ, s) = 1, for all characters φ of F × . Proof. If π is cuspidal, then so is φπ and L(φπ, s) = 1 (Corollary 24.5). Conversely, suppose that π is not cuspidal, and hence a composition factor of ιG B χ, for some character χ = χ1 ⊗ χ2 of T . The representation φπ is then a −1 composition factor of ιG B φχ, and φχ = φχ1 ⊗ φχ2 . We choose φ = χ2 , and 26.1 Theorem then gives L(φπ, s) = 1.

27. Converse Theorem

171

27.3. In the theorem, therefore, if one of the πi is in the principal series then so is the other. To prove the theorem for such representations, we show that an irreducible, principal series representation π of G is determined by the function φ → L(φπ, s) of characters φ of F × . We do this using the list of L-functions in 26.1 Theorem. We can assume to start with that π has been chosen, within the set of twists φπ, so that L(π, s) = 1. If L(π, s) has degree 2, it is of the form L(χ1 , s)L(χ2 , s), for unramiﬁed characters χi of F × . The set {χ1 , χ2 } is determined by the function L(χ1 , s)L(χ2 , s) (23.4 Corollary 1). If χ1 χ−1 2 is not one of the characters x → x±1 , the representation ιG B (χ1 ⊗χ2 ) is irreducible, hence equivalent to π, and it is the only irreducible representation of G with L-function L(χ1 , s)L(χ2 , s). If, on the other hand, χ1 χ−1 2 is one of the characters x → x±1 , then π is the one-dimensional composition factor of the reducible representation ιG B (χ1 ⊗ χ2 ). We therefore assume that the non-trivial function L(π, s) is of degree 1, i.e., L(π, s) = L(θ, s), for some unramiﬁed character θ of F × . There are only two cases giving rise to such L-functions. In the ﬁrst, π is an irreducible × representation ιG B (θ ⊗θ), where θ is a ramiﬁed character of F . In the second, −1/2 . π = θ · StG , where θ is the character x → θ(x)x We distinguish these cases as follows. In the ﬁrst, there is a ramiﬁed character φ of F × such that L(φπ, s) = 1: for example, we could take φ−1 = θ . In the second, by contrast, we have L(φπ, s) = 1 for all ramiﬁed characters φ. In the ﬁrst case, we recover the character θ as follows. We choose a ramiﬁed character φ such that L(φπ, s) = 1. Thus L(φπ, s) = L(θ , s), for a uniquely determined unramiﬁed character θ of F × . We then have θ = φ−1 θ . This completes the proof of 27.1 Theorem for non-cuspidal representations. 27.4. We turn to cuspidal representations. We start with a minor simpliﬁcation. Lemma. Let π1 , π2 be irreducible cuspidal representations of G, and suppose there exists ν ∈ F, ν = 1, such that ε(χπ1 , s, ν) = ε(χπ2 , s, ν), for all characters χ of F × . We then have ωπ1 = ωπ2 and ε(χπ1 , s, µ) = ε(χπ2 , s, µ) for all χ and all µ ∈ F, µ = 1. Proof. Let χ be a character of F × of level m(χ). Assuming m(χ) 1, there exists δ(χ) ∈ F × such that χ(1+x) = ν(δ(χ)x), x ∈ p[m(χ)/2]+1 . If δ(χ) is suﬃciently large, we have (25.7 Theorem) (δ(χ)) ε(χ ◦ det, 12 , ν). ε(χπi , 12 , ν) = ωπ−1 i The relation ε(χπ1 , 12 , ν) = ε(χπ2 , 12 , ν) implies that the characters ωπi agree on all elements of F × of suﬃciently large valuation. Therefore ωπ1 = ωπ2 . The ﬁnal assertion follows from 24.3 Proposition.

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6 Functional Equation

27.5. We need to make some preparations. We ﬁx a chain order A, with radical P, and an integer n 0 relatively prime to e = eA . We abbreviate K = KA , and put K(n) = {g ∈ K : gA = P−n }, F (n) = {x ∈ F × : υF (x) = −n}. We record some elementary identities: Lemma. (1) If x ∈ K(n), then det x ∈ F (2n/e) and υF (trA x) −[n/e]. [n/e]+1 and trA (P) = p. (2) We have det UAn+1 = UF Proof. This follows easily from the explicit forms (12.1.2).

The group K acts by conjugation on the coset space K(n)/UAn+1 : we denote the orbit space by Ad K\K(n)/UAn+1 . We have a map [n/e]+1

Ψ : Ad K\K(n)/UAn+1 −→ F (2n/e)/UF xUAn+1 −→ det x, trA x .

× p−[n/e] /p,

Proposition. (1) Let x, y ∈ K(n), and suppose that x is minimal over F . If Ψ (yUAn+1 ) = Ψ (xUAn+1 ), then y is minimal over F . (2) Let x ∈ K(n) be minimal over F . The number of cosets yUAn+1 contained in the Ad K-orbit of xUAn+1 depends only on n/e. (3) The map Ψ is injective on orbits of minimal elements; it is bijective in the case e = 2. Proof. In the case e = 2, all elements of K(n) are minimal over F . If e = 1, the minimality of x ∈ K(n) is expressed in terms of the characteristic polynomial of n x modulo p, where is a prime element of F . Assertion (1) follows. In (2), the number of cosets in question is the index in K/UAn+1 of the centralizer of xUAn+1 . Lemma 16.2 shows that this centralizer is F [x]× UAn+1 , and the assertion follows readily. In (3), we take x ∈ K(n), minimal over F . We set d = det x, t = trA x. We can assume that A is M or I. Using rational canonical form (5.3), x is G-conjugate to a matrix . x = 01 −d t a We conjugate x by a diagonal matrix 0 10 , to get a matrix

a x = 0−a −t d .

27. Converse Theorem

173

Taking a = [n/e], we get x ∈ K. Since x is minimal over F , any element of G conjugating x to x lies in K. That is, x is K-conjugate to x . Likewise, if y ∈ K(n) is minimal, it is K-conjugate to the matrix

a y = 0−a −t d , a = [n/e], where d = det y, t = trA y. The hypothesis Ψ (yUAn+1 ) = Ψ (xUAn+1 ) is then equivalent to y ≡ x (mod UAn+1 ), as required for (3). In the case e = 2, we have already remarked that K(n) consists solely of minimal elements. Surjectivity of Ψ in this case follows easily: one simply writes down a matrix of the form ( ∗0 ∗∗ ), lying in K(n), with the desired determinant and trace. 27.6. Let π be an irreducible cuspidal representation of G satisfying (π) (χπ), for all characters χ of F × . The representation π contains a cuspidal inducing datum (A, Ξ), unique up to conjugacy. Setting n = A (Ξ) (15.8.1), we have gcd(n, eA ) = 1 and (π) = n/eA . In particular, the rational number (π) determines the conjugacy class of the chain order A. It determines rather more: Lemma. Let π be an irreducible cuspidal representation of G such that (π) (χπ), for all characters χ of F × . Let (A, Ξ) be a cuspidal inducing datum contained in π, and set n = A (Ξ). We then have: if (π) = n ∈ Z, (q−1)q n dim Ξ = (n−1)/2 if (π) = n/2 ∈ / Z. (q−1)q Proof. Suppose ﬁrst that (π) ∈ / Z. Thus n is odd and we may take A = I. There is a simple stratum (I, n, α), and a cuspidal type (I, Jα , Λ) which [n/2]+1 and dim Λ = 1 (15.6). The ﬁeld induces Ξ. We have Jα = F [α]× UI extension F [α]/F is totally ramiﬁed, and dim Ξ = (KI : Jα ) = (q−1)q (n−1)/2 , as required. If (π) = n ∈ Z and n is odd, a similar calculation applies, except that the ﬁeld extension F [α]/F is unramiﬁed. Consider next the case (π) = 0. We may take A = M. The representation Ξ | UM is the inﬂation of an irreducible cuspidal representation of GL2 (k), hence of dimension q−1 (6.4). Finally, we have the case where (π) = n ∈ Z, n = 2m > 0. We can take A = M; the representation Ξ is induced from a cuspidal type (M, Jα , Λ) based m , the ﬁeld extension on a simple stratum (M, n, α). We have Jα = F [α]× UM F [α]/F is unramiﬁed, and dim Λ = q (15.6). The result follows. 27.7. If r 0 is an integer, let Γr denote the group of characters of F × which are trivial on UFr .

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6 Functional Equation

Theorem. Let π be an irreducible cuspidal representation of G such that (π) (χπ), for all characters χ of F × . Let (A, Ξ) be a cuspidal inducing datum occurring in π and write n = A (Ξ), e = eA and r = [n/e]. Let ψ be a character of F of level 1. There is a constant k > 0, depending only on (π) = n/e, such that ˇ tr Ξ(x) =k ε(χπ, 12 , cψ) χ(det x) cψ(−trA x), (27.7.1) χ ∈ Γr+1 /Γ0 , r+1 c ∈ UF /UF

for every x ∈ K(n) which is minimal over F . Proof. We choose Haar measure on G so that UA has measure 1. For χ ∈ Γr+1 and µ ∈ F of level one, (25.2.2) gives a constant k1 > 0 such that ˇ tr Ξ(x) χ(det x)−1 µ(trA x) dx = k1 ε(χπ, 12 , µ). K(n)

By 27.6 Lemma, the constant k1 depends only on (π). If, on the other hand, µ ∈ F has level 0, (25.3.2) gives ˇ tr Ξ(x) χ(det x)−1 µ(trA x) dx = 0. K(n)

For a ∈ F (2n/e)/UFr+1 and b ∈ p−r /p, we deﬁne ˇ tr Ξ(x), e(a, b) = x

where x ranges over the elements of K(n)/UAn+1 such that Ψ (x) = (a, b). By 27.5 Proposition, there is a constant k2 > 0, depending only on (π), such that ˇ tr Ξ(x) = k2 e(det x, trA x) for all minimal x ∈ K(n). We deﬁne the Fourier transform Υ(χ, µ) =

e(a, b) χ(a)−1 µ(b),

r+1 a ∈ F (2n/e)/UF , b ∈ p−r /p

for χ ∈ Γr+1 and µ ∈ (p−r /p) . There is then a constant k3 > 0, depending only on (π), such that ˇ Υ(χ, µ) = k3 tr Ξ(x) χ(det x)−1 µ(trA x) dx. K(n)

27. Converse Theorem

Therefore Υ(χ, µ) =

k4 ε(χπ, 12 , µ)

175

if µ has level one,

0 otherwise,

where again k4 > 0 depends only on (π). We take the Fourier transform of Υ on the ﬁnite abelian group Γr+1 /Γ0 × (p−r /p) . To get the result, it remains only to observe that the elements of (p−r /p) of level one are the characters cψ, for c ∈ UF /UFr+1 . 27.8. We return to the Converse Theorem. We are given irreducible cuspidal representations π1 , π2 of G, satisfying ε(χπ1 , s, ψ) = ε(χπ2 , s, ψ),

(27.8.1)

for all characters χ of F × . We may assume that ψ has level one (27.4). We may also assume that (π1 ) (χπ1 ) for all χ. (25.2.1) gives ε(χπi , s, ψ) = q 2 (χπi )( 2 −s) ε(χπi , 12 , ψ). 1

We deduce that (π1 ) = (π2 ) (χπ2 ) for all χ. Thus there is a chain order A and a cuspidal inducing datum (A, Ξi ) such that πi contains Ξi , i = 1, 2. We set n = A (Ξi ), so that (πi ) = n/eA . Theorem 27.7 gives tr Ξˇ1 (x) = tr Ξˇ2 (x),

(27.8.2)

for all x ∈ KA (n) which are minimal over F . We recall (27.4) that ωπ1 = ωπ2 .

(27.8.3)

Taking account of (27.8.2) and (27.8.3), the Converse Theorem for cuspidal representations will follow from: Proposition. Let A be a chain order in A and, for i = 1, 2, let (A, Ξi ) be a cuspidal inducing datum such that A (χΞi ) A (Ξi ) for all characters χ of F × . Suppose further that (a) A (Ξ1 ) = A (Ξ2 ) = n, say; (b) the restrictions Ξi | F × are multiples of the same character ω; (c) tr Ξ1 (x) = tr Ξ2 (x), for all F -minimal elements x ∈ KA (n). We then have Ξ1 ∼ = Ξ2 . Proof. We have to divide into a small number of cases. Suppose ﬁrst that n = 0. We may then take A = M = M2 (o). The representation Ξi | UM is the inﬂation of an irreducible cuspidal representation ξ˜i of GL2 (k). Using the notation of (6.4), we have ξ˜i = πθi , where θi is a regular character of l× and

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6 Functional Equation

l/k is the quadratic ﬁeld extension, viewed as a sub-algebra of M2 (k). Using the character table (6.4.1), we get tr πθi (ζ) = − θi (ζ)+θiq (ζ) , for every ζ ∈ l k. However, an element x ∈ KM (n) = GL2 (o) is F -minimal if and only if its image in GL2 (k) is conjugate to an element of l k. We conclude that θ1 = θ2 or θ2q and hence that Ξ1 | UM ∼ = Ξ2 | UM . Since × ∼ KM = F UM , we deduce that Ξ1 = Ξ2 , as required. We therefore assume n 1, and use the machinery of §15. There is a cuspidal type (A, J, Λ1 ) which induces Ξ1 . The hypothesis A (Ξ1 ) A (χΞ1 ) implies that there is a simple stratum (A, n, α) such that J = Jα and Λ1 ∈ [n/2]+1 . C(ψα , A) (all notation as in 15.3, 15.5). Write E = F [α], Hα1 = UE1 UA 1 The restriction Λ1 | Hα is then a multiple of a character θ. We take a Haar measure µG on G; the character of the representation Λ1 is then given by tr Ξ1 (ju) θ(u)−1 dµG (u), j ∈ Jα . (27.8.4) tr Λ1 (j) = µG (Hα1 )−1 1 Hα

Lemma. There exists j ∈ Jα ∩ KA (n) which is minimal over F and such that tr Λ1 (j) = 0. Proof. If n is odd, then dim Λ1 = 1 (15.6 Proposition 1) and tr Λ1 (j) = 0 for all j ∈ Jα . Suppose therefore that n is even. Thus E/F is unramiﬁed. According to the Tame Parametrization Theorem (20.2), the cuspidal type Λ1 is constructed from an admissible pair following the procedure of 19.4: such types certainly have the desired property. For an element j as in the Lemma, we have −1 tr Ξ2 (ju) θ(u) dµG (u) = tr Ξ1 (ju) θ(u)−1 dµG (u) = 0. 1 Hα

1 Hα

We conclude that Ξ2 contains the character θ of Hα1 and hence a cuspidal type (A, Jα , Λ2 ), with Λ2 | Jα1 ∼ = Λ1 | Jα1 (15.6). Since the central characters agree, the irreducible representations Λi | F × Jα1 are the same. However, the set Jα F × Jα1 consists of minimal elements: in all cases, it is the set of minimal elements in F × KA (n). The character formula (27.8.4), and its analogue for Λ2 , implies that the characters tr Λi also agree on Jα F × Jα1 . This implies Λ1 ∼ = Λ2 and Ξ1 ∼ = Ξ2 , as required. This ﬁnishes the proof of the Converse Theorem (27.1). We give a diﬀerent proof, for cuspidal representations, in §37 below: see, in particular, 37.3 Remark.

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177

Further reading. The material of §23 is based on Tate’s thesis [81], although the approach we take here is more representation-theoretic and suggested by ideas of Weil [86]. The global context is also treated in [81] in an illuminating manner. §24 is based on [35] and [46], but we have taken a more algebraic approach, as in §23. The arguments of [35] are valid for GLn (F ) (and more). The same applies to our modiﬁed version. The description of cuspidal local constants in terms of Gauss sums is likewise valid for GLn (F ) and has its origins in [12], [11].

7 Representations of Weil Groups

28. 29. 30. 31. 32.

Weil groups and representations Local class field theory Existence of the local constant Deligne representations Relation with -adic representations

We now transfer attention to the arithmetic of the ﬁeld F , expressed via the ﬁnite-dimensional, smooth representations of its Weil group. In order to deﬁne the Weil group WF of F , we start by recalling some basic results from Galois theory. We discuss the category of representations of WF in which we shall be interested. We then recall, without proofs, the elements of local class ﬁeld theory. The core of the chapter concerns the theory of L-functions and local constants attached to ﬁnite-dimensional, semisimple, smooth representations of WF . This theory generalizes that of §23, but in a direction diﬀerent from the main thrust of Chapter VI. Local class ﬁeld theory sets up a canonical isomorphism between the groups of characters of WF and of F × , with the result that L(ρ, s), ε(ρ, s, ψ) are deﬁned for representations ρ of WF of dimension one. The objective is to extend to arbitrary ﬁnite dimension. For the L-function, this is quite straightforward, albeit guided by global experience. The local constant is, in this situation, speciﬁed by a short list of formal properties concerning its interaction with the process of induction of representations of Weil groups (of ﬁnite extension ﬁelds of F ). The proof that such an object exists is quite elaborate. The only practical way known for dealing with it involves transferring the problem to global ﬁelds, where some strong properties of L-functions can be invoked. We cannot assemble here the

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necessary apparatus from algebraic number theory; our account is brief but, for a reader familiar with such matters, it is complete. There is another aspect. Our aim is to relate the irreducible smooth representations of G = GL2 (F ) to the 2-dimensional representations of WF : we give the precise statement in the next chapter. It turns out that the standard ole, reﬂecting the reconcept of representation of WF is inadequate to this rˆ ducibility of certain principal series representations of G. One has to expand the framework by introducing the “Deligne representations” of WF : these are ordinary representations with a further element of structure. This approach gives an imperfect idea of the signiﬁcance of Deligne representations, as they arise quite naturally in a broader context. In many areas of arithmetic or algebraic geometry, one is led to consider continuous representations of WF not over C, but over the algebraic closure Q of the ﬁeld Q of -adic numbers, = p. These so-called -adic representations of WF are parametrized by Deligne representations over C. We have included the argument, as it gives a more satisfactory reason for the introduction of the Deligne representations. It also provides a broader context in which to consider the Langlands correspondence and raise further issues in the next chapter.

28. Weil Groups and Representations We start by recalling some features of the Galois theory of F . In the ﬁrst ﬁve paragraphs, the proofs are all standard or reasonably straightforward, so we have omitted them. Throughout, p means the characteristic of the residue class ﬁeld k = o/p. 28.1. We choose a separable algebraic closure F of F . We set ΩF = Gal(F /F ), and view it as a proﬁnite group with its natural (Krull) topology. Thus ΩF = lim Gal(E/F ), ←−

where E/F ranges over ﬁnite Galois extensions with E ⊂ F . If K/F is a ﬁnite ﬁeld extension, with K ⊂ F , we denote by ΩK the open subgroup Gal(F /K) of ΩF . If F/F is some other separable algebraic closure of F , an F -isomorphism F ∼ = ΩF . This does depend on the = F induces an isomorphism Gal(F/F ) ∼ ∼ choice of isomorphism F = F , but only up to inner automorphism. In particular, it induces a canonical bijection between the sets of equivalence classes of irreducible smooth representations of the two groups Gal(F/F ), ΩF . 28.2. For the time being, we consider only extension ﬁelds E/F with E ⊂ F . With this convention, the ﬁeld F admits a unique unramiﬁed extension Fm /F

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181

of degree m, for each m 1. Let F∞ denote the composite of all these ﬁelds. Thus F∞ /F is the unique maximal unramiﬁed extension of F . The extension Fm /F is Galois and Gal(Fm /F ) is cyclic. An F -automorphism of Fm is determined by its action on the residue ﬁeld kFm ∼ = Fqm . In particular, there is a unique element φm of Gal(Fm /F ) which acts on kFm as x → xq . We set Φm = φ−1 m . The map Φm → 1 gives a canonical isomorphism Gal(Fm /F ) → Z/mZ. Taking the limit over m, we get a canonical isomorphism of topological groups Gal(F∞ /F ) ∼ = lim m1 Z/mZ, ←−

(28.2.1)

and a unique element ΦF ∈ Gal(F∞ /F ) which acts on Fm as Φm , for all m. The element ΦF is the geometric Frobenius substitution on F∞ . (Its inverse φF is the arithmetic Frobenius substitution.) An element of ΩF is called a geometric Frobenius element (over F ) if its image in Gal(F∞ /F ) is ΦF . The It is conventional to denote the projective limit in (28.2.1) by Z. Chinese Remainder Theorem gives a canonical isomorphism of topological groups ∼ Z , Z =

where ranges over all prime numbers and Z is the (additive) group of -adic integers. We set IF = Gal(F /F∞ ): this is called the inertia group of F . We have an exact sequence of topological groups → 0. 1 → IF −→ Ω F −→ Z 28.3. For each integer n 1, p n, the ﬁeld F∞ has a unique extension En /F∞ of degree n. If is a prime element of F , the extension En is generated over F∞ by an n-th root of . The extension En /F∞ is cyclic, and the Galois group Gal(En /F∞ ) admits a canonical description. Choose α ∈ En such that αn = . The map Gal(En /F∞ ) −→ µn , σ −→ σ(α)/α,

(28.3.1)

gives a canonical isomorphism of Gal(En /F∞ ) with the group µn of n-th roots of unity in F∞ . The composite E∞ of all extensions En /F∞ is the maximal tamely ramiﬁed extension of F . Taking the limit of the isomorphisms (28.3.1), we get topological isomorphisms Z . (28.3.2) Gal(E∞ /F∞ ) ∼ = = lim µn ∼ ←−

=p

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(In the limit, n ranges over all positive integers not divisible by p; in the product, ranges over all prime numbers other than p.) Let PF = Gal(F /E∞ ); one calls PF the wild inertia group of F . Any ﬁnite extension of E∞ has degree a power of p, so the proﬁnite group PF is a pro p-group. It is the unique pro p-Sylow subgroup1 of IF . The isomorphism IF /PF ∼ = =p Z (28.3.2) is uniquely determined up to multiplication by an element of Z× . The group Gal(F∞ /F ) acts on Gal(E∞ /F∞ ) by conjugation. An isomor phism (28.3.2) transfers this action to one on the group =p Z . Tracking through from (28.3.1), we get: Proposition. If t0 : IF /PF → =p Z is a topological isomorphism, then −1 t0 (σ), t0 (ΦF σΦ−1 F )=q

σ ∈ IF /PF .

(28.3.3)

28.4. Let aW F denote the inverse image in ΩF of the cyclic subgroup ΦF of Gal(F∞ /F ) generated by ΦF . Thus aW F is the dense subgroup of ΩF generated by the Frobenius elements. It is normal in ΩF and it ﬁts into an exact sequence (of abstract groups) 1 → IF −→ aW F −→ Z → 0. Deﬁnition. The Weil group WF of F (relative to F /F ) is the topological group, with underlying abstract group aWF , so that (1) IF is an open subgroup of WF , and (2) the topology on IF , as subspace of WF , coincides with its natural topology as Gal(F /F∞ ) ⊂ ΩF . Thus WF is locally proﬁnite, and the identity map ιF : WF → aW F ⊂ ΩF is a continuous injection. The deﬁnition of WF does depend on the choice of F /F , but only up to inner automorphism of ΩF . We write υF : WF → Z for the canonical map taking a geometric Frobenius element to 1 and x = q −υF (x) , x ∈ WF . 28.5. Let E/F be a ﬁnite extension, E ⊂ F . The canonical inclusion ΩE → ΩF induces a bijection a W E → a W F ∩ ΩE . This same map induces a homeomorphism of WE with an open subgroup of WF of ﬁnite index. More formally: 1

For the notions of “order” and “Sylow subgroup”, applied to a proﬁnite group, see [75].

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183

Proposition. (1) Let E/F be a ﬁnite extension, E ⊂ F . (a) The group WF has a unique subgroup WFE such that ιF (WFE ) = a W F ∩ ΩE . (b) The subgroup WFE is open and of ﬁnite index in WF ; it is normal in WF if and only if E/F is Galois. (c) The canonical map WFE \WF → ΩE \ΩF is a bijection. (d) The canonical map ιE : WE → ΩE induces a topological isomorphism WE ∼ = WFE . (2) The map E/F → WFE is a bijection between the set of ﬁnite extensions E of F inside F and the set of open subgroups of WF of ﬁnite index. We henceforward identify WE with the subgroup WFE of WF . 28.6. We consider basic aspects of the representation theory of the locally proﬁnite group WF . The group ΩF is proﬁnite, so a smooth representation of ΩF is semisimple (cf. 2.2). On the other hand, WF admits the discrete group Z as a quotient and it therefore has smooth representations which are not semisimple. As we will see, the irreducible representations of WF are quite closely related to those of ΩF , but the reducible ones exhibit a greater variety. We shall work with representations of WF over C. However, everything we say remains valid if C is replaced by any algebraically closed ﬁeld of characteristic zero which is uncountable: the key point is that the proof of Schur’s Lemma (2.6) is valid for such ﬁelds. Lemma 1. Let (ρ, V ) be an irreducible smooth representation of WF . Then ρ has ﬁnite dimension. Proof. Let v ∈ V , v = 0. Since V is irreducible, the WF -translates of v span V . However, v is ﬁxed by an open subgroup J of IF . This is of the form J = IF ∩ Ω E , for some ﬁnite extension E/F . We can assume E/F is Galois, and hence that J is normal in WF . It follows that J ⊂ Ker ρ. Let Φ ∈ WF be a Frobenius element. It acts by conjugation on the ﬁnite group IF /J , so some power Φd , d 1, acts trivially. Thus ρ(Φd ) commutes with ρ(WF ) and, since ρ is irreducible, ρ(Φd ) must be a scalar (2.6 Schur’s Lemma). It follows that the translates ρ(x)v, x ∈ WF , span a ﬁnitedimensional space U . This space U is stable under ρ(WF ), so U = V and dim V is ﬁnite. Remark. Let ρ be a ﬁnite-dimensional smooth representation of WF . The restriction of ρ to the proﬁnite group IF is thus semisimple, and the group ρ(IF ) is ﬁnite. All complications, therefore, arise from the Frobenius elements of WF .

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Let ρ be an irreducible smooth representation of ΩF . Thus ρ ◦ ιF is a smooth representation of WF which we think of as the “restriction” of ρ to WF . Lemma 2. (1) Let ρ be an irreducible smooth representation of ΩF . The representation ρ ◦ ιF of WF is then irreducible. (2) If ρ1 , ρ2 are irreducible smooth representations of ΩF , then ρ1 ∼ = ρ2 if and only if ρ1 ◦ ιF ∼ = ρ2 ◦ ιF . Proof. Since Ker ρ is an open subgroup of ΩF , we have ΩF = WF · Ker ρ and ρ(WF ) = ρ(ΩF ). Part (1) follows and, in (2), any WF -isomorphism ρ1 → ρ2 is an ΩF -isomorphism. A character χ of ΩF or of WF is called unramiﬁed if it is trivial on IF . If ρ is a smooth, ﬁnite-dimensional representation of either group, the map x → det(ρ(x)) is a character, which we denote det ρ. Proposition. Let τ be an irreducible smooth representation of WF . The following are equivalent: (1) the group τ (WF ) is ﬁnite; (2) τ ∼ = ρ ◦ ιF , for some irreducible smooth representation ρ of ΩF ; (3) the character det τ has ﬁnite order. For any irreducible smooth representation τ of WF , there is an unramiﬁed character χ of WF such that χ ⊗ τ satisﬁes (1)–(3). Proof. Let Φ ∈ WF be a Frobenius element. Assuming (1) holds, the operator there is an integer a τ (Φ) has ﬁnite order d say. For a ∈ Z, ¯ such that a ¯≡a and a (mod dZ), ¯ is uniquely determined modulo dZ. As in (28.2.1), an element and ω ∈ ΩF can be written uniquely in the form ω = Φa σ, for some a ∈ Z a σ ∈ IF . We deﬁne a smooth representation ρ of ΩF by setting ρ(Φ σ) = τ (Φ)a¯ τ (σ). The representation ρ is irreducible and it satisﬁes ρ ◦ ιF = τ , so (1) ⇒ (2). Surely (2) ⇒ (3), so we assume that det τ has ﬁnite order. We choose a Frobenius Φ ∈ WF . There is an integer k 1 such that τ (Φ)k commutes with the ﬁnite group τ (IF ) and hence with the whole of τ (WF ). Thus τ (Φ)k is a scalar. If det τ has ﬁnite order, then so does the scalar τ (Φ)k . Therefore τ (WF ) is ﬁnite, as required for (1). In any case, there is an integer k 1 such that τ (Φk ) is a scalar c, say. If χ is an unramiﬁed character of WF such that χ(Φ)k = c, then the representation χ−1 ⊗ τ satisﬁes condition (1).

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185

28.7. If E/F is a ﬁnite, separable ﬁeld extension inside F , we write IndE/F , F rather than IndW WE , for the functor of smooth induction from WE to WF . Also, if ρ is a smooth representation of WF , we put ρE = ResE/F ρ = ρ | WE . We consider the issue of semisimplicity, for representations of WF . Translating directly from 2.7 Lemma, we have: Lemma. Let E/F be a ﬁnite separable ﬁeld extension, E ⊂ F . (1) Let ρ be a smooth representation of WF ; then ρ is semisimple if and only if ρE is semisimple. (2) Let τ be a smooth representation of WE ; then τ is semisimple if and only if IndE/F τ is semisimple. For each integer n 1, we denote by Gss n (F ) the set of isomorphism classes of semisimple smooth representations of WF of dimension n. We denote by G0n (F ) the set of isomorphism classes of irreducible smooth representations of WF of dimension n. If E/F is a ﬁnite extension, E ⊂ F , the lemma shows that we have induction and restriction maps ss IndE/F : Gss n (E) −→ Gnd (F ), ss ResE/F : Gn (F ) −→ Gss n (E),

(28.7.1)

where d = [E:F ]. Remark. Many choices were made in the deﬁnition of the Weil group, but WF is, colloquially speaking, well-deﬁned up to inner automorphism of ΩF . The sets G0n (F ), Gss n (F ) are thus well-deﬁned and independent of all choices. Similarly for the maps (28.7.1). Put formally, suppose we have two separable ﬁeld extensions E/F , E /F of ﬁnite degree d and an isomorphism α : E → E ss such that α(F ) = F . We get a well-deﬁned bijection Gss n (F ) → Gn (F ), and similarly for the larger ﬁelds. Moreover, the diagram ss Gss n (E) −→ Gn (E ) ⏐ ⏐ ⏐ ⏐ IndE/F IndE /F ss Gss nd (F ) −→ Gnd (F )

commutes. Similarly for restriction. In particular, the maps (28.7.1) are deﬁned for any ﬁnite separable extension E/F , independent of any choice of F -embedding of E in F . In practice, it is often important to recognize semisimplicity of representations in terms of individual Frobenius elements. Proposition. Let (ρ, V ) be a smooth representation of WF of ﬁnite dimension, and let Φ ∈ WF be a Frobenius element. The following are equivalent:

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(1) the representation ρ is semisimple; (2) the automorphism ρ(Φ) ∈ AutC (V ) is semisimple; (3) the automorphism ρ(Ψ ) ∈ AutC (V ) is semisimple, for every element Ψ of WF . Proof. Certainly (3) ⇒ (2). Suppose that (2) holds. The group ρ(IF ) is ﬁnite, so there exists an integer d 1 such that ρ(Φ)d commutes with it. The automorphism ρ(Φ) is semisimple, so ρ(Φ)d is semisimple. The restriction of ρ to the open subgroup Φd , IF is semisimple. The lemma implies that ρ is semisimple, so (2) ⇒ (1). Suppose that (1) holds, and let Ψ ∈ WF . If Ψ is an element of the proﬁnite / IF . Thus there is a ﬁnite group IF , then ρ(Ψ ) is semisimple, so we assume Ψ ∈ extension E/F such that Ψ ∈ WE and WE is the group Ψ, IE generated by Ψ and IE . The representation ρ | WE is semisimple, by the lemma, so we may as well assume E = F . We can further assume that ρ is irreducible. By 28.6 Proposition, ρ(Ψ ) = cφ, where c ∈ C× and φ has ﬁnite order. Thus ρ(Ψ ) is semisimple and we have shown (1) ⇒ (3).

29. Local Class Field Theory We recall the broad outlines of local class ﬁeld theory. For our purposes, an axiomatic account will suﬃce. We use it to develop a theory of L-functions and local constants attached to representations of the Weil group WF . 29.1. Let WFder denote the closure of the commutator subgroup of WF , and write WFab = WF /WFder . Thus WFab is a locally proﬁnite abelian group (deﬁned independently of the choices made in the deﬁnition of WF ). Local class ﬁeld theory. There is a canonical continuous group homomorphism aF : WF −→ F × with the following properties. (1) The map aF induces a topological isomorphism WFab ∼ = F ×. (2) An element x ∈ WF is a geometric Frobenius if and only if aF (x) is a prime element of F . (3) We have aF (IF ) = UF and aF (PF ) = UF1 . (4) If E/F is a ﬁnite separable extension, the diagram WE ⏐ ⏐ WF commutes.

aE −→ −→ aF

E× ⏐ ⏐ NE/F F×

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187

(5) Let α : F → F be an isomorphism of ﬁelds. The map α induces an isomorphism α : WFab → WFab , and the diagram WFab ⏐ ⏐ aF F×

WFab ⏐ ⏐ aF −→ × α F α −→

commutes. The map aF is the Artin Reciprocity Map. Part (4) implies the more standard statements of local class ﬁeld theory: Consequences. (1) The map E/F → NE/F (E × ) gives a bijection between the set of ﬁnite abelian extensions E/F , E ⊂ F , and the set of open subgroups of F × of ﬁnite index. (2) The map aF induces an isomorphism Gal(E/F ) ∼ = F × /NE/F (E × ), for any ﬁnite abelian extension E/F . (3) Let E/F be a ﬁnite separable extension. If E1 /F is the maximal abelian sub-extension of E/F , then NE/F (E × ) = NE1 /F (E1× ). There is a further property to be added to the list. For the moment, let G be some group and H a subgroup of G of ﬁnite index. Let D(G) be the commutator subgroup of G. We choose a section t : G/H → G of the quotient map G → G/H. For x ∈ G/H and g ∈ G, deﬁne an element hx,g ∈ H by gt(x) = t(gx)hx,g . We deﬁne a map verG/H : G/D(G) −→ H/D(H), hx,g D(H). g D(G) −→ x∈G/H

The map verG/H , called the “transfer” or “Verlagerung”, is a homomorphism of abelian groups G/D(G) → H/D(H). If G is locally compact, we write Gder for the closure of D(G) in G and Gab = G/Gder . If H is a closed subgroup of G of ﬁnite index, the transfer then induces a continuous homomorphism Gab → H ab , which we continue to denote verG/H . In particular, if E/F is a ﬁnite separable ﬁeld extension, the transfer gives ab . a map verE/F : WFab → WE Transfer theorem. (6) If E/F is a ﬁnite separable ﬁeld extension, the diagram × aE E −→ ⏐ verE/F ⏐ −→ WFab aF F × ab WE

commutes, where the map F × → E × is inclusion.

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29.2. In particular, the Artin Reciprocity Map aF gives an isomorphism χ → χ ◦ aF of the group of characters of F × with that of WF . Because of property (3), unramiﬁed characters of F × correspond with unramiﬁed characters of WF , and tamely ramiﬁed characters of F × correspond with characters of WF trivial on PF . We henceforward identify the two sorts of character and omit aF from the notation whenever possible. In particular, if ρ ∈ Gss n (F ), we often view det ρ as a character of F × . If K/F is a separable extension of ﬁnite degree d and 1K denotes the trivial character of WK , we can form the regular representation RK/F = IndK/F 1K ∈ Gss d (F ),

(29.2.1)

κK/F = det RK/F .

(29.2.2)

and its determinant 2 = 1. It arises in the following Thus κK/F is a character of F × satisfying κK/F context:

Proposition. Let K/F be a separable extension of degree d, let ρ ∈ Gss m (K) ss and put σ = IndK/F ρ ∈ Gmd (F ). Then m ⊗ det ρ | F × . det σ = κK/F

Proof. This follows from the Transfer Theorem in 29.1 and the following general fact. Lemma. Let G be a group, let H be a subgroup of G of ﬁnite index, and ρ a ﬁnite dimensional representation of H. Set σ = IndG H ρ. Then det σ = det(RG/H )dim ρ ⊗ (det ρ ◦ verG/H ), where RG/H = IndG H 1H . 29.3. If χ is a character of F × , the quantities L(χ, s), ε(χ, s, ψ) have been deﬁned in §23. We translate these deﬁnitions to characters χ ◦ aF of WF by setting L(χ ◦ aF , s) = L(χ, s), (29.3.1) ε(χ ◦ aF , s, ψ) = ε(χ, s, ψ). We have to extend the deﬁnition (29.3.1) to ﬁnite-dimensional, semisimple, smooth representations of WF . The L-function is straightforward. First, let σ ∈ G0n (F ). The case n = 1 is given by (29.3.1); otherwise, we set L(σ, s) = 1,

σ ∈ G0n (F ), n 2.

(29.3.2)

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189

We then use the rule L(σ1 ⊕ σ2 , s) = L(σ1 , s)L(σ2 , s) to extend the deﬁnition to σ ∈ Gss n (F ). Remark. There is a more uniform version of the deﬁnition of L(σ, s). Let (σ, V ) be a ﬁnite-dimensional, semisimple smooth representation of WF . The space V IF of IF -ﬁxed vectors in V then carries a natural representation σI of WF . If Φ is a geometric Frobenius in WF , we have L(σ, s) = det(1 − σI (Φ)q −s )−1 .

(29.3.3)

The formula (29.3.3) is the standard deﬁnition of the Artin L-function. Exercise. Let K/F be a ﬁnite separable extension, let ρ ∈ Gss n (K), and put σ = IndK/F ρ. Show that L(σ, s) = L(ρ, s). 29.4. The deﬁnition of the local constant is more involved. If E/F is a ﬁnite separable extension, we again write 1E for the trivial character of WE . For ss We also write Gss (F ) = ψ ∈ F, we set ψE = ψ ◦ TrE/F ∈ E. n1 Gn (F ). Theorem. Let ψ ∈ F, ψ = 1, and let E/F range over ﬁnite extensions inside F . There is a unique family of functions Gss (E) −→ C[q s , q −s ]× , ρ −→ ε(ρ, s, ψE ), with the following properties: (1) If χ is a character of E × , then ε(χ ◦ aE , s, ψE ) = ε(χ, s, ψE ). (2) If ρ1 , ρ2 ∈ Gss (E), then ε(ρ1 ⊕ ρ2 , s, ψE ) = ε(ρ1 , s, ψE ) ε(ρ2 , s, ψE ). (3) If ρ ∈ Gss n (E) and E ⊃ K ⊃ F , then ε(RE/K , s, ψK )n ε(IndE/K ρ, s, ψK ) = . ε(ρ, s, ψE ) ε(1E , s, ψE )n

(29.4.1)

Remark. The uniqueness properties imply that the function ε is natural with respect to isomorphisms of ﬁelds and so it is deﬁned independently of choices made in the construction of the Weil groups. For the same reason, the restriction to subﬁelds of F is unnecessary.

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The quantity ε(ρ, s, ψ), ρ ∈ Gss (F ), is called the Langlands-Deligne local constant of ρ, relative to the character ψ ∈ F and the complex variable s. We list some of its useful properties. Proposition. Let ψ ∈ F, ψ = 1, and let ρ ∈ Gss (F ). (1) There is an integer n(ρ, ψ) such that ε(ρ, s, ψ) = q n(ρ,ψ)( 2 −s) ε(ρ, 12 , ψ). 1

(2) Let a ∈ F × . Then: 1

ε(ρ, s, aψ) = det ρ(a) a dim(ρ)(s− 2 ) ε(ρ, s, ψ), n(ρ, aψ) = n(ρ, ψ) + υF (a) dim ρ. In particular, n(ρ, ψ) depends only on ρ and the level of ψ. (3) The local constants satisfy the functional equation ε(ρ, s, ψ) ε(ˇ ρ, 1−s, ψ) = det ρ(−1). (4) There is an integer nρ such that, if χ is a character of F × of level k nρ , then ε(χ ⊗ ρ, s, ψ) = det ρ(c(χ))−1 ε(χ, s, ψ)dim ρ , for any c(χ) ∈ F × such that χ(1+x) = ψ(c(χ)x), x ∈ p[k/2]+1 . We prove these assertions in the next section.

30. Existence of the Local Constant We prove the Theorem and Proposition of 29.4. The proof has several steps, one of which makes essential use of machinery from the theory of global ﬁelds. We have separated out that part and put it at the end of the section. The ideas in it intervene nowhere else. 30.1. We start with some abstract machinery. Let G be a proﬁnite group, and let K0 G be the Grothendieck group of G. Thus K0 G is the free abelian group on symbols [ρ], where ρ ranges over the set of isomorphism classes of irreducible smooth representations of G. We regard the set of isomorphism classes of ﬁnite-dimensional smooth representations of G as contained in K0 G: if ρ is such a representation, we write it as a sum of irreducible representations ρ = ρ1 ⊕ ρ2 ⊕ · · · ⊕ ρr , to obtain an element [ρ] =

r i=1

[ρi ] ∈ K0 G.

30. Existence of the Local Constant

191

We invariably abuse notation and write [ρ] = ρ. There is a dimension map K0 G → Z, deﬁned in the obvious way. We put

0G = K K0 H, H

where H ranges over the open subgroups of G. We denote the elements of 0 G as pairs (H, ρ), where H is an open subgroup of G and ρ ∈ K0 H. K The group K0 G contains the set Γ (G) = Hom(G, C× ). We write

0 G, Γ(G) = Γ (H) ⊂ K H

where H ranges over the open subgroups of G. Suppose we have another proﬁnite group G and a continuous surjective homomorphism G → G. Using inﬂation of representations, we can regard 0 G. 0 G as a subset of K K Remark. Write G = lim Gi , where Gi ranges over the ﬁnite quotients of G. ←−

0 G is then the union of its subsets K 0 Gi . This enables us to reduce The set K 0 G to the case of a ﬁnite group G. most issues concerning K Deﬁnitions. Let A be an abelian group and G a proﬁnite group. (1) An induction constant on G (with values in A) is a function 0 G −→ A F :K with the following properties: (a) for each open subgroup H of G, the map F | K0 H is a group homomorphism; (b) if H ⊂ J are open subgroups of G and (H, ρ) ∈ K0 H has dimension zero, then F(J, IndJH ρ) = F(H, ρ). (2) A division on G (with values in A) is a function D : Γ(G) → A. An induction constant F on G deﬁnes a division ∂F on G by restriction: we call ∂F the boundary of F. We shall be more interested in the converse relation: a division D on G will be called pre-inductive on G if it is of the form D = ∂F, for some induction constant F on G. The starting point is a simple uniqueness property: Lemma 1. Let F be an induction constant on G. The division ∂F determines F uniquely, and F takes its values in the abelian group generated by the values of ∂F.

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7 Representations of Weil Groups

Proof. It is enough to treat the case where G is ﬁnite. We recall2 : Brauer induction theorem. Let G be a ﬁnite group and ρ an irreducible representation of G of dimension m 1. There then exist elements (Hi , χi ) ∈ Γ(G), 1 i r, such that [ρ] − m[1G ] = IndG Hi [χi ] − [1Hi ] , 1ir

where 1H denotes the trivial character of a group H. In the lemma, let ρ be an irreducible representation of H ⊂ G of dimension m. We use the Brauer induction theorem for H to write IndH (30.1.1) [ρ] − m[1H ] = Hi [χi ] − [1Hi ] , 1ir

for various (Hi , χi ) ∈ Γ(H). Writing A multiplicatively, this yields F(H, ρ) = ∂F(H, 1H )m ∂F(Hi , χi ) ∂F(Hi , 1Hi )−1 . 1ir

The value F(H, ρ) is thereby expressed in terms of values of ∂F. 0 G. Let 1H Remark. Let F be an induction constant on G and let (H, ρ) ∈ K G be the trivial character of H and set RG/H = IndH 1H . We then have F([ρ] − m[1H ]) = F([IndG H ρ] − m[RG/H ]), where m = dim ρ (cf. (29.4.1)). The uniqueness property of Lemma 1 enables us to reduce to the case of ﬁnite groups: Lemma 2. Let G be a proﬁnite group and let D be a division on G. Suppose there is a family H of open normal subgroups H of G such that (a) the canonical map G −→ lim ←−

H∈H

G/H

is an isomorphism, and (b) the restriction DG/H of D to Γ(G/H) is pre-inductive on G/H, for all H ∈ H. The division D is then pre-inductive on G. If F is the induction constant on 0 G/H. G with boundary D, then DG/H is the boundary of F | K 2

For this version of the well-known result, see [26].

30. Existence of the Local Constant

193

30.2. We will use this apparatus in the following context. We take a ﬁnite Galois extension L/F of local ﬁelds and set G = Gal(L/F ). Via local class ﬁeld theory, we think of Γ(G) as the set of pairs (E, χ), where E ranges over the ﬁelds between L and F , and χ over the characters of E × which are null on NL/E (L× ). The core result to be proved is: Theorem. Let L/F be a ﬁnite Galois extension, and set G = Gal(L/F ). There exists ψ ∈ F, ψ = 1, such that the division on G given by L/F

Dψ

(E, χ) ∈ Γ(G),

: (E, χ) −→ ε(χ, s, ψE ),

is pre-inductive on G. The proof is deferred to the end of the section. In preparation for it, however, we record a technical local result. Lemma. Let L/F be a ﬁnite Galois extension, and put G = Gal(L/F ). There exists an integer nL/F 1 such that, if α is a character of F × of level nL/F and (E, χ) ∈ Γ(G), the level of α ◦ NE/F is greater than twice the level of χ. (E)

⊂ Proof. For F ⊂ E ⊂ L, let (E) 1 be an integer such that UE NL/E (L× ). If (E, χ) ∈ Γ(G), the character χ then has level < (E). We n(E) ⊂ set m(E) = 2(E)+1. We choose an integer n(E) such that UF m(E) NE/F (UE ). The integer nL/F = maxE n(E), as E/F ranges over subextensions of L/F , then has the required property. 30.3. For the time being, we assume 30.2 Theorem and deduce the other assertions of 29.4. We ﬁrst remove the restriction on ψ in 30.2 Theorem. By 30.1 Lemma 2, the division (E, χ) → ε(χ, s, ψE ) is pre-inductive on ΩF , being (by deﬁnition) the boundary of the induction constant (ΩE , ρ) → ε(ρ, s, ψE ). Take a ∈ F × , and consider the function (s− 12 ) dim ρ

(E, ρ) −→ det ρ(a) a E

,

0 ΩF . (E, ρ) ∈ K

This is an induction constant on ΩF , whence so is the function (s− 12 ) dim ρ

(E, ρ) −→ det ρ(a) a E

ε(ρ, s, ψE ).

The boundary of this induction constant is s− 12

(E, χ) −→ χ(a) a E

ε(χ, s, ψE ) = ε(χ, s, aψE ).

This division is therefore pre-inductive, and the boundary of (by deﬁnition) the induction constant (E, ρ) → ε(ρ, s, aψE ). Thus 30.2 Theorem holds for all

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7 Representations of Weil Groups

ψ ∈ F, ψ = 1, and this ﬁnishes the proof of 29.4 Theorem for representations of Galois groups. We next prove parts (1) and (2) of 29.4 Proposition for representations of Galois groups. Part (1) is an immediate consequence of the corresponding property of local constants of characters and 30.1 Lemma 1. The ﬁrst identity in (2) has just been proved, and the second follows immediately. 30.4. We have to extend the arguments to cover representations of Weil groups. We ﬁx a prime element of F . Let φ be an unramiﬁed character of F × . We write φ() = q −s(φ) , for some s(φ) ∈ C. If E/F is a ﬁnite exten−s(φ) . Thus, if χ is a sion and E is a prime of E, we also have φE (E ) = qE × character of E , we have ε(χφE , s, ψE ) = ε(χ, s+s(φ), ψE ). 0 ΩF , and let φ be an unramiﬁed character of Proposition. Let (ΩE , ρ) ∈ K × F of ﬁnite order. Then ε(φE ⊗ ρ, s, ψE ) = ε(ρ, s+s(φ), ψE ). Proof. The maps (ΩE , ρ) → ε(φE ⊗ ρ, s, ψE ), (ΩE , ρ) → ε(ρ, s+s(φ), ψE ) are induction constants, with the same boundary. They are therefore the same. Let E/F be a ﬁnite extension, let 1E be the trivial character of WE and set RE/F = IndE/F 1E . Deﬁne λE/F (s, ψ) =

ε(RE/F , s, ψ) . ε(1E , s, ψE )

(30.4.1)

Corollary. The function λE/F (s, ψ) is constant in s. Proof. Let φ be an unramiﬁed character of F × of ﬁnite order. We have φ ⊗ RE/F ∼ = IndE/F φE , so ε(IndE/F φE , s, ψ) ε(φE , s, ψE ) ε(RE/F , s+s(φ), ψ) = ε(1E , s+s(φ), ψE ) = λE/F (s+s(φ), ψ).

λE/F (s, ψ) =

Thus λE/F (s+s(φ), ψ) = λE/F (s, ψ), for all unramiﬁed characters φ of ﬁnite order. That is, λE/F (s+ζ, ψ) = λE/F (s, ψ) for all roots of unity ζ ∈ C. Since 1 λE/F (s, ψ) is a constant times a power of q 2 −s , the result follows. We therefore write λE/F (s, ψ) = λE/F (ψ): it is called the Langlands constant.

30. Existence of the Local Constant

195

Remark. In terms of the Langlands constant, the deﬁning relation (29.4.1) reads ε(IndE/K ρ, s, ψK ) = λE/K (ψK )n , ρ ∈ Gss (30.4.2) n (E). ε(ρ, s, ψE ) The functional equation 29.4 Proposition (3) reads λE/F (ψ)2 = κE/F (−1),

(30.4.3)

so λE/F (ψ) is a 4-th root of unity. 30.5. Let ρ be an irreducible smooth representation of WE . There is an unramiﬁed character φ of WF such that φE ⊗ ρ factors through a representation ρ0 of ΩE . Because of 30.4 Proposition, we can unambiguously deﬁne ε(ρ, s, ψE ) = ε(ρ0 , s−s(φ), ψE ).

(30.5.1)

The inductive properties required by 29.4 Theorem follow from 30.4 Corollary. The uniqueness property is proved by applying the Brauer induction theorem to ρ0 , as in 30.1. 30.6. We now prove 29.4 Proposition. We have already proved parts (1) and (2) for representations of Galois groups. The general case of (1) follows directly from the deﬁnition (30.5.1). The ﬁrst assertion in (2) follows from the Galois case and the deﬁnition, and there is nothing to do in the second. In parts (3) and (4), it is enough to consider representations of Galois groups. The functions

τ , 1−s, ψE ), ε(τ, s, ψE ) ε(ˇ (ΩE , τ ) −→ det τ (−1), on K0 ΩF are both induction constants with boundary ˇ 1−s, ψE ) = φ(−1) (E, φ) −→ ε(φ, s, ψE ) ε(φ, (23.4.2). Thus they are the same, as required for (3). In part (4), it is enough to consider an irreducible representation ρ of ΩF , of dimension m, say. We write [ρ] − m[1F ] =

r

IndEi /F [φi ] − [1Ei ] ,

(30.6.1)

i=1

for characters φi of Ei× , for various ﬁnite extensions Ei /F . We then have ε(χ ⊗ ρ, s, ψ) ε(χEi φi , s, ψEi ) . = ε(χ, s, ψ)m ε(χEi , s, ψEi ) i=1 r

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7 Representations of Weil Groups

If χ has suﬃciently large level, the right hand side reduces to giving (cf. 23.8) ε(χ ⊗ ρ, s, ψ) = ε(χ, s, ψ)m φi (c(χ))−1 .

r i=1

φi (c(χ))−1 ,

i

The determinant of the deﬁning relation (30.6.1) and 29.2 Lemma give φi (c(χ)) = det ρ(c(χ)), i

as required. 30.7. We turn to the proof of 30.2 Theorem. We ﬁrst recall, with extreme brevity, some matters concerning global ﬁelds. Let K be a global ﬁeld; thus K is a ﬁnite extension of either the rational ﬁeld Q or a ﬁeld Fq (t) of rational functions, in one variable, over a ﬁnite ﬁeld. We ﬁx a separable algebraic closure K/K and write ΩK = Gal(K/K). Let AK denote the ad`ele ring of K, and IK its id`ele group. Thus K × embeds naturally in IK as a discrete subgroup; we denote by CK = IK /K × the id`ele class group of K. Let P (K) denote the set of places of K. For v ∈ P (K), we denote by Kv the completion of K at v. There is a canonical embedding of Kv× in CK , for each v ∈ P (K). Thus, if χ is a continuous homomorphism CK → C× , the restriction of χ to the image of Kv× gives a character χv of Kv× . If v is non-Archimedean, we can form the L-function L(χv , s) as in §23. If v is Archimedean, L(χv , s) is given by an explicit formula involving elementary functions which need not detain us. The product L(χv , s) (30.7.1) L(χ, s) = v∈P (K)

converges in a half-plane Re s > s0 . It admits analytic continuation to a meromorphic function on the whole s-plane, where it satisﬁes a functional equation (30.7.2) L(χ, s) = ε(χ, s) L(χ−1 , 1−s). The factor ε(χ, s) is an exponential function, having neither poles nor zeros. There are canonical embeddings of K and of Kv in AK , for each v ∈ P (K). In particular, the image of K is a discrete subgroup of AK . Let ψ be a nontrivial unitary character of AK /K. We can restrict to get a character ψv of Kv for each v. The character ψv is invariably non-trivial. The ε-factor in (30.7.2) then satisﬁes ε(χv , s, ψv ). (30.7.3) ε(χ, s) = v∈P (K)

30. Existence of the Local Constant

197

In this expression, the factor ε(χv , s, ψv ) (for v non-Archimedean) is that discussed in §23. (The deﬁnition of ε(χv , s, ψv ), for Archimedean v, need not detain us.) 30.8. For each v ∈ P (K), choose a separable algebraic closure K v /Kv and write Ωv = Gal(K v /Kv ). There is then an embedding of Ωv in ΩK as a closed subgroup, determined up to inner automorphism of ΩK . If ρ is a ﬁnitedimensional, smooth representation of ΩK , then ρv = ρ | Ωv is a smooth representation of Ωv . Using (29.3.3) (which applies equally when v is Archimedean), we can deﬁne L(ρv , s), and put L(ρ, s) = L(ρv , s). (30.8.1) v∈P (K)

This is the Artin L-function of the representation ρ. The basic facts here are: Lemma. (1) Let L/K be a ﬁnite extension, L ⊂ K, and let τ be a smooth representation of ΩL = Gal(K/L) of ﬁnite dimension. Then L(IndL/K τ, s) = L(τ, s). (2) Let ρ be a ﬁnite-dimensional smooth representation of ΩK . The product (30.8.1) converges in some half-plane and admits analytic continuation to a meromorphic function on the whole s-plane. It satisﬁes a functional equation L(ρ, s) = ε(ρ, s) L(ˇ ρ, 1−s), (30.8.2) for a uniquely determined exponential function ε(ρ, s) having neither zeros nor poles. (3) In the situation of (1) we have ε(IndL/K τ, s) = ε(τ, s).

(30.8.3)

Sketch proof. Part (1) is given by an elementary argument, based on the deﬁnitions (30.8.1), (29.3.3). Part (2) then follows from the Brauer Induction Theorem (as in 30.1) and the case recalled in 30.7. Part (3) is an immediate consequence. 30.9. We return to the situation of 30.2, with a ﬁnite Galois extension L/F of non-Archimedean local ﬁelds and G = Gal(L/F ). We make a connection with the global situation: Lemma 1. There is a ﬁnite Galois extension L/F of global ﬁelds, and a nonArchimedean place v0 of F , with the following properties:

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7 Representations of Weil Groups

(1) There is a unique place u0 of L lying over v0 and a topological isomorphism Lu0 ∼ = L which induces a topological isomorphism Fv0 ∼ = F. (2) The ﬁeld F has no embedding in R. Proof. In the case where F has characteristic zero, we use the Weak Approximation Theorem in Q to ﬁnd a negative x ∈ Q× which is a square in F . We choose a global ﬁeld E which is dense in L and contains a square root of x in F . In the case where F has positive characteristic, we simply choose the global ﬁeld E to be dense in L. In either case, we let L be the composite of the ﬁelds E σ , σ ∈ Gal(L/F ), and set F = L ∩ F . The extension L/F is then Galois, and restriction induces an isomorphism Gal(L/F ) → Gal(L/F ). Moreover, when F has characteristic √ zero, F contains the imaginary quadratic ﬁeld Q( x), with the result that L satisﬁes (2). The inclusion F → F deﬁnes a place v0 of F and an embedding Fv0 → F . Likewise, the inclusion L → L deﬁnes a place u0 of L lying over v0 and, since L is dense in L, an isomorphism Lu0 ∼ = L. The place u0 is stable under Gal(L/F ) = Gal(L/F ), hence is the only place of L lying over v0 . The extensions L/F , L/F have the same degree, so Fv0 = F , as required. In particular, we have a canonical isomorphism G ∼ = Gal(L/F ). So, for any intermediate ﬁeld E, F ⊂ E ⊂ L, there is a unique ﬁeld E, F ⊂ E ⊂ L, with closure E in L = Lu0 . For each non-Archimedean place v of F , let ov denote the discrete valuation ring in Fv and pv its maximal ideal: we use similar notations for extension ﬁelds. For each non-Archimedean place v = v0 of F , we choose an integer nv 0 as follows. (30.9.1) For each v ∈ P (F ), let u be a place of L lying over v. (1) If v is unramiﬁed in L, set nv = 0, and (2) choose nv nLu /Fv if v is ramiﬁed in L. (In (2), we use the notation of 30.2 Lemma.) We need one more deep result3 : Lemma 2. There exists a character α of CF , of ﬁnite order, such that (1) αv0 = 1, and (2) the level lv of αv satisﬁes lv nv , for every non-Archimedean v = v0 . For an intermediate ﬁeld F ⊂ E ⊂ L, we set αE = α ◦ NE/F . We ﬁx a non-trivial character Ψ of AF /F and, for E as before, set ΨE = Ψ ◦ TrE/F . We put ψ = Ψv0 . 3

This is a simple instance of the Grunwald-Wang Theorem, for which see [2].

30. Existence of the Local Constant

199

For each non-Archimedean place v of F , we choose cv ∈ Fv such that [l /2]+1 , with the understanding that cv = 1 if αv (1+x) = Ψv (cv x), x ∈ pv v lv = 0. Thus cv = 1 for almost all non-Archimedean v. If v is Archimedean, we put cv = 1. Let c denote the id`ele (cv ). Let us adjust our notation slightly, so that G = Gal(L/F ), G = Gal(L/F ). Under the canonical isomorphism G ∼ = G, we have a bijection Γ(G) ∼ = Γ(G), which we denote (E, χ) → (E, χ). In this notation, χ is a character of CE trivial on norms NL/E (CL ). Let (E, χ) ∈ Γ(G). Let w range over the non-Archimedean places of E, with w0 being the unique place of E lying over v0 . We obtain:

χw (cv )−1 ε(αE,w , s, ΨE,w ), w = w0 , ε(χw αE,w , s, ΨE,w ) = ε(χ, s, ψE ), w = w0 . In the ﬁrst identity, v denotes the place of F lying under w. When v is unramiﬁed in L, the character χw is also unramiﬁed and the identity follows immediately from the formulas in 23.5. Otherwise, the ﬁrst identity follows from the Stability Theorem 23.8 and 30.2 Lemma. The statement is trivial at the Archimedean places, since χw and αv are then both trivial. The second identity comes from the deﬁnition of α. Now we expand using the product formula (30.7.3) ε(χαE , s) = ε(χw αEw , s, ΨE,w ) w∈P (E)

= ε(χ, s, ψE ) χ(c)−1 a(E), where a(E) =

ε(αEw , s, ΨE,w ).

w=w0

The division (E, χ) → ε(χαE , s) on G is pre-inductive (30.8.3). Also, the division (E, χ) → χ(c)−1 is pre-inductive: it is ∂F, where F(E, ρ) = det ρ(c)−1 . As the notation indicates, the quantity a(E) depends only on the ﬁeld E. The map (E, χ) → a(E) is a pre-inductive division, being the boundary of the induction constant (Gal(L/E), ρ) → a(E)dim ρ . The division (E, χ) → ε(χ, s, ψE ) on G is therefore pre-inductive: it is the boundary of an induction constant which we denote (E, ρ) −→ ε(ρ, s, ψE ). Because of 30.1 Lemma 2, this proves 30.2 Theorem, for the particular char acter Ψv0 = ψ ∈ F.

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7 Representations of Weil Groups

31. Deligne Representations We introduce a more highly-structured family of representations of WF . In this section, we take a pragmatic and ad hoc approach. A more convincing motivation is given in §32. 31.1. A Deligne representation of WF is a triple (ρ, V, n), in which (ρ, V ) is a ﬁnite-dimensional, smooth representation of WF and n ∈ EndC (V ) is nilpotent satisfying ρ(x) n ρ(x)−1 = x n, x ∈ WF . One deﬁnes isomorphism between such triples in the obvious way. A Deligne representation (ρ, V, n) is called semisimple if the smooth representation (ρ, V ) of WF is semisimple (cf. 28.7 Proposition). We shall write Gn (F ) for the set of equivalence classes of n-dimensional, semisimple, Deligne representations of WF . We can identify Gss n (F ) with the set of triples (ρ, V, 0) ∈ Gn (F ), so G0n (F ) ⊂ Gss n (F ) ⊂ Gn (F ). Example. Let V = Cn , n 1. Let n ∈ Mn (C) be the standard Jordan block of rank n−1: thus, if {v0 , v1 , . . . , vn−1 } is the standard basis of V , we have nvn−1 = 0 and nvi = vi+1 , 0 i < n−1. We deﬁne a smooth representation ρ0 of WF by ρ0 (x)vi = x i vi , 0 i n−1, x ∈ WF . We set ρ(x) = x (1−n)/2 ρ0 (x). The triple (ρ, Cn , n) is then a semisimple Deligne representation of WF , denoted Sp(n). 31.2. The standard constructions of representation theory have analogues for Deligne representations, although they are not always the obvious ones. For example, if (ρ, V, n) is a Deligne representation, we can form the contragredient ˇ ∈ EndC (Vˇ ) of n. However, one sets (ˇ ρ, Vˇ ) of (ρ, V ) and the transpose n ρ, Vˇ , −ˇ n). (ρ, V, n)∨ = (ˇ

(31.2.1)

Likewise, the deﬁnition of tensor product is not quite obvious: (ρ1 , V1 , n1 ) ⊗ (ρ2 , V2 , n2 ) = (ρ1 ⊗ ρ2 , V1 ⊗ V2 , Id1 ⊗ n2 + n1 ⊗ Id2 ), (31.2.2) where Idj is the identity endomorphism of Vj . The deﬁnition of direct sum (ρ1 , V1 , n1 ) ⊕ (ρ2 , V2 , n2 ) = (ρ1 ⊕ ρ2 , V1 ⊕ V2 , n1 ⊕ n2 ) is, however, the obvious one. The reasons for these forms of standard deﬁnitions emerge with 32.7 Theorem below.

32. Relation with -adic Representations

201

Exercise. Say that a semisimple Deligne representation of WF is indecomposable if it cannot be written as a direct sum of two non-zero Deligne representations. Show that the indecomposable, semisimple Deligne representations of WF are those of the form ρ ⊗ Sp(n), for n 1 and ρ ∈ G0n (F ). 31.3. We extend the machinery of L-functions and local constants to the class of semisimple Deligne representations as follows. Given a triple (ρ, V, n) ∈ Gn (F ), for some n, the space Vn = Ker n carries a semisimple representation ρn of WF . We set L((ρ, V, n), s) = L(ρn , s).

(31.3.1)

We form the contragredient (ˇ ρ, Vˇ , −ˇ n) of (ρ, V, n) as above, and put ε((ρ, V, n), s, ψ) = ε(ρ, s, ψ)

L(ˇ ρ, 1−s) L(ρn , s) . L(ρ, s) L(ˇ ρnˇ , 1−s)

(31.3.2)

Example. Consider the case (ρ, V, n) = Sp(2). Setting ζF (s) = (1−q −s )−1 , we have ρ, s), L(ρ, s) = ζF (s− 12 ) ζF (s+ 12 ) = L(ˇ while ρnˇ , s). L(ρn , s) = ζF (s+ 12 ) = L(ˇ Taking ψ of level one, we get ε(ρ, s, ψ) = q 2s−1 (23.5 Proposition), and hence ε(Sp(2), s, ψ) = q 2s−1

ζF ( 12 −s)ζF ( 32 −s) ζF (s+ 12 ) ζF (s− 12 )ζF (s+ 12 ) ζF ( 32 −s) (1−q 2 −s ) 1

=q

2s−1

1 (1−q s− 2 )

1

= −q s− 2 .

32. Relation with -adic Representations In this section, we give some motivation for the introduction of Deligne representations in §31. Let be a prime number. Most of the representations of ΩF and WF which arise “in nature” are actually -adic representations: they act continuously on vector spaces over extensions of the ﬁeld Q . The primary examples of this phenomenon are provided by ´etale cohomology groups. We now explain how such representations can be classiﬁed in terms of complex Deligne representations. We assume throughout that = p, the residual characteristic of F . The case = p is more complicated and we say nothing of it.

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7 Representations of Weil Groups

32.1. Let G be a locally proﬁnite group, and let C be a ﬁeld of characteristic zero. A representation π : G → AutC (V ) of G on a C-vector space V is deﬁned to be smooth if every v ∈ V has open stabilizer in G, just as in the standard case C = C. Smooth representations of G on C-vector spaces form a category RepC (G). Let C be another ﬁeld of characteristic zero, and ι : C → C a ﬁeld embedding. If (π, V ) is a smooth representation of G over C, we get a representation πC of G on C ⊗C V by πC (g) : 1 ⊗ v −→ 1 ⊗ π(g)v. This representation is smooth, and we have a functor RepC (G) −→ RepC (G), (π, V ) −→ (πC , C ⊗C V ). If ι is an isomorphism, this functor is an equivalence of categories. Example. We consider the case where C is an algebraically closed ﬁeld of characteristic zero, equipped with an isomorphism C ∼ = C. (We have in mind the case where C is an algebraic closure Q of the ﬁeld Q of -adic numbers, for a prime = p.) We take G = GL2 (F ). We have an equivalence of categories Rep(G) ∼ = RepC (G), and hence a bijection between the sets of equivalence classes of irreducible smooth representations of G over C and over C. As before, let B be the group of upper triangular matrices in G. We also have an equivalence Rep(B) ∼ = RepC (B), and a commutative diagram Rep(B) −→ RepC (B) IndG B ↓

↓ IndG B

Rep(G) −→ RepC (G) × Note that we cannot use normalized induction ιG B because, for x ∈ F , the 1/2 1/2 is a power of q and has no clear meaning in C. quantity x So, if we re-write the classiﬁcation theorem 9.11 in terms of Ind rather than ι, it makes sense and remains true for representations over C. Similarly, the equivalences of categories induced by the isomorphism C ∼ = C are compatible with the compact induction functor, so the classiﬁcation theorem for cuspidal representations (15.5) holds for representations over C. Informally speaking, the classiﬁcation of irreducible smooth representations of G is essentially independent of the coeﬃcient ﬁeld. In particular, there is no need to attempt to reproduce or replace the proofs of the classiﬁcation theorems by working directly over the ﬁeld C. Observe, however, that concepts like unitarity and square-integrability make no sense at all for representations over C.

32. Relation with -adic Representations

203

32.2. Similarly, the notion of a Deligne representation of WF makes sense over any ﬁeld C of characteristic zero: a Deligne representation of WF on a ﬁnite-dimensional C-vector space V is a triple σ = (ρ, V, n), where (ρ, V ) is a smooth representation of WF on V and n ∈ EndC (V ) is nilpotent, satisfying ρ(x) n ρ(x)−1 = x n,

x ∈ WF .

(Observe that x , as deﬁned in 28.4, is a power of q and so has an unambiguous meaning as an element of C.) A homomorphism (ρ1 , V1 , n1 ) → (ρ2 , V2 , n2 ) of Deligne representations over C is a C-linear map f : V1 → V2 such that f ◦ ρ1 (g) = ρ2 (g) ◦ f , for all g ∈ WF , and f ◦ n1 = n2 ◦ f . The class of Deligne representations of WF over C forms a category, which we denote D-RepC (WF ). A ﬁeld embedding ι : C → C gives rise to a functor D-RepC (WF ) −→ D-RepC (WF ), (ρ, V, n) −→ (ρC , C ⊗ V, 1 ⊗ n). If C → C is an isomorphism, this is an equivalence of categories. 32.3. We shall make frequent use of an elementary device. Let C be a ﬁeld of characteristic zero and V a ﬁnite-dimensional C-vector space. If n ∈ EndC (V ) is nilpotent, the series nj exp n = 1 + j! j1

has only ﬁnitely many terms, and represents a unipotent element of AutC (V ). In the opposite direction, if u = 1+n ∈ AutC (V ) is unipotent, we get a nilpotent element nj (−1)j−1 log u = j j1

of EndC (V ). We have the standard relations log(exp n) = n, exp(log u) = u. 32.4. Let be a prime number. We take as coeﬃcient ﬁeld an algebraic closure Q of Q . In particular, Q is a topological Q -algebra, the topology being deﬁned by a valuation Q → Q ∪ {∞}. This deﬁnes a metric on Q , but Q is not complete with respect to this metric. Let V be a Q -vector space of ﬁnite dimension d. The choice of a basis d identiﬁes V with Q and AutQ (V ) with GLd (Q ). Thus AutQ (V ) inherits a topology, and it is independent of the choice of basis. If G is a locally proﬁnite group, a representation (π, V ) of G on V is said to be continuous if the implied map π : G → AutQ (V ) is continuous, relative to this topology on AutQ (V ). Observe that a smooth representation of G on V is automatically continuous, but the converse statement does not apply.

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7 Representations of Weil Groups

32.5. Henceforth, we assume = p. We take G = WF . We recall (28.3.1) that there is a continuous surjection t : IF → Z . Its kernel ﬁts into an exact sequence Zm → 0, 1 → PF −→ Ker t −→ m prime, m=,p

and so has pro-order prime to . Such a map t is therefore unique up to multiplication by an element of Z× , and so it satisﬁes the analogue of (28.3.3): t(gxg −1 ) = g t(x),

x ∈ IF , g ∈ WF .

(32.5.1)

Moreover, the kernel of t contains no open subgroup of IF . The following result is the key to analyzing -adic representations of WF . Theorem. Let (σ, V ) be a ﬁnite-dimensional, continuous representation of WF over Q , = p. There is a unique nilpotent endomorphism nσ ∈ EndQ (V ) such that σ(x) = exp(t(x)nσ ), for all elements x of some open subgroup of IF . Proof. The uniqueness of nσ is straightforward. For, suppose there is a nilpotent n ∈ EndQ (V ) and an open subgroup H of IF such that

σ(y) = exp(t(y)n),

y ∈ H.

The character t | H is non-zero and, for any y ∈ H with t(y) = 0, the operator σ(y) ∈ AutQ (V ) is unipotent. We get n = t(y)−1 log σ(y). The issue is therefore the existence of nσ . Let O denote the integral closure d of Z in Q . Fixing a basis of V allows us to identify V with Q , d = dimQ V , and AutQ (V ) with GLd (Q ). For an integer m 1, let Km denote the sub group 1 + m Md (O ) of GLd (Q ). Then Km is an open subgroup of GLd (Q ) normalizing Km+1 . The quotient Km /Km+1 is a discrete abelian group of exponent . We view σ as a continuous homomorphism WF → GLd (Q ). Let J denote the set of g ∈ Ker t such that σ(g) ∈ K2 . Thus J is an open subgroup of Ker t and σ(J) ⊂ K2 . The image of σ(J) in K2 /K3 is an abelian group of exponent ; since Ker t has pro-order relatively prime to , this implies σ(J) ⊂ K3 . Inductively, we deduce that σ(J) ⊂ Km for all m 2, so σ(J) is trivial. Since J is an open subgroup of Ker t, there is an open subgroup H0 of IF such that H0 ∩ Ker t ⊂ J. Shrinking H0 if necessary, we can assume σ(H0 ) ⊂ K2 . There is an open, normal subgroup H of WF , of ﬁnite index in WF , such that H0 ⊃ H ∩ IF .

32. Relation with -adic Representations

205

The restriction of σ to H ∩ IF therefore factors through a continuous homomorphism φ : t(H ∩ IF ) → K2 , that is, σ(h) = φ(t(h)), h ∈ H ∩ IF . Combining this with (32.5.1), we have σ(ΦhΦ−1 )q = σ(h),

h ∈ H ∩ IF ,

(32.5.2)

for every Frobenius element Φ ∈ WF . Let h ∈ H ∩ IF , and let v ∈ V be an eigenvector for σ(h), with eigenvalue α, say. Thus σ(Φ)v is an eigenvector for σ(ΦhΦ−1 ) with eigenvalue α. By (32.5.2), αq is also an eigenvalue for σ(h). The set of eigenvalues for σ(h) is therefore invariant under the map α → αq . As σ(h) is invertible, this implies that the eigenvalues of σ(h) are all roots of unity in Q . Since σ(h) ∈ K2 , the element (σ(h)−1V )/2 is integral (over Z ). Lemma. (1) If α ∈ Q is a root of unity such that (α−1)/2 is integral, then α = 1. (2) For h ∈ H ∩ IF , the matrix σ(h) ∈ GLd (Q ) is unipotent. Proof. Suppose ﬁrst that α has order r , r 1. The extension Q (α)/Q is then totally ramiﬁed and α−1 is prime in Q (α). Thus (α−1)/ is not integral except in the case = 2 and r = 1. In all cases, therefore, (α−1)/2 is not integral. Suppose next that α = 1, and that the order of α is divisible by a prime number other than . We can write α = βγ, for roots of unity β = 1 and γ, of order prime to and a power of respectively. We have α−1 = β(γ−1)+(β−1). The quantity β−1 is a unit in Q (α) and (γ−1) is not. So, α−1 is a unit and (α−1)/2 cannot be integral. This proves (1), and (2) follows immediately. We now choose h0 ∈ H ∩ IF such that t(h0 ) = 0 and set n = t(h0 )−1 log σ(h0 ). This element n is nilpotent and σ(h0 ) = φ(t(h0 )) = exp(t(h0 )n). Now put A = Z t(h0 ) ⊂ Z ; we have two continuous homomorphisms A → GLd (Q ), namely x → φ(x) and x → exp(xn). They coincide on h0 , hence on Zh0 and also on the closure A of Zh0 . Putting H = t−1 (A) we have σ(y) = exp(t(y)n), for y ∈ H . Finally, H is open in IF , since A is open in . Z . Remark. In the context of the theorem, the representation (σ, V ) is smooth if and only if the nilpotent element nσ is zero. In particular, if dim σ = 1, then σ is smooth. 32.6. We draw some conclusions. Let (σ, V ) be a continuous, ﬁnite-dimensional representation of WF over Q . The uniqueness of nσ and (32.5.1) together imply (32.6.1) σ(g) nσ σ(g)−1 = g nσ , g ∈ WF .

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7 Representations of Weil Groups

In particular, nσ commutes with σ(IF ). Fixing a Frobenius element Φ ∈ WF , we put σΦ (Φa x) = σ(Φa x) exp(−t(x)nσ ),

a ∈ Z, x ∈ IF .

(32.6.2)

The map σΦ : WF → AutQ (V ) is a homomorphism which, by the theorem, is trivial on some open subgroup of IF . It therefore provides a smooth representation of WF . By (32.6.1), the triple (σΦ , V, nσ ) is a Deligne representation of WF on V . Theorem. Let RepfQ (WF ) be the category of ﬁnite-dimensional, continuous

representations of WF over Q . Let Φ ∈ WF be a Frobenius element, and t : IF → Z a continuous surjection. The assignment (σ, V ) → (σΦ , V, nσ ) of (32.6.2) is functorial, and induces an equivalence of categories RepfQ (WF ) −→ D-RepQ (WF ).

(32.6.3)

The isomorphism class of the Deligne representation (σΦ , V, nσ ) depends only on the isomorphism class of (σ, V ), and not on the choices of Φ and t. Proof. We ﬁrst check that the association (σ, V ) → (σΦ , V, nσ ) is given by a functor. If (ρ, U ) is a continuous representation of WF on a ﬁnite-dimensional Q vector space U , and if φ : V → U is a WF -homomorphism, the uniqueness property gives nρ ◦ φ = φ ◦ nσ . Thus φ induces a WF -homomorphism (σΦ , V, nσ ) → (ρΦ , U, nρ ) of Deligne representations. In the opposite direction, let (τ, V, n) be a Deligne representation of WF over Q . The formula τ Φ (Φa x) = τ (Φa x) exp(t(x)n),

a ∈ Z, x ∈ IF ,

deﬁnes a representation of WF on V . It is continuous because τ is smooth and the homomorphism IF −→ AutC (V ), x −→ exp(t(x)n), is continuous. Moreover, n = n(τ Φ )Φ . If (τ1 , V1 , n1 ) is another Deligne representation and φ : (τ, V, n) → (τ1 , V1 , n1 ) is a homomorphism, then the map φ : V → V1 gives a homomorphism (τ Φ , V ) → (τ1Φ , V1 ). Thus (τ, V, n) → (τ Φ , V ) is a functor, inverse to the ﬁrst.

32. Relation with -adic Representations

207

Now let Φ = Φx, x ∈ IF , be some other Frobenius. We deﬁne an automorphism A ∈ AutQ (V ) by

A = exp(λt(x)nσ ), where λ = (q−1)−1 . For y ∈ IF , we have A ◦ σΦ (y) ◦ A−1 = σ(y) exp (λt(x)−t(y)−λt(x))nσ = σΦ (y), and A ◦ σΦ (Φ) ◦ A−1 = exp(λt(x)nσ ) σ(Φ) exp(−λt(x)nσ ) = σ(Φ) exp(λt(x)qnσ ) exp(−λt(x)nσ ) = σ(Φ) exp(λ(q − 1)t(x)nσ ) = σ(Φ) exp(t(x)nσ ). On the other hand, σΦ (Φ) = σΦ (Φ x−1 ) = σ(Φ) exp(t(x)nσ ) = A ◦ σΦ (Φ) ◦ A−1 . Thus A ◦ σΦ (g) ◦ A−1 = σΦ (g), for all g ∈ WF . This shows that, up to isomorphism, (σΦ , V, nσ ) is independent of Φ. We next have to show that the isomorphism class of (σΦ , V, nσ ) is unchanged on replacing the character t by αt, for some α ∈ Z× . To do this, it is enough to produce an automorphism B of V , which commutes with σ(WF ), such that Bnσ B −1 = αnσ . Choose a positive integer a such that σ(Φa ) is central in σ(WF ). Let Vλ be the “generalized λ-eigenspace” of σ(Φa ), that is, Vλ = Ker (σ(Φa )−λ1V )dim V . The relation Φa nσ = Φ a nσ Φa implies nσ Vλqa ⊂ Vλ . Each Vλ is a σ(WF )subspace of V . We deﬁne B by Bv = µλ v,

v ∈ Vλ ,

a for a family of elements µλ ∈ Z× satisfying αµλq = µλ . This completes the proof.

Remark. The deﬁnitions in 31.2 are chosen to correspond, via the equivalence of the theorem, to the standard constructions in the category RepfQ (WF ).

The theorem gives a canonical bijection between the set of isomorphism classes of ﬁnite-dimensional continuous representations of WF over Q and the set of isomorphism classes of Deligne representations of WF over Q . As in 32.1, the choice of an isomorphism Q ∼ = C gives further bijection of these sets with the set of isomorphism classes of Deligne representations of WF over C.

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7 Representations of Weil Groups

32.7. It is necessary to have a more reﬁned version. As in 31.1, a Deligne representation (ρ, V, n) over Q is called semisimple if the underlying smooth representation (ρ, V ) is semisimple, in the sense of 28.7. A continuous, ﬁnite-dimensional representation (σ, V ) of WF is deﬁned to be Φ-semisimple if the associated Deligne representation (σΦ , V, nσ ) is semisimple. Proposition. Let (σ, V ) be a ﬁnite-dimensional continuous representation of WF over Q . The following conditions are equivalent: (1) (σ, V ) is Φ-semisimple; (2) there is a Frobenius element Ψ ∈ WF such that σ(Ψ ) is semisimple; (3) σ(g) is semisimple, for every g ∈ WF IF . Proof. We form the Deligne representation (σΨ , V, nσ ), as in 32.6. As σΨ (Ψ ) = σ(Ψ ), the equivalence of (1) and (2) follows from 28.7 Proposition. Surely (3) implies (2). Conversely, take g ∈ WF IF . If g is of the form Ψ x, x ∈ WF , the operators σ(g), σ(Ψ ) are conjugate, as in the proof of 32.6 Theorem. Thus σ(g) is semisimple. In general, g = Ψ a x, for some x ∈ IF and some integer a = 0. Restricting to WE , where E/F is unramiﬁed of degree |a|, the same argument shows that σ(g) is semisimple, as required. As a consequence, we get: Theorem. Let be a prime number, = p, and let n 1 be an integer. The sets of isomorphism classes of the following objects are in canonical bijection: (1) n-dimensional, Φ-semisimple, continuous representations of WF over Q ; (2) n-dimensional, semisimple, Deligne representations of WF over Q . The choice of an isomorphism Q ∼ = C induces a bijection between these sets and the set of isomorphism classes of n-dimensional, semisimple, Deligne representations of WF over C. Further reading The concept of Weil group originates in [85]. A formal summary, incorporating the more subtle global version, is given in [83]. Local class ﬁeld theory can be found in many texts, of which [74], [76] have been particularly inﬂuential. More recent and extremely compact, is [68]. Older works tend to deﬁne the Artin Reciprocity isomorphism as the inverse of the one used now: it takes arithmetic Frobenius elements of the Weil group to prime elements of F . Langlands’ original proof of the existence of the local constant used only local methods, but was long and only ever published informally [56]. Deligne

32. Relation with -adic Representations

209

[27] discovered a brief global argument. The version we give here is adapted from Tate’s Durham lectures [82] on a variant of Deligne’s proof. Various attempts, based on “canonical” versions of the Brauer induction theorem, have been made to provide a more manageable local construction, but they are not useful in our context. The algebraic number theory of §30 is standard fare: [54] is probably closest to what we use, particularly as it covers the Artin L-function (for Galois representations). Most of the key facts are also in [68]. The careful summary in [61] may be found helpful. The ﬁrst part (30.7) is a summary of the global part of Tate’s thesis [81]. The -adic context is summarized in [83]. The key result 32.5 is due to Grothendieck and is outlined in an appendix to [78]. The paper [78] discusses a signiﬁcant area in which -adic representations of Weil groups arise naturally.

8 The Langlands Correspondence

33. The Langlands correspondence 34. The tame correspondence 35. The -adic correspondence

We have to hand all of the concepts necessary for the statement of the central result. It concerns the existence of a canonical bijection, at the level of isomorphism classes, between the two-dimensional, semisimple Deligne representations of the Weil group WF and the irreducible smooth representations of G = GL2 (F ). This bijection, the Langlands correspondence, is speciﬁed in terms of relations between the L-functions and local constants of the two kinds of representation. It will be clear from the L-function relations that the correspondence must match the non-irreducible representations of WF with the principal series representations of G. Given the classiﬁcation theory in Chapter III and the calculations in Chapter VI, it is straightforward to write down the correspondence for such representations: this is the ﬁrst step below. The interest is in the matching of the irreducible representations of WF with the irreducible cuspidal representations of G. When the residual characteristic p of the base ﬁeld F is odd, we get the result very quickly by using the Tame Parametrization Theorem of Chapter V. All that is required is a little Galois theory and some elementary Gauss sum manipulations. The case p = 2 is considerably more involved and interesting: it requires several chapters to itself. Before going on to that, we look back at §32, particularly 32.7 Theorem. This shows that the Langlands correspondence implies a bijection between classes of irreducible smooth representations of G and classes of the appropriate two-dimensional -adic representations of WF . However, it relies on

212

8 The Langlands Correspondence

choosing an isomorphism C ∼ = Q : the rationality properties of the Langlands correspondence, relative to automorphisms of the coeﬃcient ﬁeld C, therefore become relevant. As it happens, they are not quite what is required. We show how to modify the original (complex) correspondence to get a canonical -adic correspondence. (We observe that §34, §35 are independent of each other, and could be read in either order.)

33. The Langlands Correspondence We recall that G2 (F ) denotes the set of equivalence classes of 2-dimensional, semisimple, Deligne representations of the Weil group WF . Analogously, let A2 (F ) denote the set of equivalence classes of irreducible smooth representations of G = GL2 (F ). 33.1. We arrive at the formal statement of the motivating result. As before, if χ is a character of F × , we write simply χ for the corresponding character χ ◦ aF of WF . Langlands Correspondence. Let ψ ∈ F, ψ = 1. There is a unique map π : G2 (F ) −→ A2 (F ) such that

L(χπ(ρ), s) = L(χ ⊗ ρ, s), ε(χπ(ρ), s, ψ) = ε(χ ⊗ ρ, s, ψ),

(33.1.1)

for all ρ ∈ G2 (F ) and all characters χ of F × . The map π is a bijection, and (33.1.1) holds for all ψ ∈ F, ψ = 1. The map π is the Langlands correspondence for GL2 (F ). We use the notation πF when we need to specify the base ﬁeld F . 33.2. We make some preliminary remarks. First, there is at most one map π satisfying (33.1.1), because of the Converse Theorem (27.1). We have a decomposition G2 (F ) = G12 (F ) ∪ G02 (F ), where G02 (F ) is the set of equivalence classes of irreducible smooth representations of WF of dimension two, and G12 (F ) consists of those classes of semisimple Deligne representations (ρ, V, n) ∈ G2 (F ) for which the representation ρ of WF is reducible. Likewise, A2 (F ) = A12 (F ) ∪ A02 (F ), where A02 (F ) (resp. A12 (F )) consists of the classes of representations π ∈ A2 (F ) which are cuspidal (resp. non-cuspidal).

33. The Langlands Correspondence

213

Proposition. (1) Let π ∈ A2 (F ); then π ∈ A02 (F ) if and only if L(χπ, s) = 1 for all characters χ of F × . (2) Let ρ ∈ G2 (F ); then ρ ∈ G02 (F ) if and only if L(χ ⊗ ρ, s) = 1 for all characters χ of F × . Proof. The ﬁrst assertion is 27.2 Proposition. The second follows immediately from the deﬁnitions in 29.3 and 31.3. 33.3. Any map π satisfying (33.1.1) must therefore take G12 (F ) to A12 (F ) and G02 (F ) to A02 (F ). We deal with the ﬁrst half of the problem. Theorem. There is a unique map π 1 : G12 (F ) −→ A12 (F ) with the property L(χπ 1 (ρ), s) = L(χ ⊗ ρ, s), for all ρ ∈ G12 (F ) and all characters χ of F × . The map π 1 is bijective, and it satisﬁes π 1 (χ ⊗ ρ) = χπ 1 (ρ), ε(π 1 (ρ), s, ψ) = ε(ρ, s, ψ), for all ρ, χ, and all ψ ∈ F, ψ = 1. Proof. We simply exhibit the map π 1 : the required properties are easily veriﬁed from the deﬁnitions in 29.3, 31.3, and the list of L-functions and local constants in 26.1 Theorem. So, let (ρ, V, n) ∈ G12 (F ). Since ρ is semisimple, it is of the form χ1 ⊕ χ2 , for characters χi of F × . We form the character χ = χ1 ⊗ χ2 of the group T of diagonal matrices in G, and consider the induced representation π = ιG Bχ of G. If π is irreducible, the criterion in 9.11 Lemma implies that n = 0. We set π 1 (ρ) = π = ιG B χ. This leaves the case where ιG B χ is not irreducible. Thus there is a character × φ of F such that χ1 (x) = φ(x)x−1/2 and χ2 (x) = φ(x)x1/2 . There are two elements (ρ, V, n) of G12 (F ) in which ρ has WF -composition factors χ1 , χ2 ; they are distinguished by whether or not n is zero. We put π 1 (ρ, V, 0) = φ ◦ det in the one case and (cf. 31.3 Example) π 1 (ρ, V, n) = φ · StG , in the other.

n = 0

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8 The Langlands Correspondence

33.4. The heart of the matter is therefore: Theorem. Let ψ ∈ F, ψ = 1. There is a unique map π : G02 (F ) −→ A02 (F ) with the property ε(χ ⊗ ρ, s, ψ) = ε(χπ(ρ), s, ψ),

(33.4.1)

for all ρ ∈ G02 (F ), all characters χ of F × . Moreover, the map π is a bijection, and (33.4.1) holds for all ψ ∈ F, ψ = 1. Again, the uniqueness of the map follows from 27.1, 27.2. We can make some helpful simpliﬁcations. Proposition. Let π be a map satisfying (33.4.1) relative to some ψ ∈ F, ψ = 1. Then: (1) If ρ ∈ G02 (F ) and π = π(ρ), then ωπ = det ρ. (2) The map π satisﬁes (33.4.1) for all ψ ∈ F, ψ = 1. Proof. The ﬁrst assertion follows from comparing the stability theorem 25.7 with its analogue 29.4 Proposition (4). The second then follows from 24.3 Proposition and 29.4 Proposition (2). Exercise. Let π be a map satisfying the requirements of the theorem. (a) Show that π(χ ⊗ ρ) = χπ(ρ), for ρ ∈ G02 (F ) and any character χ of F × . ρ) = π ˇ. (b) If ρ ∈ G02 (F ) and π = π(ρ), show that π(ˇ

34. The Tame Correspondence In this section, we prove the theorem of 33.4 in almost all cases, using an explicit construction based on that of Chapter V. Throughout, p denotes the characteristic of the ﬁnite ﬁeld k = o/p. We allow the case p = 2, although the results are complete only when p = 2. 0 0 34.1. We deﬁne a subset Gnr 2 (F ) of G2 (F ) as follows: the class ρ ∈ G2 (F ) lies nr in G2 (F ) if there is an unramiﬁed character χ = 1 of WF such that χ ⊗ ρ ∼ = ρ. (F ), we say that ρ is totally ramiﬁed. If ρ ∈ G02 (F ) Gnr 2 Let P2 (F ) denote the set of isomorphism classes of admissible pairs, as in 18.2. If (E/F, ξ) ∈ P2 (F ), we can view ξ as a character of WE and form the induced representation ρξ = IndE/F ξ of WF .

34. The Tame Correspondence

215

Theorem. If (E/F, ξ) is an admissible pair, the representation ρξ of WF is irreducible. The map (E/F, ξ) → ρξ induces a bijection ≈

P2 (F ) −→ G02 (F ) ≈

P2 (F ) −→

Gnr 2 (F )

if p = 2, or

(34.1.1)

if p = 2.

Proof. Let (E/F, ξ) ∈ P2 (F ) and σ ∈ Gal(E/F ), σ = 1. Since ξ does not factor through NE/F , the characters ξ, ξ σ of E × are distinct. The Artin map ab ∼ WE = E × is Gal(E/F )-equivariant (29.1 property (5)) so the characters ξ, σ ξ of WE are distinct and the representation ρξ = IndE/F ξ is consequently irreducible. The equivalence class of ρξ surely only depends on the isomorphism class of (E/F, ξ). Lemma. Let (E/F, ξ) be an admissible pair, set ρξ = IndE/F ξ. Let κ = κE/F be the non-trivial character of F × which is null on NE/F (E × ). We then have nr ρξ ∼ = κ ⊗ ρξ . In particular, ρξ ∈ G2 (F ) if and only if E/F is unramiﬁed. Proof. We have κ ⊗ ρ ∼ = IndE/F (κE ⊗ ξ), where κE = κ ◦ NE/F = 1. This gives one implication in the ﬁnal assertion. Conversely, suppose ρξ ∈ Gnr 2 (F ). × Let χ be the unramiﬁed quadratic character of F . If σ generates Gal(E/F ), then ξ σ /ξ = χE . This implies that ξ | UE1 factors through NE/F , whence, by the deﬁnition of admissible pair, the extension E/F is unramiﬁed. We now show that the map (34.1.1) is injective. Let (Ei /F, ξi ) ∈ P2 (F ), i = 1, 2, and suppose that ρξ1 ∼ = ρξ2 . If E1 /F ∼ = E2 /F , we may as well take E1 = E2 = E. The restriction ρξ1 | WE is ξ1 ⊕ ξ1σ , where σ ∈ Gal(E/F ), σ = 1. It follows that ξ2 = ξ1 or ξ1σ , and (E/F, ξ2 ) ∼ = (E/F, ξ1 ) in either case. We therefore assume E1 = E2 and let L = E1 E2 . At least one of the Ei /F is totally ramiﬁed, so [L:F ] = 4 and the maximal unramiﬁed sub-extension E/F of L/F has degree 2. Set κi = κEi /F . The representation ρ = ρξi is ﬁxed under tensoring with κ1 and κ2 , hence also with κE/F . That is, ρ ∈ Gnr 2 (F ). The lemma now implies that E1 = E2 = E, contrary to hypothesis. We conclude that the map (34.1.1) is injective. We have to prove it is surjective. If ρ ∈ Gnr 2 (F ), elementary considerations show that ρ is induced from a character ξ of WE , where E/F is unramiﬁed quadratic. The pair (E/F, ξ) is then admissible and ρ ∼ = ρξ . We therefore assume ρ totally ramiﬁed; in particular, p = 2. Since dim ρ = 2, the restriction of ρ to the pro p-group PF decomposes as a sum of characters. It follows that there is a ﬁnite, tamely ramiﬁed Galois extension K/F such that ρ | WK decomposes as a sum of characters, ρ | WK = θ ⊕ θ , say. Suppose ﬁrst that θ = θ . The WF -stabilizer of θ is therefore WL , for some quadratic extension L/F contained in K. The natural representation of

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8 The Langlands Correspondence

WL in the θ-isotypic subspace of ρ provides a character ξ of WL such that ρ = IndL/F ξ. We view ξ as a character of L× ; we have to show that the pair (L/F, ξ) is admissible. Let σ ∈ Gal(L/F ), σ = 1. The character ξ σ | WK is θ , so ξ = ξ σ and ξ does not factor through NL/F . Consider ξ | UL1 . The extension L/F is totally ramiﬁed, since ρ is totally ramiﬁed. Thus, if ξ | UL1 factors through NL/F , the character ξ σ /ξ is trivial on UL = UF UL1 , and is unramiﬁed: ξ σ = χL ξ, for some unramiﬁed character χ of F × , χ = 1. Thus nr ρ∼ = χ ⊗ ρ, and ρ ∈ G2 (F ), contrary to hypothesis. We conclude that, in this case, the pair (L/F, ξ) is admissible and ρ ∼ = ρξ . It remains only to exclude the possibility θ = θ . Let L/F be the maximal unramiﬁed sub-extension of K/F . Since K/L is cyclic, θ admits extension to a character ξ of WL , such that ξ occurs in ρ | WL . In other words, we could have taken K/F unramiﬁed. But this would imply ρ ∈ Gnr 2 (F ), contrary to hypothesis. 34.2. For p = 2, we have canonical bijections P2 (F ) −→ A02 , (E/F, ξ) −→ πξ ,

P2 (F ) −→ G02 (F ), (E/F, ξ) −→ ρξ ,

given by 20.2 and 34.1 respectively. The implied bijection ≈

G02 (F ) −→ A02 (F ), ρξ −→ πξ ,

(34.2.1)

is not the map π demanded by 33.4. For example, if (E/F, ξ) ∈ P2 (F ), the representation ρξ has determinant κE/F ⊗ξ|F × while (see 20.2) πξ has central character ξ|F × , contrary to the requirement of 33.4 Proposition. When p = 2, we have a parallel situation for unramiﬁed admissible pairs. We obtain the map π of 33.4 by systematically modifying the bijection (34.2.1). 34.3. If K/F is a ﬁnite separable extension and ψ ∈ F, ψ = 1, let λK/F (ψ) =

ε(RK/F , s, ψ) ε(1K , s, ψK )

denote the Langlands constant, as in (30.4.1), 30.4 Corollary. Proposition. Let ψ ∈ F have level one, and let K/F be a tamely ramiﬁed quadratic extension. (1) If K/F is unramiﬁed, then κK/F is unramiﬁed of order 2 and λK/F (ψ) = −1.

34. The Tame Correspondence

217

(2) If K/F is totally ramiﬁed, then κK/F is the non-trivial character of F × /NK/F (K × ) and λK/F (ψ) = τ (κK/F , ψ)/q 1/2 . In particular, λK/F (ψ)2 = κK/F (−1). Proof. We have RK/F = IndK/F 1K = 1F ⊕ κK/F . The declared formulæ are then given by the proposition and theorem of 23.5, the ﬁnal assertion being (30.4.3). We observe that, in (2), κK/F is of order 2, and ramiﬁed of level zero. Thus κK/F (−1) is given by the Jacobi symbol: −1 . (34.3.1) κK/F (−1) = q 34.4. We take an admissible pair (E/F, ξ) ∈ P2 (F ). We associate to this pair a character ∆ = ∆ξ of E × of level zero. First: Deﬁnition. Let (E/F, ξ) be an admissible pair in which E/F is unramiﬁed. Deﬁne ∆ξ to be unramiﬁed, of order 2. The ramiﬁed case is more involved. We recall that µF denotes the group of roots of unity in F of order prime to p. Let E/F be a totally tamely ramiﬁed quadratic extension, let be a prime element of E, and let β ∈ E × . Since UE = µE UE1 = µF UE1 , there is a unique root of unity ζ(β, ) ∈ µF such that β−υE (β) ≡ ζ(β, )

(mod UE1 ).

(34.4.1)

Proposition-Deﬁnition. Let ψ ∈ F have level one. (1) Let (E/F, ξ) ∈ P2 (F ) be a minimal pair such that E/F is totally ramiﬁed. Let n be the level of ξ and let α ∈ p−n E satisfy ξ(1+x) = ψE (αx), n x ∈ pE . There is a unique character ∆ = ∆ξ of E × such that: ∆ | UE1 = 1,

∆ | F × = κE/F ,

∆() = κE/F (ζ(α, )) λE/F (ψ)n ,

(34.4.2)

for any prime element of E. The deﬁnition of ∆ξ is independent of the choices of ψ and α. (2) Let (E/F, ξ) ∈ P2 (F ) and suppose that E/F is totally ramiﬁed. Write ξ = ξ · χE , for a minimal pair (E/F, ξ ) and a character χ of F × . Deﬁne (34.4.3) ∆ξ = ∆ξ . The deﬁnition of ∆ξ is independent of the choice of decomposition ξ = ξ · χE .

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8 The Langlands Correspondence

Proof. There is certainly at most one character ∆ of E × satisfying the conditions (34.4.2). If we ﬁx the character ψ, the deﬁnition of ∆ depends only on the coset αUE1 . If we replace ψ by another character ψ ∈ F of level one, then ψ = uψ, for some u ∈ UF . The element α is replaced by u−1 α, and κE/F (ζ(u−1 α, )) λE/F (uψ)n = κE/F (ζ(α, )) λE/F (ψ)n κE/F (u)n−1 , while κE/F (u)n−1 = 1, since n is odd. In other words, the character (assuming it exists) is deﬁned independently of the choices of ψ and α within the permitted ranges. To establish its existence, we have to check the consistency conditions ∆(u) = ∆(u)∆(), 2

u ∈ UE ,

2

∆( ) = ∆() . For the ﬁrst, we write u = ηv, with η ∈ µF and v ∈ UE1 . This gives ζ(α, u) ≡ ηζ(α, ) (mod µ2F ) and ∆(u) = κE/F (η) κE/F (ζ(α, )) λE/F (ψ)n = ∆(u)∆(). For the second, we note that 2 ≡ −NE/F () (mod UE1 ), whence by (34.3.1), ∆(2 ) = ∆(−NE/F ()) = κE/F (−1) = ∆()2 , as required. The ﬁnal assertion is immediate.

Lemma. (1) If (E/F, ξ) is an admissible pair, the pair (E/F, ∆ξ ξ) is admissible and its isomorphism class depends only on that of (E/F, ξ). The character ∆ξ satisﬁes ∆2ξ = 1, except when E/F is totally ramiﬁed and q ≡ 3 (mod 4). In the exceptional case, ∆ξ has order 4. (2) The map P2 (F ) −→ P2 (F ), (E/F, ξ) −→ (E/F, ∆ξ ξ), is bijective. Proof. (1) follows directly from the deﬁnitions, 34.3 Proposition and (34.3.1). Part (2) follows from the observation that ∆ξ is tamely ramiﬁed, depending only on E/F and ξ | UE1 .

34. The Tame Correspondence

219

Tame Langlands Correspondence. (1) Suppose p = 2. For ρ ∈ G02 (F ), deﬁne π(ρ) = π∆ξ ξ for any (E/F, ξ) ∈ P2 (F ) such that ρ ∼ = ρξ . The map π : G02 (F ) −→ A02 (F ) is a bijection satisfying ε(χ ⊗ ρ, s, ψ) = ε(χπ(ρ), s, ψ),

(34.4.4)

for all characters χ of F × and all ψ ∈ F, ψ = 1. (2) In the case p = 2, the map ρξ → π∆ξ ξ is a bijection nr π : Gnr 2 (F ) −→ A2 (F )

satisfying the analogue of (34.4.4). (3) In both cases, the map π satisﬁes π(χ ⊗ ρ) = χ π(ρ)

and π(ˇ ρ) = π(ρ)∨ ,

(34.4.5)

for all ρ and all characters χ of F × . Remark. Because of the uniqueness properties (33.2), π is the Langlands correspondence when p = 2, and is the restriction to Gnr 2 (F ) of the Langlands correspondence when p = 2. Proof. First we note that π is bijective because of the lemma, 34.1 Theorem and 20.2 Theorem, while (34.4.5) follows by construction and 20.2 Theorem (3). The construction also gives ωπ(ρ) = det ρ, so it is enough to check (34.4.4) for one choice of ψ ∈ F. We take ψ to have level one. Further, n(ρ, ψ) = n(π(ρ), ψ) (as follows readily from 20.2 Theorem (1)), so we have only to check the ε-relation at s = 12 . We divide into cases. Case 1. Let ρ = IndE/F ξ, where (E/F, ξ) is minimal and E/F totally ramiﬁed. Thus ξ has odd level n = 2m+1. We choose α such that ξ(1+x) = ψE (αx), . Using the standard reduction technique (23.6.4), we get x ∈ pm+1 E ˇ ε(ξ, 12 , ψE ) = ξ(α)ψ E (α), ε(ρ, 12 , ψ) = ε(ξ, 12 , ψE ) λE/F (ψ) ˇ = ξ(α)ψ E (α) λE/F (ψ). Setting π = π(ρ), 25.5 Corollary gives ˇ ε(π, 12 , ψ) = ∆ˇξ (α)ξ(α)ψ A (α).

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8 The Langlands Correspondence

ˇξ (α) = λE/F (ψ). We have to check that ∆ We choose a prime element of E and ζ ∈ µF such that αn ≡ ζ (mod UE1 ). The deﬁnition of ∆ = ∆ξ gives ∆(α) = κE/F (ζ) ∆(−n ). Since n is odd and λE/F (ψ) is a 4-th root of unity, we get κE/F (ζ) ∆(−n ) λE/F (ψ) = κE/F (ζ)1−n λE/F (ψ)1−n = 1, 2

as required. Case 2. Let ρ = IndE/F ξ be as in Case 1, and let χ be a character of F × of level l 1. We consider the representation χ ⊗ ρ. This is obtained, by induction, from the admissible pair (E/F, ξχE ). If l m, the pair (E/F, ξχE ) is minimal and there is nothing more to do. Otherwise, χE has level 2l > n and there exists c ∈ F such that χ(1+x) = ψ(cx), x ∈ p[l/2]+1 . Thus χE (1+y) = ψE (cy), l y ∈ pl+1 E . Further, ξ(1+x) = ψE (αx), x ∈ pE . From (23.6.4), we therefore get ε(ξχE , 12 , ψE ) = q −1/2

ξˇχ ˇE (α+c)(1+y) ψE (α+c)(1+y)

y∈plE /pl+1 E

ˇ = q −1/2 ξ(c+α) χ ˇE (1+c−1 α) ψE (α)

χ ˇE (c(1+y)) ψE (c(1+y))

y

ˇ = ξ(c+α) χ ˇE (1+c−1 α) ψE (α) ε(χE , 12 , ψE ). Therefore, via (30.4.2) and 23.6, ˇ ε(χ ⊗ ρ, 12 , ψ) = ξ(c+α) χ ˇE (1+c−1 α) ψE (α) ε(χE , 12 , ψE ) λE/F (ψ) ˇ = ξ(c+α) χ ˇE (1+c−1 α) ψE (α) ε(χ, 1 , ψ) ε(χκE/F , 1 , ψ) 2

2

ˇ = ξ(c+α) χ ˇE (1+c−1 α) ψE (α) ε(χ, 12 , ψ)2 κE/F (c). On the other hand, with π = π(ρ) as before, the same techniques ((23.6.2), (25.6.1) and 25.5 Corollary) yield −1 ˇ ˇξ(c+α) χ(det(1+c ˇ α)) ψA (α) ε(χ, 12 , ψ)2 . ε(χπ, 12 , ψ) = ∆

ˇ However, ψA (α) = ψE (α) and, since ∆ is tamely ramiﬁed, we have∆(c+α) = ˇ ∆(c) = κE/F (c). We deduce ε(χ ⊗ ρ, 12 , ψ) = ε(χπ, 12 , ψ), as required. Case 3. Let ρ = IndE/F ξ, where (E/F, ξ) is minimal and E/F is unramiﬁed.

35. The -adic Correspondence

221

If the level n of ξ is odd, the argument is a simpler version of Case 1, so we omit the details. We therefore assume n is even. The case n = 0 follows from the identity (25.4.1) and straightforward manipulations. We therefore assume n = 2m > 0. We have (23.6.2), (23.6.4), ˇ ε(ξ, 12 , ψE ) = q −1 ξ(α(1+x)) ψE (α(1+x)), x∈pm /pm+1 E E

for α chosen suitably, in the usual way. On the other hand (25.5 Corollary), tr Λ∨ (α(1+y)) ψA (α(1+y)). ε(π, 12 , ψ) = q −3 y∈pm M/pm+1 M

(Here, we use the notation of the construction of πξ in 19.4.) m m Lemma. Let x ∈ UM ; then tr Λ(αx) = 0 unless αx is UM -conjugate to an m m+1 element of αUE UM .

This is a restatement of 22.3 Lemma. Taking account of the number of elements of the conjugacy classes, we get ˇ ε(π, 12 , ψ) = −q −1 ξ(α(1+y)) ψA (α(1+y)). y∈pm /pm+1 E E

= ε(ρ,

1 2 , ψ),

as desired. The remaining cases, of representations χ ⊗ ρ, where ρ is unramiﬁed and given by a minimal pair, follow much the same course as Case 2, so we omit the details. We have completed the proof of 34.4 Theorem. We have also proved 33.1 Theorem in the case p = 2.

35. The -adic Correspondence We return to the Langlands correspondence π : G2 (F ) → A2 (F ) in the general setting of 33.1. For the purposes of this section, we assume that 33.1 Theorem has been proved in all cases. 35.1. We consider representations over the ﬁeld Q , = p, as in 32.4 et seq. Let G2 (F, Q ) denote the set of equivalence classes of semisimple, Deligne representations of WF , over Q and of dimension 2. We recall (32.7) that G2 (F, Q ) is in canonical bijection with the set of equivalence classes of 2-dimensional, Φ-semisimple, continuous representations of WF over Q .

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8 The Langlands Correspondence

Let A2 (F, Q ) denote the set of equivalence classes of irreducible smooth representations of G = GL2 (F ) over Q . The choice of an isomorphism ι : C ∼ = Q gives bijections G2 (F ) ∼ = G2 (F, Q ), σ → ι σ

A2 (F ) ∼ = A2 (F, Q ), π → ι π.

The Langlands correspondence thus gives a bijection G2 (F, Q ) ∼ = A2 (F, Q ). However, such a bijection is not canonical, in that it depends on the choice of ι. Put another way, let Aut C denote the group of (not necessarily continuous) automorphisms of the ﬁeld C. The constructions of 32.1, 32.2 deﬁne actions of Aut C on the sets G2 (F ), A2 (F ). We denote these ϕ : η −→ ϕ η, for ϕ ∈ Aut C and η ∈ G2 (F ) or A2 (F ). The problem arises from the fact that the Langlands correspondence does not respect this action. To get a canonical correspondence for representations over Q , one has to modify the ideas slightly. Deﬁnition. For ρ ∈ Gss ˜ ∈ Gss 2 (F ), deﬁne ρ 2 (F ) by ρ˜ : x −→ x− 2 ρ(x), 1

x ∈ WF .

For σ = (ρ, V, n) ∈ G2 (F ), deﬁne σ ˜ = (˜ ρ, V, n). Theorem. The map

ΠC : G2 (F ) −→ A2 (F ), σ −→ π(˜ σ ),

(35.1.1)

is a bijection which commutes with the natural actions of Aut C: ΠC (ϕ ρ) = ϕ Π C (ρ),

ρ ∈ G2 (F ), ϕ ∈ Aut C.

Moreover, ΠC is the unique map G2 (F ) → A2 (F ) which satisﬁes L(χΠC (σ), s) = L(χ ⊗ σ, s− 12 ), ε(χΠC (σ), s, ψ) = ε(χ ⊗ σ, s− 12 , ψ),

(35.1.2)

for all characters χ of F × and all ψ ∈ F, ψ = 1. Note that the condition (35.1.2) speciﬁes ΠC uniquely via the Converse Theorem, as in the standard case. We sketch a proof of the theorem in the following paragraphs. Before passing on to that, we record the desired consequence:

35. The -adic Correspondence

223

-adic Langlands Correspondence. There is a unique bijection Π : G2 (F, Q ) −→ A2 (F, Q ), such that Π (ι σ) = ι Π C (σ), for all σ ∈ G2 (F ) and all ﬁeld isomorphisms ι : Q ∼ = C. 35.2. We outline a proof of 35.1 Theorem. We ﬁrst have to investigate the eﬀect of an automorphism ϕ ∈ Aut C on the constructions of §23. We let ϕ act on the rings C((X)), C(X), C(q −s ) etc. via the coeﬃcients, ﬁxing X and q −s . In the deﬁnition of Z(Φ, χ, X), we choose a Haar measure µ∗ on F × for which µ∗ (UF ) ∈ Q. The measures of compact open subgroups of F × then are all rational. The map Φ → ϕ ◦ Φ is an additive automorphism of Cc∞ (F ). For a character χ of F × and Φ ∈ Cc∞ (F ), we then have Z(ϕ ◦ Φ, ϕ χ, X) = ϕ Z(Φ, χ, X). Consequently L(ϕ χ, s) = ϕ L(χ, s).

(35.2.1) Now take ψ ∈ F of level k. The composite ϕ ◦ ψ is again a smooth homomorphism F → C× , and so lies in F. The character ϕ ◦ ψ has level k, and hence ψ, ϕ ◦ ψ give rise to the same self-dual Haar measure on F . However, to compare the Fourier transforms Fψ , Fϕ◦ψ on Cc∞ (F ), we have to remember that this self-dual measure µψ = µϕ◦ψ does not necessarily take rational values on compact open subgroups. Indeed, µψ is rational on compact open subgroups √ if and only if q is a square or if k is even. Otherwise, these values lie in q Q. We deduce Fϕ◦ψ (ϕ ◦ Φ) = α(ϕ, ψ) · ϕ ◦ (Fψ (Φ)), where

Φ ∈ Cc∞ (F ),

√ √ α(ϕ, ψ) = (ϕ q/ q)k .

It follows that ε(ϕ χ, s, ϕ ◦ ψ) = α(ϕ, ψ) ϕε(χ, s, ψ). 35.3. The deﬁnitions of L(σ, s), σ ∈ Gn (F ), give L(ϕ σ, s) = ϕ L(σ, s). The machinery of induction constants, especially the uniqueness properties, implies ε(ϕ σ, s, ϕ ◦ ψ) = α(ϕ, ψ)dim σ · ϕ ε(σ, s, ψ). Therefore ε(ϕ σ, s, ϕ ◦ ψ) = ϕ ε(σ, s, ψ),

σ ∈ G2 (F ).

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8 The Langlands Correspondence

35.4. For π ∈ A2 (F ), f ∈ C(π) and Φ ∈ Cc∞ (A), we similarly get Z(ϕ ◦ Φ, ϕf , X) = ϕ Z(Φ, f, X), and L(ϕ π, s− 12 ) = ϕ L(π, s− 12 ). In this case, the self-dual Haar measure µA ψ on A, relative to any non-trivial ψ ∈ F , takes rational values on compact open subgroups of A. In notation analogous to that of 35.2, we therefore have Fϕ◦ψ (ϕ ◦ Φ) = ϕ ◦ Fψ (Φ),

Φ ∈ Cc∞ (A),

which implies ε(ϕ π, s− 12 , ϕ ◦ ψ) = ϕ ε(π, s− 12 , ψ). The theorem now follows from the characterization (35.1.2).

Further reading. We brieﬂy discuss the Langlands Correspondence in higher dimensions. To state it precisely, one needs the L-function and the local constant of a pair of irreducible smooth representations πi of GLni (F ), i = 1, 2. These are denoted L(π1 × π2 , s), ε(π1 × π2 , s, ψ) respectively. As before, the L-function is of the form Pπ1 ×π2 (q −s )−1 , where Pπ1 ×π2 (t) is a polynomial with constant term 1, and ε(π1 × π2 , s, ψ) is a monomial in q −s . The original account is in [48], but see also [79]. The crucial property is that, if π is an irreducible cuspidal representation of GLn (F ) (notation: π ∈ A0n (F )), then π is determined, up to equivalence, by the function σ → ε(σ × π, s, ψ), as σ ranges over the sets A0m (F ), m n−1 [42]. If dim σ = 1, the deﬁnition gives ε(σ × π, s, ψ) = ε(σπ, s, ψ) in the sense we use. The Langlands Conjecture states that, for each n 1, there is a unique bijection π n : G0n (F ) → A0n (F ) such that ε(ρ1 ⊗ ρ2 , s, ψ) = ε(π1 × π2 , s, ψ), for all ρi ∈ G0ni (F ), πi = π ni (ρi ). As we have noted in the Introduction, the conjecture is proved in all dimensions: the positive characteristic case is in [58], the characteristic zero case in [38] (or [43], building on [37]). The technique of §34 can likewise be used to give an explicit account of the Langlands correspondence in dimension n ≡ 0 (mod p): see [17], [18] (but the results are not complete at the time of writing). The considerations of §35 likewise apply in higher dimension: see [15].

9 The Weil Representation

36. 37. 38. 39. 40.

Whittaker and Kirillov models Manifestation of the local constant A metaplectic representation The Weil representation A partial correspondence

Having proved the existence of the Langlands correspondence in the “tame” case p = 2 (34.4), we now need only concern ourselves with the case where the residual characteristic p of F is 2. To get started on this, however, we use a general construction valid for all p, and there is no advantage in imposing any restrictions. In fact, there is some overlap between the implications of this chapter and what we have done already. We get another deﬁnition of the Langlands correspondence when p is odd, and another proof of the Converse Theorem (27.1) for cuspidal representations. However, much of what one approach yields easily is quite diﬃcult in the other, and it seems preferable to have both. We ﬁrst return to the mirabolic subgroup M = SN of G, encountered in §8, Chapter III. The reason is that, if π is an irreducible cuspidal representation of G, the restriction of π to M is the unique irreducible smooth representation of M of inﬁnite dimension. This can be realized, in an obvious way, on the space Cc∞ (F × ). Thus Cc∞ (F × ) carries a representation of G, equivalent to π and restricting to M in the speciﬁed fashion. Via the Bruhat decomposition in G, the cuspidal representation π is determined by its central character ωπ ∞ × and operator π(w) on Cc (F ), where w is the standard Weyl element 0 1the −1 0 . The miracle is that the operator π(w) can be described in terms of the local constants ε(χπ, s, ψ), as χ ranges over the characters of F × .

226

9 The Weil Representation

There is a parallel strand. One starts with a separable quadratic extension E/F and constructs, from E × and G, a group G = GE/F . This is a locally proﬁnite group, which has a subgroup structure and a Bruhat decomposition analogous to those of G. One uses this to construct a standard representation of G on the space Cc∞ (E): the process imitates the action of G on Cc∞ (F × ) given by a cuspidal representation of G, but the Weyl element w acts as a Galois-twisted Fourier transform. The compact group KE/F = Ker NE/F is a central subgroup of G, so one can decompose the standard representation according to characters of KE/F . If Θ is a character of E × , non-trivial on KE/F , the component of the standard representation corresponding to Θ | KE/F gives rise to an irreducible cuspidal representation πE/F (Θ) of G. This is the Weil representation deﬁned by the datum (E/F, Θ). Its detailed structure is accessible to the methods of the ﬁrst part of the chapter. The hypothesis Θ | KE/F = 1 is equivalent to the representation ρΘ = IndE/F Θ of WF being irreducible. The process ρΘ → πE/F (Θ) then gives a well-deﬁned map from a subset of G02 (F ) to A02 (F ), with the properties demanded of a Langlands correspondence. When the residual characteristic p of F is odd, it is deﬁned on all of G02 (F ) and is the Langlands correspondence. (In light of the results of 34.4, we do not give the complete argument.) In the more diﬃcult case p = 2, the existence of this “partial correspondence” is the starting point for the work of the later chapters XI and XII. It is indeed only the existence which matters for that. There will be no reason to re-visit the methods of this chapter: we give a self-contained statement of the outcome in §40. Our discussion of GE/F and its representations is an adaptation of a small part of an extensive general theory. We say nothing of this, beyond allowing the broader context to inﬂuence the choice of vocabulary.

36. Whittaker and Kirillov Models Let ϑ be a non-trivial character of the group N of upper triangular unipotent matrices in G = GL2 (F ). 36.1. Let (π, V ) be an irreducible smooth representation of G; we consider the space Vϑ , as in 8.1. If dim V = 1, then π = φ ◦ det, for some character φ of F × . Clearly, Vϑ = 0 in this case. Otherwise: Theorem. Let ϑ be a non-trivial character of N . If (π, V ) is an irreducible smooth representation of G of inﬁnite dimension, then dim Vϑ = 1.

36. Whittaker and Kirillov Models

227

Before proving this result, we derive some consequences. Frobenius Reciprocity (2.4) gives canonical isomorphisms G ∼ HomN (Vϑ , ϑ) ∼ = HomM (V, IndM N ϑ) = HomG (V, IndN ϑ).

(36.1.1)

The theorem asserts that the ﬁrst of these spaces has dimension one, so we deduce: Corollary 1. Let (π, V ) be an irreducible smooth representation of G of inﬁnite dimension. There is a non-zero G-homomorphism φ : V → IndG N ϑ which is, moreover, uniquely determined up to a constant scalar factor. The map φ is injective, since π is irreducible. It follows that there is a unique G-subspace W (π, ϑ) of IndG N ϑ which is isomorphic to π. It is called the Whittaker model of π. The other isomorphism in (36.1.1) yields the ﬁrst assertion of: Corollary 2. Let (π, V ) be an irreducible smooth representation of G of inﬁnite dimension. (1) There is a non-zero M -homomorphism f : V → IndM N ϑ which is, moreover, uniquely determined up to a constant scalar factor. (2) The map f is injective. (3) The image of f contains c-IndM N ϑ, and f induces an isomorphism VN ∼ = Im f /c-IndM N ϑ. In particular, π is cuspidal if and only if f (V ) = c-IndM N ϑ. Proof. Part (1) has already been noted. Since dim Vϑ = 1, the composite ∼ map V → IndM N ϑ → ϑ induces an isomorphism Vϑ = ϑ. It follows that (Ker f )ϑ = 0. Therefore (Ker f )(N ) = 0 (8.1 Corollary 1) and so N acts trivially on Ker f . Thus Ker f = 0 (9.2 Exercise 1). Since f is injective, the space f (V (N )) is non-zero, and M f (V (N )) ⊂ (IndM N ϑ)(N ) = c-IndN ϑ

(8.2 Proposition (1)). Since c-IndM N ϑ is irreducible (8.2 Corollary), we deduce that f (V ) ⊃ f (V (N )) = c-IndM N ϑ, Therefore f induces an M -isomorphism VN = V /V (N ) ∼ = f (V )/c-IndM N ϑ, and the ﬁnal assertion follows. Therefore: Corollary-Deﬁnition. Let (π, V ) be an irreducible smooth representation of G of inﬁnite dimension, and let ϑ be a non-trivial character of M .

228

9 The Weil Representation

(1) There is a unique M -subspace K = K(π, ϑ) of IndM N ϑ which is M isomorphic to V . (2) There is a unique homomorphism πK : G → AutC (K) such that (a) πK | M induces the natural action of M on K, and (b) πK ∼ = π. The representation (πK , K(π, ϑ)) is called the Kirillov model of π. Restating Corollary 2, we have K(π, ψ) ⊃ c-IndM N ϑ, with equality if and only if π is cuspidal. The canonical isomorphism of Frobenius Reciprocity M ∼ HomG (V, IndG N ϑ) = HomM (V, IndN ϑ)

is given by restricting functions to M . We deduce: Corollary 3. Restriction of functions from G to M induces an M -isomorphism W (π, ϑ) ∼ = K(π, ϑ), and hence a G-isomorphism W (π, ϑ) ∼ = (πK , K(π, ϑ)). 36.2. We start the proof of 36.1 Theorem with the case of a principal series representation. Let χ be a character of T and consider the induced representation (Σ, X) = IndG B χ. Let W = X(N ) be the kernel of the canonical B-map X → XN : this is the same as the space W = V (N ) in 9.7. The space XN has dimension 2 and N acts trivially, so Wϑ = Xϑ (8.1.3). However, W is irreducible as representation of M (9.7) so W ∼ = c-IndM N ϑ (8.3 Corollary), whence dim Wϑ = dim Xϑ = 1 (8.2 Proposition). Let (π, V ) be a G-composition factor of (Σ, X), of inﬁnite dimension. If π = Σ is irreducible, we have Vϑ = Xϑ and this has dimension 1. Otherwise, Σ has G-composition length 2: one factor is V and the other, call it Y , has dimension 1. Thus Yϑ = 0 and dim Vϑ = dim Xϑ = 1, as required. 36.3. We now assume that (π, V ) is cuspidal. Thus VN = 0, whence (8.1 Corollary 1) Vϑ = 0, for any ϑ = 1. We have to show that dim Vϑ = 1. This assertion is insensitive to twisting by characters of F × . We therefore assume (π) (χπ), for all characters χ of F × . We take a cuspidal type (A, J, Λ) occurring in π; thus π = c-IndG J Λ. Lemma 1. (1) The natural map (T ∩ J)\T → J\G/N is a surjection. (2) Let t ∈ T . The dimension of the space t HomN c-IndN N ∩t−1 Jt (Λ ), ϑ −1

is the multiplicity in Λ of the character ϑt

| N ∩ J.

36. Whittaker and Kirillov Models

229

Proof. In all cases, there is a ﬁeld extension E/F , of degree 2, such that E × ⊂ J. Let B be the group of upper triangular matrices in G, as always. In particular, B = T N and B is the G-stabilizer of the line F e1 , e1 = ( 10 ), in F 2 . Let g ∈ G and consider the element ge1 . The group E × acts transitively on the set of non-zero elements of F 2 , so there exists x ∈ E × such that xge1 = e1 . Therefore xg ∈ B, and we deduce that G = E × B = E × T N . Since J ⊃ E × , it follows that the natural map (T ∩ J)\T −→ J\G/N, t

−→ JtN,

is surjective, as required for (1). In part (2), we note that N ∩ t−1 JT is an open subgroup of N ; the result now follows directly from Frobenius Reciprocity (2.5.2). Working explicitly with the functions in c-Ind Λ, we ﬁnd: g c-IndN π|N = N ∩g −1 Jg Λ , g∈J\G/N

HomG (π, IndG N

ϑ) = HomN (π, ϑ) =

g HomN (c-IndN N ∩g −1 Jg Λ , ϑ).

g∈J\G/N

According to Lemma 1, we can take the coset representatives g from T . For t ∈ T , we have N t t−1 ∼ ). HomN (c-IndN N ∩t−1 Jt Λ , ϑ) = HomN (c-IndN ∩J Λ, ϑ

We therefore have to show: Lemma 2. Let ϑ1 be a non-trivial character of N such that ϑ1 | N ∩ J occurs in Λ. (1) The character ϑ1 | N ∩ J occurs in Λ with multiplicity one. (2) If ϑ2 is a non-trivial character of N such that ϑ2 | N ∩ J occurs in Λ, then ϑ2 is conjugate to ϑ1 under T ∩ J. Proof. We shall use our standard notation j Nj = 1 p , j ∈ Z. 0 1

Suppose ﬁrst that (π) = 0. We may take J = F × GL2 (o). We have N ∩ J = N0 . The restriction of Λ to GL2 (o) is the inﬂation of an irreducible cuspidal representation of GL2 (k). It follows (cf. (6.4.1)) that Λ | N0 is the direct sum of the non-trivial characters of N0 which are trivial on N1 , each occurring with multiplicity one. Any two characters of N which are trivial on N1 , but

230

9 The Weil Representation

not on N0 , are conjugate under T ∩ GL2 (o) (cf. 1.7), and the result follows in this case. We therefore assume (π) > 0, and ﬁx a character ψ ∈ F of level one. There is a simple stratum (A, n, α), n 1, such that Λ ∈ C(ψα , A). Setting [(n+1)/2] . We will work explicitly, so we assume E = F [α], we have J = E × UA that A = M or I. Lemma 3. [(n+1)/2]

. (1) We have N ∩ J = N ∩ UA (2) Let ϑ be a character of N such that ϑ | N ∩ J occurs in Λ | N ∩ J. [n/2]+1 [n/2]+1 = ψα | UA . Then ϑ | N ∩ UA (3) The restriction of ψα to UAn ∩ N is non-trivial. [n/2]+1 (4) Any two characters of N , agreeing with ψα on N ∩ UA , are con[(n+1)/2] jugate under T ∩ UA ⊂ T ∩ J. Proof. Write P = rad A. Any unipotent element of J is contained in UA ∩ [(n+1)/2] . Writing N = 1+N , let x ∈ N ∩ (oE +Pm ), where m = J = UE UA [(n + 1)/2]. We accordingly write x = y+z, with y ∈ oE , z ∈ Pm . We choose integers j, k maximal for the properties y ∈ pjE , z ∈ Pk . If j k, there is nothing to do. Otherwise, j < k and x2 = 0 ∈ P2j P1+2j , which is nonsense. This proves (1). [n/2]+1 is a Assertion (2) simply re-iterates the deﬁning fact that Λ | UA multiple of ψα , and part (3) is given by the explicit form of simple elements. j = Nj and N ∩ UIj = N[j/2] . Part (4) If j 0 is an integer, we have N ∩ UM now follows from a simple calculation. Lemma 3 reduces us to checking that, if ϑ is a character of N , then ϑ | J ∩ N occurs in Λ | J ∩ N with multiplicity at most one. If dim Λ = 1, this is trivial. We therefore assume that n = 2m is even, so A = M. We have dim Λ = q, m , and N ∩ J = Nm . If θ is a character of Nm occurring in Λ | Nm , J = E × UM m , all of which occur in Λ | J ∩ N . there are q distinct conjugates θx , x ∈ T ∩ UM The restriction Λ | J ∩ N is therefore multiplicity-free. This completes the proof of 36.1 Theorem.

37. Manifestation of the Local Constant Let (π, V ) be an irreducible cuspidal representation of G. In this section, we give a description of the local constant ε(π, s, ψ) of π (24.2) in terms of the Kirillov model of π. The arguments can be extended to apply to any irreducible smooth representation (π, V ) of inﬁnite dimension, but we shall have no need for the non-cuspidal case.

37. Manifestation of the Local Constant

231

37.1. Let ψ ∈ F, ψ = 1, and deﬁne a character ϑ of N by ψ(x) = ϑ 10 x1 , x ∈ F. We henceforward write W (π, ψ) = W (π, ϑ),

K(π, ψ) = K(π, ϑ).

Since (π, V ) is cuspidal, we have K(π, ψ) = c-IndM N ϑ. As in 8.2 Gloss, restriction of functions to the group x0 × ∼ : x ∈ F S= = F× 0 1 induces a linear isomorphism c-Ind ϑ ∼ = Cc∞ (F × ). Combining with 36.1 Corollary 3, we have linear isomorphisms W (π, ψ) ∼ = K(π, ψ) = c-Ind ϑ ∼ = Cc∞ (F × ). The composite map transfers the actions of G on W (π, ψ), K(π, ψ) to one on Cc∞ (F × ). This is isomorphic to π (or πK ), so we continue to denote it πK : G −→ AutC (Cc∞ (F × )). Remark. One often also refers to the representation (πK , Cc∞ (F × )) as the Kirillov model of π. This version of the representation πK satisﬁes the following identities. If φ ∈ Cc∞ (F × ), a, b ∈ F × and x ∈ F , then πK a0 01 φ : b −→ φ(ab), (37.1.1) πK 10 x1 φ : b −→ ψ(bx)φ(b), πK a0 a0 φ = ωπ (a)φ. The Bruhat decomposition (5.2) G = B ∪ BwN,

w=

0 1 −1 0

,

taken together with (37.1.1), shows that the operators πK (g), g ∈ G, can be expressed purely in terms of πK (w) and πK | B. The representation πK is therefore completely determined by ωπ and the operator πK (w) on Cc∞ (F × ). We will need to know how the Kirillov model behaves with respect to twisting:

232

9 The Weil Representation

Lemma. Let π be an irreducible cuspidal representation of G. Let χ be a character of F × and deﬁne tχ : Cc∞ (F × ) −→ Cc∞ (F × ), f

−→ χ−1 f.

The map tχ is a G-isomorphism (χπ)K → χπK . Proof. We deﬁne a map tχ on IndG N ϑ in the analogous way. Let πW denote the natural representation of G on the function space W (π, ψ). For f ∈ W (χπ, ψ), tχ ◦ (χπ)W (g)f : x −→ χ−1 (det x)f (xg) = χ(det g) πW (g) ◦ tχ f (x). That is, tχ ◦ (χπ)W = χπW ◦ tχ . The lemma follows on restricting functions, as in 36.1 Corollary 3. 37.2. We show how to express the operator πK (w) in terms of local constants. We have to re-interpret the discussion of 24.4 et seq. in the present context. Set A = M2 (F ). For m ∈ Z, let Gm be the set of g ∈ G such that υF (det g) = m. As in (24.4.3), we put

zm (Φ, f ) = Φ(x)f (x) d∗ x, Gm

where Φ ∈ Cc∞ (A), f ∈ C(π), and d∗ x is a ﬁxed Haar measure on G. The series zm (Φ, f ) X m Z(Φ, f, X) = m∈Z

is then a Laurent polynomial in X (24.5 Lemma (3)). By 24.5 Corollary, 24.2 Corollary, there is a monomial c(π, X, ψ) = c(π, ψ)X n(π,ψ) such that ˆ fˇ, 1/q 2 X) = c(π, X, ψ) Z(Φ, f, X). Z(Φ,

(37.2.1)

We have c(π, q − 2 −s , ψ) = ε(π, s, ψ) (24.6.3) . 1

37.3. For a character χ of F × and k ∈ Z, we deﬁne a function ξχ,k ∈ Cc∞ (F × ) by χ(x) if υF (x) = k, ξχ,k (x) = 0 otherwise. The set of functions ξχ,k provides a basis of Cc∞ (F × ). Theorem. Let χ be a character of F × and n ∈ Z; then πK (w) ξχ,k = c(χ−1 π, q −1 , ψ) ξχ−1 ωπ ,−n(χ−1 π,ψ)−k .

(37.3.1)

37. Manifestation of the Local Constant

233

The proof of the theorem will occupy the rest of the section. Remark. The equation (37.3.1) can be written in the form πK (w) ξχ,k = ε(χ−1 π, 12 , ψ) ξχ−1 ωπ ,−n(χ−1 π,ψ)−k .

(37.3.2)

Once one knows that the local constants ε(χπ, s, ψ) determine ωπ (from, for instance, 27.4 Lemma), the theorem yields another proof of the Converse Theorem 27.1 for cuspidal representations. 37.4. To prove the theorem, we apply the functional equation (37.2.1) to carefully chosen functions Φ and f . Let H be a compact open subgroup of G. The map

W (h) d∗ h, W ∈ W (π, ψ), W −→ H

is a smooth linear form. So, for W ∈ W (π, ψ), the function

W (hg) d∗ h, g ∈ G, fW,H : g −→ µ∗ (H)−1 H

is a coeﬃcient of π. For W ∈ W (π, ψ) and Φ ∈ Cc∞ (A), we choose a compact open subgroup H of G which ﬁxes Φ under left and right translation, and set zm (Φ, fW,H ) X m . Z(Φ, W, X) = Z(Φ, fW,H , X) = m∈Z

In particular,

zm (Φ, fW,H ) =

Φ(x)W (x) d∗ x.

Gm

This deﬁnition is independent of the choice of H. We can perform analogous constructions using the function W ∨ : g → W (g −1 ) in place of W . In these terms, the functional equation (37.2.1) reads ˆ W ∨ , 1/q 2 X) = c(π, X, ψ) Z(Φ, W, X). Z(Φ,

(37.4.1)

37.5. Let ψ ∈ F have level ν, so that ψ is trivial on pν but not on pν−1 . Let n be a positive integer (to be chosen “suﬃciently large”), and let Φn be the function deﬁned by

ν−n ν−n p p supp Φn = Sn = , pn 1+pn

and Φn

a b c d

The function Φn lies in Cc∞ (A).

¯ = ψ(b),

a b c d

∈ Sn .

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9 The Weil Representation

Let N denote the group of lower triangular unipotent matrices in G, and consider the “big cell” a b ∈ G : d = 0 . BN = c d The cell BN is the image of F × F × × F × × F under the bijective map a+ty t 1 0 a0 =z , (37.5.1) (t, z, a, y) −→ z 10 1t y 1 y 1 0 1 Further, Sn ∩ G is contained in BN and is the image (under (37.5.1)) of the set pν−n × UFn × (pν−n ∩ F × ) × pn . For x ∈ Sn ∩ G, written according to the decomposition (37.5.1), we have ¯ ¯ = ψ(t). Φn (x) = ψ(zt) n . For x ∈ Sn ∩ G, written in the form Let W ∈ W (π, ψ) be ﬁxed by Kn = UM (37.5.1), we have W (x) = ψ(t) W a0 10 , Φn (x)W (x) = W a0 10 .

Corresponding to (37.5.1), the Haar measure d∗ x on BN decomposes thus: d∗ x = a −1 dµ(t) dµ× (z) dµ× (a) dµ(y), for Haar measures µ on F and µ× on F × (cf. 7.6 Exercises). We compute

qm X m W a0 10 dµ× (a), Z(Φn , W, X) = k mν−n

m UF

where is a prime element of F and k = µ(pν−n ) µ(pn ) µ× (UFn ). ˆn . This has support 37.6. We consider the Fourier transform Φ pn pν−n , Σn = 1+pn pν−n and ˆn Φ

a b c d

= ψ(d),

a b c d

∈ Σn .

(37.5.2)

37. Manifestation of the Local Constant

We have to compute Z(Φˆn , W ∨ , X) =

X

m

235

ˆn (x)W (x−1 ) d∗ x. Φ

Gm

m∈Z

The set G ∩ Σn is the image of the map pν−n × UFn × (pν−n ∩ F × ) × pn −→ G, −y −a−ty 1 0 1 0 1 t (t, z, a, y) −→ w−1 z y 1 = z . 0a 01 1 t ˆn (x) = ψ(t), while For such an element x of Σn ∩ G, we have Φ 0 ¯ ωπ (a)−1 W (( a 0 ) w) , ¯ W 1 −1 w = ψ(t) W (x−1 ) = ψ(t) 0 1 0a giving

ˆn (x)W (x−1 ) = ωπ (a)−1 W (( a 0 ) w) . Φ 0 1

Therefore

ˆn , W ∨ , X) = k Z(Φ

m

q X

m m UF

mν−n

ωπ (a)−1 W (( a0 10 ) w) dµ× (a),

with the same k as before (37.5.2). 37.7. For W ∈ W (π, ψ), we put

Xm Ψ(W, X) = m UF

m∈Z

Ψ(W, X) =

m∈Z

X

m m UF

W ( a0 10 ) dµ× (a), ωπ (a)−1 W ( a0 10 ) dµ× (a).

Since the function a → W ( a0 10 ) has compact support in F × , the expressions Ψ(W, X), Ψ(W, X) are Laurent polynomials in X. For n suﬃciently large, we have Z(Φn , W, X) = k Ψ(W, qX), ˆn , W ∨ , X) = k Ψ(π(w)W, qX). Z(Φ The functional equation (37.4.1) therefore reduces to Ψ(π(w)W, X) = c(π, 1/qX, ψ) Ψ(W, 1/X).

(37.7.1)

We now take W so that W ( a0 10 ) = ξ1,m (a), for some integer m, where 1 denotes the trivial character of F × . Then Ψ(W, X) = µ× (UF ) X m .

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9 The Weil Representation

We deduce from (37.7.1) that Ψ(π(w)W, X) = µ∗ (UF ) c(π, 1/q, ψ) X −m−n(π,ψ) . (37.7.2) Under the action of the group U0F 10 , it is clear from the choice of W that a −→ π(w)W ( a0 10 ) transforms according to the character ωπ . It follows that πK (w)ξ1,m is a linear combination of the functions ξωπ ,k , k ∈ Z. The equation (37.7.2) implies πK (w) ξ1,m = c(π, 1/q, ψ) ξωπ ,−n(π,ψ)−m , and the theorem follows in the case χ = 1. Applying this special case to the representation χ−1 π, we get (χ−1 π)K (w) ξ1,m = c(χ−1 π, 1/q, ψ) ξχ−2 ωπ ,−n(χ−1 π,ψ)−m . The theorem now follows from 37.1 Lemma.

38. A Metaplectic Representation Let E/F be a separable quadratic extension, with Gal(E/F ) = {1, σ}. In this section, we ﬁrst attach to E/F a locally proﬁnite group G = GE/F . We then ﬁx a character ψ ∈ F, ψ = 1. We construct a smooth representation ηψG of the group GE/F on the vector space Cc∞ (E). 38.1. Let κ = κE/F be the character of F × with kernel NE/F (E × ) and let Gκ denote the kernel of the map g → κ(det g), g ∈ G. Let K = KE/F = Ker NE/F ⊂ UE . We deﬁne G = GE/F = (g, h) : g ∈ G, h ∈ E × , det g = NE/F (h)−1 . This is a closed subgroup of G × E × , and so is locally proﬁnite. Projecting on the two factors, we get exact sequences 1 → K −→ G −→ Gκ → 1, 1 → SL2 (F ) −→ G −→ E × → 1.

38. A Metaplectic Representation

237

38.2. The group G = GE/F has distinguished subgroups similar to those of G. It has particular families of elements: n(t) = (( 10 1t ) , 1) , NE/F (x) 0 −σ , x , a(x) = 0 1 ζ 0 −1 , z(ζ) = 0 ζ ,ζ

t ∈ F, x ∈ E×,

(38.2.1)

×

ζ∈F .

(Here, x−σ means (x−1 )σ = (xσ )−1 .) The sets N = {n(t) : t ∈ F },

A = {a(x) : x ∈ E × },

Z = {z(ζ) : ζ ∈ F × },

are then subgroups of G. The set ZA is also a group; if we let A1 be the group of elements a(x), x ∈ K, then A1 ∼ = K and ZA1 is the centre of G. The product map Z × A × N −→ G is injective. Its image, denoted B, is given by B = ZAN = {(b, x) ∈ B × E × : det(b) NE/F (x) = 1} ; B is a closed subgroup of G. We write w=

0 1 −1 0

.

Thus (w, 1) ∈ G. We also denote this element w. The Bruhat decomposition for G implies: Lemma. The group G is the disjoint union G = B ∪ BwN . The map B × N −→ BwN , (b, n) −→ bwn, is a bijection. 38.3. The representation theory of the group G provides a link between representations of Gκ and characters of E × . Let (π, V ) be an irreducible smooth representation of the group G = GE/F . Schur’s Lemma (2.6) applies, so the centre ZA1 of G acts on V via a character θ. The restriction of θ to A1 = K can be extended to a character Θ of E × . The composition of Θ with the projection G → E × deﬁnes a character ΘG of G. Consider the representation ΘG−1 ⊗ π. This is an irreducible representation of G, in which A1 = K acts trivially. Thus ΘG−1 ⊗ π is the inﬂation of an irreducible smooth representation π0 of G/K = Gκ . That is,

238

9 The Weil Representation

Proposition. (1) Let π be an irreducible smooth representation of G. There is an irreducible smooth representation π0 of Gκ and a character Θ of E × such that π is equivalent to the representation π0 Θ : (g, x) −→ Θ(x)π0 (g),

(g, x) ∈ G ⊂ G × E × .

(2) For i = 1, 2, let πi be an irreducible smooth representation of Gκ and Θi a character of E × ; then π1 Θ1 ∼ = π2 Θ2 if and only if there exists a × character χ of F such that Θ2 = Θ1 ⊗ χE and π2 is equivalent to the representation g → χ(det g)−1 π1 (g), g ∈ Gκ . In (2) we have used the notation χE = χ ◦ NE/F . 38.4. We ﬁx a character ψ ∈ F, ψ = 1, and use it to deﬁne a representation ηψB of B on Cc∞ (E). For f ∈ Cc∞ (E) and y ∈ E, we set: n(t)f : y −→ ψ t NE/F (y) f (y),

t ∈ F,

a(x)f : y −→ f (xy),

x ∈ E×,

z(ζ)f : y −→ f (y),

(38.4.1)

×

ζ∈F .

Proposition. There is a unique representation ηψB of B on Cc∞ (E) satisfying the relations (38.4.1). The representation ηψB is smooth. Proof. In each case b = n(t), a(x), z(ζ), the function bf lies in Cc∞ (E). The only non-trivial relation among the generators (38.2.1) is a(x) n(t) a(x)−1 = n(t NE/F (x)),

t ∈ F, x ∈ E × .

To show that (38.4.1) deﬁnes a representation of B, we therefore need only check that a(x) n(t) a(x)−1 f : y −→ ψ(t NE/F (xy))f (y),

f ∈ Cc∞ (E).

This is straightforward. Let f ∈ Cc∞ (E); then f is surely ﬁxed, under the representation ηψB , by Z and an open subgroup Af of A. Further, since f has compact support, it is ﬁxed by an open subgroup Nf of N . The set ZAf Nf is open in B, and it generates an open subgroup of B which ﬁxes f . It follows that ηψB is smooth. 38.5. We consider the general problem of extending representations of B to representations of G. The Bruhat decomposition (38.2 Lemma) gives:

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239

Lemma. For i = 1, 2, let (ρi , V ) be a representation of G, and suppose that ρ1 (b) = ρ2 (b) for all b ∈ B. Suppose also that ρ1 (w) = ρ2 (w). We then have ρ1 = ρ2 . Proposition. Let (ρ, V ) be a representation of B, and let A ∈ AutC (V ). The following are equivalent: (1) there is a representation (˜ ρ, V ) of G, satisfying ρ˜(b) = ρ(b), b ∈ B, and ρ˜(w) = A; (2) the operator A satisﬁes A2 = ρ z(−1)a(−1) , A ρ z(ζ) = ρ z(ζ) A, ζ ∈ F ×, (38.5.1) x ∈ E×, A ρ a(x) = ρ z(NE/F x)a(x−σ ) A, A ρ n(t) A = ρ z(−t)a(−t−1 )n(−t) A ρ n(−t−1 ) , t ∈ F × . If ρ is smooth and these conditions hold, then ρ˜ is smooth. Proof. In G, we have the relations w2 = z(−1)a(−1), w z(ζ) = z(ζ) w,

ζ ∈ F ×,

w a(x) = z(NE/F x) a(x−σ ) w,

x ∈ E×,

w n(t) w = z(−t) a(−t−1 ) n(−t) w n(−t−1 ),

t ∈ F ×.

The Bruhat decomposition shows that these relations, taken together with the obvious relations inside B, give a presentation of G. The ﬁrst assertion follows immediately. For the second, take v ∈ V . There is an open subgroup H of B which ﬁxes both v and Av. Thus v is ﬁxed by the subgroup of G generated by H and wHw−1 . This subgroup is open, so ρ˜ is smooth, as required. 38.6. We now construct a representation ηψG of the group G = GE/F on Cc∞ (E). We set ψE = ψ ◦ TrE/F , and write f → fˆ for the Fourier transform operation on Cc∞ (E) deﬁned using the Haar measure on E which is self-dual with respect to ψE . We recall that κ = κE/F is the character of F × with kernel NE/F (E × ). Let λE/F (ψ) be the Langlands constant, as in 30.4. Theorem. There is a unique representation η = ηψG of G on Cc∞ (E) satisfying: η n(t) f : y −→ ψ t NE/F (y) f (y), t ∈ F, 1/2 x ∈ E×, η a(x) f : y −→ x E f (xy), (38.6.1) η z(ζ) f : y −→ κ(ζ)f (y), ζ ∈ F ×, η(w)f : y −→ λE/F (ψ)fˆ(y σ ), for f ∈ Cc∞ (E) and y ∈ E. The representation ηψG is smooth.

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9 The Weil Representation

Proof. The ﬁrst three relations in (38.6.1) deﬁne a representation η of B which is the twist, by a character, of the representation ηψB of 38.4 Proposition. Explicitly, 1/2 a b , x ∈ B. η(g) = κ(d) ad−1 F ηψB (g), g = 0 d It is therefore enough to check the relations (38.5.1) for this representation η and the operator A deﬁned by Af : y −→ λE/F (ψ) fˆ(y σ ),

f ∈ Cc∞ (E), y ∈ E.

For f ∈ Cc∞ (E), the function A2 f is that given by y −→ λE/F (ψ)2 f (−y) = κ(−1)f (−y) = η z(−1)a(−1) f (y), as required for the ﬁrst relation in (38.5.1). The second relation is immediate, so we turn to the third. For f ∈ Cc∞ (E) and t ∈ E × , let f t temporarily denote the function y → f (ty). We then have ˆ −1 y). ft (y) = t −1 E f (t This gives us −1/2 Aη a(x) f : y −→ λ(E/F, ψ) x E fˆ(x−1 y σ ) while, on the other hand, −1/2 η z(NE/F x)a(x−σ ) Af : y −→ λ(E/F, ψ) x E fˆ(x−1 y σ ), since κ(NE/F x) = 1. This proves the third relation in (38.5.1). 38.7. The fourth relation in (38.5.1) is a consequence of a property of the Fourier transform f → fˆ on Cc∞ (E). This is a sort of twisted Fourier inversion formula, which we now develop. Set q = ψ ◦ NE/F : E −→ C× and x, y = ψE (xy σ ),

x, y ∈ E.

These functions satisfy the relation q(x+y) = q(x) q(y) x, y,

x, y ∈ E.

We also abbreviate [f ](y) = fˆ(y σ ),

f ∈ Cc∞ (E), y ∈ E.

The function q is locally constant on E, so qf ∈ Cc∞ (E) for any f ∈ Cc∞ (E).

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241

Proposition. For f ∈ Cc∞ (E) and a Haar measure dx on E, we have

q(x) [f ](x) dx = λE/F (ψ) q(x)−1 f (x) dx. E

E

Proof. The assertion is independent of the choice of Haar measure dx, so we take dx self-dual with respect to ψE . We ﬁrst prove an approximate version of the proposition. Lemma. There is a constant γ = γE/F (ψ) ∈ C× such that

q(x) [f ](x) dx = γ q(x)−1 f (x) dx, E

for f ∈

E

Cc∞ (E).

Proof. We deﬁne a linear form I on Cc∞ (E) by

q(x) [qf ](x) dx. I(f ) = E

We ﬁx y ∈ E and consider the function h = fy : x → f (x+y). We have

qh(t)t, x dt [qh](x) =

E = q(t)f (t+y)t, x dt E

= q(t−y)f (t)t−y, x dt, E

so

q(x) [qh](x) =

q(x)q(t−y)f (t)t−y, x dt = q(x+t−y)f (t) dt E

= q(x−y) q(t)f (t)x−y, t dt = q(x−y) [qf ](x−y). E

E

Integrating over x ∈ E, we get I(fy ) = I(f ), so I is a scalar multiple of Haar measure. The map f → q [qf ] is bijective on Cc∞ (E), so the functional I is nonzero and the lemma is proved. The proof of the proposition continues in the next paragraph. 38.8. It remains to show that γE/F (ψ) = λE/F (ψ). We ﬁrst observe:

(38.8.1)

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9 The Weil Representation

Lemma. The constant γ = γE/F (ψ) satisﬁes γ¯ γ = 1. Proof. A simple calculation shows that, for f ∈ Cc∞ (E), we have [f ](y) = f¯ (−y). Also, ¯q = q−1 and q(−x) = q(x). So, applying complex conjugation to the deﬁning relation for γ, we get (in abbreviated notation)

q−1 f¯ = γ¯ qf¯ = γ −1 qf¯, E

so γ¯ = γ

−1

E

E

, as required.

The Langlands constant λE/F (ψ) is a fourth root of unity (30.4.3), so also |λE/F (ψ)| = 1. It is therefore enough to prove that γE/F (ψ) = c λE/F (ψ), for some c > 0. In the following argument, c denotes a positive real number, varying from line to line. To compute γ, we let f be the characteristic function of pdE , for a large positive integer d. We let µ be the level of ψE . We have

q−1 f (x) dx = c, E

so

γ=c

ψ(NE/F x) dx

pµ−d E

=c

mµ−d

mU E E

ψ(NE/F x) x E d× x,

for a prime element E of E and a Haar measure d× x on E × . The integrand is constant on cosets of KE/F . Since x E = NE/F x F , we get

γ=c ψ(y) y F d× y mµ−d

=c

mk

mU ) NE/F ( E E

mU F F

(1+κ)(y) ψ(y) y F d× y,

where k = (µ−d)f (E|F ), F is a prime element of F , and d× y is a Haar measure on F × . The integral

ψ(y) y F d× y = ψ(y) dy mk

mU F F

pk

38. A Metaplectic Representation

243

is zero for k < ν, where ν is the level of ψ. It is therefore zero for d suﬃciently large. This leaves

κ(y)ψ(y) dy.

γ=c

(38.8.2)

pk

When E/F is ramiﬁed, the character κ is ramiﬁed and the integral gives the Gauss sum formula for ε(κ, 1, ψ) (23.5 Exercise). Since ε(κ, 1, ψ) = c λE/F (ψ), we have the result in this case. Suppose, therefore, that E/F is unramiﬁed. The integral (38.8.2) is then c ζ(Ψˆ , κ, 1), where Ψ is the characteristic function of 1+pµ−k . The function ζ(Ψ, κ, s) is a positive constant (for d suﬃciently large). The L-function L(κ, s) is (1+q −s )−1 , and takes only positive values on the real axis. The functional equation (23.4.4) therefore gives γ = c ε(κ, 1, ψ) = c λE/F (ψ). This proves (38.8.1) and completes the proof of 38.7 Proposition. Corollary. Let f ∈ Cc∞ (E) and put g = q [f ], h = q−1 f . We then have [g](y) = λE/F (ψ) q−1 (y) [h](−y),

y ∈ E.

Proof. When y = 0, this is given by the proposition. In general, we apply the proposition to the function f−y : x → f (x−y), and note that

[g](y) = q [f−y ], E

[h](−y) = q(y) q−1 f−y . E

38.9. We apply 38.8 Corollary to the task of verifying the fourth identity in (38.5.1) for the operator A : f → λE/F (ψ) [f ]. In this paragraph, we treat the case t = 1. Taking f ∈ Cc∞ (E), g = q [f ], h = q−1 f , and abbreviating λ = λE/F (ψ), we have η n(1) Af = λq [f ] = λg, and so

Aη n(1) Af = λ2 [g] = κ(−1) [g]. On the other hand, η n(−1) f = q−1 f = h and Aη n(−1) f = λ [h], so that η z(−1)a(−1)n(−1) Aη n(−1) f (y) = κ(−1) λ q−1 [h](−y)

which, by 38.8 Corollary, is κ(−1) [g](y), as required.

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9 The Weil Representation

38.10. The general case of the relation will follow from a variant of the same argument. For f ∈ Cc∞ (E), we have Af = λ [f ], so η n(t) Af = λqt [f ], where qt denotes the function y → ψ(t NE/F (y)). Thus A η n(t) Af = λ2 [qt [f ]] = κ(−1) [qt [f ]]. On the other hand, η n(−t−1 ) f = q−t−1 f , so A η n(−t−1 ) f = λ [q−t−1 f ]. and η z(−t)a(−t−1 )n(−t) A η n(−t−1 ) f : −1/2

y −→ κ(−t) λ t E

q−t (−t−1 y) [q−t−1 f ](−t−1 y).

We therefore want to prove −1/2

[qt [f ]](y) = κ(t) λ t E

q−t (−t−1 y) [q−t−1 f ](−t−1 y).

We ﬁx t ∈ F × and consider the character tψ : x → ψ(tx) in place of ψ. We denote by F the normalized Fourier transform on Cc∞ (E) relative to the character tψ ◦ TrE/F . Thus 1/2 F(f )(y) = t E fˆ(ty).

) as the function y → F(f )(y σ ), we get Deﬁning F(f )(y) = t 1/2 [f ](ty). F(f E for ϕ ∈ Cc∞ (E), we obtain We apply 38.8 Corollary to tψ and F; −1 F qt F(ϕ) (y) = λ(E/F, tψ) q−1 t (y) F qt ϕ (−y). We set [ϕ]t (y) = [ϕ](ty). Since λ(E/F, tψ) = κ(t)λ(E/F, ψ), this last relation reads F qt [ϕ]t (y) = κ(t) λ q−t (−y) [q−t ϕ]t (−y). We deﬁne ϕ by ϕ(y) = t E f (ty), so that [ϕ]t = [f ]. We get 1/2

t E [qt [f ]](ty) = κ(t) λ q−t (−y) [q−t ϕ]t (−y), so it is enough to prove [q−t ϕ](−y) = [q−t−1 f ](−t−1 y). This, however, follows from the deﬁnition of ϕ.

39. The Weil Representation

245

39. The Weil Representation Again, let E/F be a separable quadratic extension with Gal(E/F ) = {1, σ}. We let Θ be a character of E × which is E/F -regular, in the sense that Θσ = Θ. Using the representation ηψG of G = GE/F deﬁned in §38, we construct an irreducible cuspidal representation πE/F (Θ) canonically associated with (E/F, Θ). 39.1. To start with, let θ be a non-trivial character of K = KE/F . Let Cc∞ (E, θ) be the space of functions f ∈ Cc∞ (E) such that f (xy) = θ(x)f (y),

x ∈ K, y ∈ E.

(39.1.1)

In the notation of (38.6.1), condition (39.1.1) can be re-written as: ηψG (a(x))f = θ(x)f,

x ∈ K.

(39.1.2)

The function f is constant on a neighbourhood of 0, so (39.1.1) implies f (0) = 0. That is: (39.1.3) Cc∞ (E, θ) ⊂ Cc∞ (E × ). As K is contained in the centre of G, Cc∞ (E, θ) is a G-subspace of (ηψG , Cc∞ (E)). That is, Cc∞ (E, θ) carries the representation of G restricted from Cc∞ (E). We choose a character Θ of E × such that Θ | K = θ and form the representation ξ(Θ, ψ), Cc∞ (E, θ) = Θ−1 ηψG , Cc∞ (E, θ) , where Θ−1 ηψG is the representation (g, x) → Θ(x)−1 ηψG (g, x), (g, x) ∈ G. The representation ξ(Θ, ψ) is trivial on K and is therefore the inﬂation of a representation of G/K = Gκ , for which we use the same notation. 39.2. Let f ∈ Cc∞ (E, θ); the function Θ−1 f is constant on cosets of K and so is of the form Θ−1 f (x) = f 0 (NE/F (x)), for a uniquely determined function f 0 on the group NE/F (E × ) = Ker κ, which we now denote Fκ× . Indeed, f → f 0 gives a linear isomorphism Cc∞ (E, θ) ∼ = Cc∞ (Fκ× ) which we use to transport ξ(Θ, ψ) to a representation ξκ (Θ, ψ) of Gκ on Cc∞ (Fκ× ). Under the representation ξκ (Θ, ψ), the group B ∩Gκ acts on f ∈ Cc∞ (Fκ× ) as follows: 1 t f : y −→ ψ(ty)f (y), t ∈ F, 01 ζ 0 0 ζ

x0 0 1

f : y −→ κ(ζ)Θ(ζ)f (y), 1/2

f : y −→ x F Θ(x)f (xy),

ζ ∈ F ×, x ∈ Fκ× .

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9 The Weil Representation

The space Cc∞ (Fκ× ) admits a linear automorphism ϕ → ϕ , where ϕ (x) = x F Θ(x)ϕ(x). 1/2

(39.2.1)

Composing with this, we obtain an equivalent representation πκ = πκ (Θ, ψ) of Gκ on Cc∞ (Fκ× ). It acts on ϕ ∈ Cc∞ (Fκ× ) as follows: → ψ(ty)ϕ(y), πκ ( 10 1t ) ϕ : y − ζ 0 πκ 0 ζ ϕ : y −→ κ(ζ)Θ(ζ)ϕ(y), πκ ( x0 10 ) ϕ

: y −→ ϕ(xy),

t ∈ F, ζ ∈ F ×, x∈

(39.2.2)

Fκ× .

We deﬁne π(Θ, ψ) = IndG Gκ πκ (Θ, ψ), where, as usual, Ind is the functor of smooth induction. Note. The character θ of K determines Θ up to tensoring with χE = χ◦NE/F , for a character χ of Fκ× . In the opposite direction, a character Θ of E × satisﬁes Θ | K = 1 if and only if Θ is E/F -regular. In particular, the representation π(Θ, ψ) is deﬁned for any regular character Θ of E × . Proposition. Let Θ be a regular character of E × , and let ψ ∈ F, ψ = 1. (1) The representation π(Θ, ψ) of G is irreducible and cuspidal. (2) If χ is a character of F × , then π(χE Θ, ψ) ∼ = χπ(Θ, ψ). In particular,

κE/F π(Θ, ψ) ∼ = π(Θ, ψ).

(3) The central character ωπ of π(Θ, ψ) is given by ωπ = κE/F ⊗ Θ | F × . Proof. Parts (2) and (3) are direct consequences of the deﬁnitions. Deﬁne a character ϑ of N by ϑ 10 x1 = ψ(x). Write Mκ = M ∩ Gκ . Writing π = π(Θ, ψ), the relation π = IndG Gκ πκ implies (π |M ). Comparing (39.2.2) with (37.1.1), we get π | M ∼ π | M = IndM = κ κ Mκ M c-IndN ϑ. Thus indeed π is irreducible and cuspidal (36.1 Corollary 2).

39. The Weil Representation

247

39.3. We reach the crucial property of the representation π(Θ, ψ). Theorem. Let ψ ∈ F, ψ = 1. Let Θ be a regular character of E × and set ρ = IndE/F Θ, π = π(Θ, ψ). Then π is an irreducible cuspidal representation of G satisfying ε(χπ, s, ψ) = ε(χ ⊗ ρ, s, ψ), for all characters χ of F × . Proof. By 39.2 Proposition (2), it is enough to treat the case χ = 1. We calculate ε(π, s, ψ) using 37.3. We have therefore to track the action of the Weyl element w ∈ Gκ under the isomorphisms of 39.2. We abbreviate πκ = πκ (Θ, ψ). We recall that ψE = ψ ◦ TrE/F . We have ( 1 −s)n(Θ,ψE )

ε(Θ, s, ψE ) = qE2

ε(Θ, 12 , ψE ),

for an integer n(Θ, ψE ) (23.4 Corollary 2). Lemma. Let ϕ ∈ Cc∞ (Fκ× ) be the characteristic function of UF ∩ Fκ× . The support of πκ (w)ϕ is the set of x ∈ Fκ× with υF (x) = −n(Θ, ψE )f (E|F ). If υF (x) = −n(Θ, ψE )f (E|F ), then πκ (w)ϕ(x) = λE/F (ψ) ε(Θ, 12 , ψE ) Θ(x).

(39.3.1)

Proof. Let f ∈ Cc∞ (E, θ) be the function with support UE such that f (y) = Θ−σ (y), y ∈ UE . The composite of the two isomorphisms of 39.2 takes f to the characteristic function ϕ of UF ∩ Fκ× . From (38.6.1), we get ηψG (w)f (x) = λE/F (ψ) fˆ(xσ ). Expanding, with Φ0 denoting the characteristic function of UE ,

σ ˆ f (y) ψE (yxσ ) dy f (x ) = E

= Φ0 (y) Θ−1 (y σ ) ψE (xσ y) dy

E = Φ0 (y) Θ−1 (y) ψE (xy) dy E

= Φ0 (y) Θ−1 (y) ψE (xy) d× y E

= Θ(x) Θ−1 (xy) ψE (xy) d× y, UE

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9 The Weil Representation

where d× y = y −1 E dy. By 23.5 Exercise, the last integral vanishes unless υE (x) = −n(Θ, ψE ). In that case

Θ−1 (xy) ψE (xy) d× y = ε(Θ, 1, ψE ), UE

and so

ηψG (w)f (x) =

λE/F (ψ) Θ(x) ε(Θ, 1, ψE ) if υE (x) = −n(Θ, ψE ), 0

otherwise.

Tracking back to Cc∞ (Fκ× ), the lemma follows. Within the Kirillov model of π = π(Θ, ψ), the natural representation of Gκ on the subspace Cc∞ (Fκ× ) of Cc∞ (F × ) is equivalent to πκ . Comparing the lemma with 37.3 Theorem, we get n(Θ, ψE )f (E|F ) = n(π, ψ) = n, say, and ε(π, 12 , ψ) ωπ (x) = λE/F (ψ) ε(Θ, 12 , ψE ) Θ(x), for any x ∈ Fκ× of valuation −n. For such x, we have ωπ (x) = Θ(x) (39.2 Proposition (3)), and the result follows. 39.4. We extract some complementary detail. Corollary. Let E/F be a separable quadratic extension, and let Θ be an E/F regular character of E × . Let ψ ∈ F, ψ = 1, and put ρ = IndE/F Θ, π = π(Θ, ψ). We then have ε(χπ, s, ψ ) = ε(χ ⊗ ρ, s, ψ ), for all characters χ of F × and all ψ ∈ F, ψ = 1. Consequently, π(Θ, ψ) ∼ = π(Θ, ψ ). Proof. Because of 39.2 Proposition (2), it is enough to treat the case χ = 1. Part (3) of the same proposition gives ωπ = det ρ, and the ﬁrst assertion follows from 24.3 Proposition and 29.4 Proposition (2). Setting π = π(Θ, ψ ), we have ε(χπ , s, ψ) = ε(χ ⊗ ρ, s, ψ) = ε(χπ, s, ψ) for all χ, whence (27.1) π ∼ = π, as desired. Thus, if E/F is a separable quadratic extension and Θ is an E/F -regular character of E × , the isomorphism class of π(Θ, ψ) does not depend on ψ. We therefore denote it π(Θ), or πE/F (Θ), and call it the Weil representation deﬁned by (E/F, Θ).

40. A Partial Correspondence

249

40. A Partial Correspondence We can now use the Weil representation, as deﬁned in §39, to give a partial construction of the map G02 (F ) → A02 (F ) demanded by Theorem 33.4. 40.1. Let ρ ∈ G02 (F ); one says that ρ is imprimitive if there exists a separable quadratic extension E/F and a character ξ of E × such that ρ ∼ = IndE/F ξ. (F ) denote the set of imprimitive equivalence classes ρ ∈ G02 (F ). Let Gim 2 Theorem. Let ψ ∈ F, ψ = 1. There exists a unique map 0 π : Gim 2 (F ) −→ A2 (F )

with the property (40.1.1)

ε(χ ⊗ ρ, s, ψ) = ε(χπ(ρ), s, ψ)

× for all ρ ∈ Gim 2 (F ) and all characters χ of F . Indeed,

π(χ ⊗ ρ) ∼ = χπ(ρ), for all ρ, χ, and (40.1.1) holds for all ψ ∈ F, ψ = 1. Proof. Let ρ ∈ Gim 2 (F ); choose a separable quadratic extension E/F and a 0 × character ξ of E such that ρ ∼ = IndE/F ξ. Let π = πE/F (ξ) ∈ A2 (F ) be the Weil representation deﬁned by (E/F, ξ). This has the property ε(χπ, s, ψ) = ε(χ ⊗ ρ, s, ψ) for all χ and ψ (39.3). By the Converse Theorem 27.1, it is the only element of A02 (F ) with this property. The assignment IndE/F ξ → 0 πE/F (ξ) thus gives a well-deﬁned map Gim 2 (F ) → A2 (F ), and it is the only map with the desired properties. The next assertion is 39.2 Proposition (2), and the ﬁnal one is 39.4 Corollary. Remark. When the residual characteristic p of F is odd, we have Gim 2 (F ) = G02 (F ) (34.1 Theorem). The uniqueness property shows that π is the same as the tame Langlands correspondence π of 34.4. In this case, therefore, π gives a bijection G02 (F ) → A02 (F ). If p = 2, the same reasoning shows that π ∼ nr extends the bijection Gnr 2 (F ) = A2 (F ) of 34.4. In the case p = 2, we have im 0 Gnr 2 (F ) G2 (F ) G2 (F ), and there is considerably more to do. Further reading The proof of 36.1 Theorem (36.3) has been engineered to illustrate the relation with the structure of cuspidal representations. The standard source for such results in general is [70], but the method of [5] is more straightforward in our context. The rest of the chapter is a simpliﬁed version of [46], except for the treatment of the Kirillov model and its relation with the Godement-Jacquet functional equation: this owes something to [47].

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9 The Weil Representation

The Weil representation occurs in a much more general context: see, for example, [62]. This generalization however does not lead to the Langlands correspondence in higher dimension. There is an alternative method of constructing the Langlands correspondence, in arbitrary dimension, on irreducible representations of WF which are “imprimitive” in the right sense. It uses the technique of automorphic induction [44] and is reasonably transparent. It does, however, rely heavily on global constructions and so is not admissible here. The process is worked out fully in [40], using a slightly older version of the technology.

10 Arithmetic of Dyadic Fields

41. Imprimitive representations 42. Primitive representations 43. A converse theorem

We assume, throughout this chapter, that the residual characteristic p of the base ﬁeld F is 2. This chapter is devoted to a careful examination of the irreducible two-dimensional representations of WF . Our account is quite detailed, but we give only what we need for the construction of the Langlands correspondence in Chapter XII. In particular, we do not establish the existence of primitive representations. While not particularly hard, it would require a further digression, and the situation becomes completely transparent once we have the Langlands correspondence. We also prove an analogue of the Converse Theorem of 27.1: an imprimitive representation ρ ∈ G02 (F ) is determined, up to isomorphism, by the function χ → ε(χ ⊗ ρ, s, ψ) of characters χ of F × .

41. Imprimitive Representations We ﬁx a separable algebraic closure F /F (as required for the deﬁnition of WF ) and, for the time being, consider only ﬁeld extensions E/F with E ⊂ F . In particular, E/F is always assumed to be separable. 41.1. We review some basic facts concerning the arithmetic of those quadratic ﬁeld extensions E/F which are totally wildly ramiﬁed. Lemma. Let E/F be quadratic and totally ramiﬁed. Let E be a prime element of E and let f (X) = X 2 +aX+b be the minimal polynomial of E over F . Then:

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(1) υF (b) = 1, υF (a) 1 and oE = o[E ]; (2) the relative discriminant dE/F of the ﬁeld extension E/F is of the form dE/F = pd+1 , where d = min{2υF (a)−1, υF (4)}. In particular, 1 d υF (4), and d is odd except in the case where F has characteristic zero and d = υF (4). (3) Let be a prime element of E; then = (d+1)/2 if d is odd, υF (TrE/F ) 1+υF (2) if d = υF (4). (4) Let be any prime element of E and let σ generate Gal(E/F ); then υE (−σ ) = d+1. The proof is an elementary exercise. Observe that, if F has characteristic 2, then υF (4) = ∞ and d = 2υF (a)−1 is odd. (The case a = 0, d = ∞, arises exactly when E/F is purely inseparable, and is irrelevant to our purposes.) We will normally use the notation dE/F = pd+1 ,

d = d(E|F ),

(41.1.1)

when E/F is a separable quadratic extension. 41.2. Let E/F be a totally ramiﬁed quadratic extension with d(E|F ) = d. Let ϕ = ϕE/F be the inverse Herbrand function for the extension E/F . Thus ϕ is the function of non-negative integers given by j if 0 j d, ϕ(j) = 2j−d if j d. We then have the following properties: Proposition. (1) Let k ∈ Z; then k+d+1 . l= 2

TrE/F (pkE )

=

plF ,

The trace map TrE/F induces an isomorphism ≈

/p2m−d+1 −→ pm /pm+1 , p2m−d E E for all m ∈ Z.

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253

(2) Suppose j 0, j = d; the ﬁeld norm NE/F induces an isomorphism ϕ(j) 1+ϕ(j) ∼ j 1+j UE /UE = UF /UF . (3) The norm map NE/F induces a homomorphism UEd /UE1+d → UFd /UF1+d with kernel and cokernel of order 2. −(1+d)

. By deﬁnition, this Proof. The inverse diﬀerent of E/F is D−1 E/F = pE is the largest fractional ideal of oE with trace contained in o, and (1) follows easily. Under the canonical inclusion UF → UE , we get UF /UF1 = UE /UE1 ∼ = k× . × × 2 The map k → k , induced by the norm, is x → x and so it is bijective. This proves (2) in the case j = 0. Let 1 j d−1, so that j = ϕ(j). For x ∈ pjE , we have NE/F (1+x) = 1 + TrE/F (x) + NE/F (x) ≡ 1 + NE/F (x)

(mod pj+1 ).

The elements uj , u ∈ o/p, provide a set of representatives for pjE /pj+1 E , and their norms u2 NE/F ()j give a set of representatives for pj /pj+1 . This proves part (2) of the proposition when 1 j d−1. ϕ(j) For j d+1 and x ∈ pE , we get NE/F (1+x) = 1 + TrE/F (x) + NE/F (x) ≡ 1 + TrE/F (x) (mod pj+1 ), and this case of part (2) follows from part (1). Let σ ∈ Gal(E/F ), σ = 1. The well-known “Hilbert’s Theorem 90” says that the kernel of the norm map NE/F consists of all elements x/xσ , x ∈ E × . If is a prime element of E, then σ / ≡ 1 (mod pdE ), by 41.1 Lemma (4). On the other hand, if x ∈ UE = UF UE1 , then xσ /x ∈ UEd+1 . In (3), we certainly have NE/F (UEd ) ⊂ UFd . If K denotes the kernel of NE/F , then K ⊂ UEd and the group K/K ∩ UEd+1 ∼ = KUEd+1 /UEd+1 has order 2: the non-trivial element of this group is the coset of /σ , for a prime element of E. On the other hand, it follows from (2) that NE/F (UEd+1 ) = UFd+1 . So, if x ∈ UEd satisﬁes NE/F (x) ∈ UFd+1 , we may write x = yz, with z ∈ UEd+1 and y ∈ K. The kernel of the norm map UEd /UEd+1 → UFd /UFd+1 thus has order 2, whence so does its cokernel. For later use, we record a more detailed comment on the structure of the group Ker NE/F . : x → x/xσ induces an isomorphism Lemma. Let m 0, m ∈ Z. The map N UE2m+1 /UE2m+2 −→ UE2m+d+1 /UE2m+d+2 .

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Proof. Let be a prime element of E. The elements x = 1+y, y ∈ pm /pm+1 , give a set of coset representatives for UE2m+1 /UE2m+2 . As in 41.1 Lemma (4), υE (−σ ) = d+1, so x/xσ ≡ 1 + y(−σ ) (mod UE2m+d+2 ). (U 2m+2 ) ⊂ U 2m+d+2 and the lemma Since UE2m+2 = UFm+1 UE2m+3 , we have N E F follows. 41.3. Let E/F be a separable quadratic extension and ξ a character of WE . The representation IndE/F ξ of WF is then irreducible if and only if ξ = ξ σ , where σ generates Gal(E/F ). If we view ξ as a character of E × via class ﬁeld theory, this irreducibility property is equivalent (by Hilbert 90) to ξ not factoring through the norm map NE/F . Deﬁnition. Let ρ ∈ G02 (F ); deﬁne T(ρ) to be the group of characters χ of WF such that χ ⊗ ρ ∼ = ρ. Comparing determinants, we see that χ ∈ T(ρ) implies χ2 = 1. In particular, T(ρ) is an elementary abelian 2-group. In practice, it will be more convenient to view the elements of T(ρ) as characters of F × , via class ﬁeld theory. If E/F is a separable quadratic extension, then κE/F (29.2.2) is the unique non-trivial character of F × trivial on NE/F (E × ). Thus κE/F is unramiﬁed if and only if E/F is unramiﬁed. Otherwise, κE/F has level d(E|F ) (as follows from 41.2 Proposition). Class ﬁeld theory (29.1) shows that, if χ is a character of F × of order 2, there is a unique quadratic extension E/F (inside F ) such that χ = κE/F . Proposition. For any ρ ∈ G02 (F ), the group T(ρ) has order dividing 4. Proof. We start with: Lemma. Let ρ ∈ G02 (F ) and let E/F be a separable quadratic extension. The following are equivalent: (1) κE/F ∈ T(ρ); (2) there is a character ξ of WE such that ρ ∼ = IndE/F ξ. If condition (2) holds, the character ξ is uniquely determined by ρ, up to conjugation by Gal(E/F ). Proof. All assertions follow from standard Cliﬀord theory. Suppose that |T(ρ)| = 1, and let κ = κE/F ∈ T(ρ). Choose a character ξ of E × such that ρ ∼ = IndE/F ξ.

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If |T(ρ)| > 2, choose χ ∈ T(ρ), χ = 1, κ. The equation ρ ∼ = χ ⊗ ρ implies ∼ ρ = IndE/F (χE ⊗ ξ), where χE = χ ◦ NE/F (or χ | WE ). It follows (Lemma) that χE ⊗ ξ = ξ or ξ σ , where σ ∈ Gal(E/F ), σ = 1. By hypothesis, χE = 1, so χE ⊗ ξ = ξ σ , that is, χE = ξ σ /ξ. There are, at most, two characters χ of F × satisfying χE = ξ σ /ξ, namely χ itself and κχ, so T(ρ) = {1, κ, χ, κχ}. We will make repeated use of one intermediate step in the preceding argument. We therefore exhibit it explicitly: Corollary. Let ρ = IndE/F ξ ∈ G02 (F ). A character φ of F × lies in T(ρ) if and only if φE = 1 or φE = ξ/ξ σ , where σ generates Gal(E/F ). One says that ρ is ⎫ primitive⎪ ⎬ simply imprimitive triply imprimitive

⎪ ⎭

if |T(ρ)| =

⎧ ⎪ ⎨ 1, 2, ⎪ ⎩ 4.

In particular, an irreducible representation IndE/F ξ is triply imprimitive if and only if ξ/ξ σ factors through NE/F . 41.4. We recall that Gnr 2 (F ) denotes the set of classes of representations ρ ∈ G02 (F ) such that T(ρ) contains an unramiﬁed character χ = 1 (just as in 34.1). If ρ ∈ G02 (F ) Gnr 2 (F ), then ρ is called totally ramiﬁed. Let E/F be a separable, totally ramiﬁed, quadratic extension. Let ξ be a character of E × , of level l(ξ) 1. We say that ξ is minimal over F if l(ξ) l(χE ξ), for all characters χ of F × . l(ξ) Equivalently, ξ is minimal over F if and only if ξ | UE does not factor through NE/F . In particular, if ξ is minimal over F , the representation IndE/F ξ is irreducible. If ξ is a character of E × which does not factor through NE/F , then clearly ξ = χE ξ0 , for a character χ of F × and some ξ0 which is minimal over F . When analyzing induced representations ρ = IndE/F ξ or the groups T(ρ), therefore, it is enough to treat the case of ξ minimal. Lemma. Let d(E|F ) = d 1, and let ξ be a character of E × . (1) If ξ is minimal over F , then l(ξ) d. (2) Suppose that l(ξ) 1+d; then ξ is minimal over F if and only if l(ξ) ≡ d (mod 2). Proof. If ξ has level l, 1 l < d, then ξ | UEl certainly factors through the norm (41.1 Proposition (2)), whence follows (1). Part (2) is a direct consequence of 41.1 Proposition (2).

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Using the same notation as in the lemma: Proposition. Suppose that ξ is minimal of level l, and put ρ = IndE/F ξ. (1) The representation ρ lies in Gnr 2 (F ) if and only if l = d. (2) If d < l < 2d, the representation ρ is triply imprimitive, and T(ρ) contains a character of level l−d. Proof. Let σ be the generator of Gal(E/F ). If ξ has level d, the character ξ/ξ σ is trivial on UE1 (41.2 Lemma) and hence on UE = UF UE1 . It is therefore unramiﬁed (and non-trivial). Thus ξ/ξ σ = χE , where χ is a non-trivial unramiﬁed character of F × . By 41.3 Corollary, χ ∈ T(ρ) and so ρ ∈ Gnr 2 (F ). Suppose now that ξ has level l > d. Since l ≡ d (mod 2), any x ∈ UEl satisﬁes x ≡ y σ /y (mod UEl+1 ), for some y ∈ UEl−d (41.2 Lemma). We deduce that ξ/ξ σ has level l−d 1. In particular, ξ/ξ σ cannot be of the form χE , for an unramiﬁed character χ of F × . Therefore ρ is totally ramiﬁed, and we have proved (1). If, however, d < l < 2d, the character ξ/ξ σ has level l−d < d and so it factors through NE/F , say ξ/ξ σ = φ ◦ NE/F , for a character φ of F × . The level of φ is l−d, so φ = κE/F while φ ∈ T(ρ). Thus ρ is triply imprimitive, as required. 41.5. We can phrase this diﬀerently, using the Langlands-Deligne local constant. Let ρ ∈ G02 (F ) and ψ ∈ F , ψ = 1. Recall that ε(ρ, s, ψ) = q n(ρ,ψ)( 2 −s) ε(ρ, 12 , ψ), 1

for some n(ρ, ψ) ∈ Z (29.4 Proposition). For ρ ﬁxed, n(ρ, ψ) depends only on the level of ψ loc. cit. We write (ρ) = n(ρ, ψ), for some ψ of level one: we call (ρ) the level of ρ. We say ρ is minimal if (ρ) (χ ⊗ ρ), for every character χ of F × . Lemma. Let ρ ∈ G02 (F ) be totally ramiﬁed and imprimitive. Then ρ is minimal if and only if (ρ) is odd. Proof. Write ρ = IndE/F ξ, d = d(E|F ), and let ψ ∈ F have level 1. It follows, from the inductive properties of the local constant and 30.4 Corollary, that n(ξ, ψE ) = n(ρ, ψ). The character ψE = ψ ◦ TrE/F has level 1−d by 41.1 Proposition (1). Thus ξ has level (ρ)−d (23.5), and the lemma follows from 41.4 Lemma. Theorem. Let ρ ∈ G02 (F ) be totally ramiﬁed, minimal and imprimitive. The group T(ρ) then contains a character χ of level d 1 satisfying 3d (ρ).

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257

Proof. Set = (ρ) and take χ ∈ T(ρ), χ = 1. Let d be the level of χ, so that χ = κE/F for a quadratic extension E/F with d(E|F ) = d. We can write ρ = IndE/F ξ, for a character ξ of E × . The character ξ has level l(ξ) = −d. The minimality of ρ implies that ξ is minimal over F . If l(ξ) 2d, there is nothing more to do. We therefore assume the contrary, that is, d < l(ξ) < 2d. By 41.4 Proposition, T(ρ) contains a character φ of level d = l(ξ)−d < d. We have (ρ) = l(ξ)+d = d +2d > 3d , as required. It is sometimes useful to have a little more detail. Suppose ρ is triply imprimitive, so that T(ρ) is non-cyclic of order 4. There are two possibilities: either all non-trivial χ ∈ T(ρ) have the same level l(χ) = d 1, and the characters χ | UFd are distinct, or else there is a unique non-trivial element with minimal level l(χ). The proof of the theorem gives: Corollary. Let ρ be totally ramiﬁed, triply imprimitive and minimal, of level n. Let χi , 1 i 3, be the non-trivial elements of T(ρ) and let l(χi ) be the level of χi . The characters may be numbered so that either (1) l(χ1 ) < l(χ2 ) = l(χ3 ) and 3l(χ2 ) > n > 3l(χ1 ), or else (2) l(χ1 ) = l(χ2 ) = l(χ3 ) and n = 3l(χi ). In the second case, setting l = l(χi ), the characters χi | UFl are distinct. Exercise. Suppose, for the moment, that p = 2. Let ρ ∈ G02 (F ) be minimal and triply imprimitive. Show that ρ has level zero. Classify the minimal triply imprimitive elements of G02 (F ).

42. Primitive Representations We give a preliminary analysis of the primitive 2-dimensional representations of WF . By deﬁnition, a primitive representation ρ ∈ G02 (F ) is totally ramiﬁed. 42.1. Let ρ ∈ G02 (F ). If K/F is a ﬁnite ﬁeld extension, with K ⊂ F as always, we put ρK = ρ | WK . Observe that if χ is a character of WF , then χ | WK corresponds to χK = χ ◦ NK/F , via class ﬁeld theory. Proposition. Let ρ ∈ G02 (F ) be totally ramiﬁed, and let K/F be a ﬁnite, tamely ramiﬁed ﬁeld extension. (1) The representation ρK is irreducible and totally ramiﬁed. (2) The canonical map T(ρ) −→ T(ρK ), χ −→ χK , is injective.

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(3) If ρ ∈ G02 (F ) satisﬁes ρK ∼ = ρK , then ρ ∼ = χ ⊗ ρ, for a uniquely deter× mined character χ of F which is trivial on NK/F (K × ). Proof. In (1), we suppose for a contradiction that there exists a ﬁnite, tamely ramiﬁed extension such that ρK is reducible. Since ρK is semisimple (28.7 Lemma), it is a sum of characters, ρK = θ1 ⊕ θ2 . We may as well assume that K/F is Galois. We are then in the situation of the proof of 34.1 Theorem. As there, if θ1 = θ2 , then ρ is induced from a character of WL , where L/F is quadratic and tamely ramiﬁed, hence unramiﬁed. If θ1 = θ2 , the same argument as in 34.1 shows that ρ ∈ Gnr 2 (F ). Either of these outcomes is contrary to hypothesis, so ρK is irreducible. Let L/K be unramiﬁed quadratic. If ρK were not totally ramiﬁed, then ρL would be a sum of characters (41.3 Lemma), contrary to the ﬁrst argument. In part (2), let χ ∈ T(ρ), χ = 1. By hypothesis, χ is not unramiﬁed; since χ2 = 1, it is not tamely ramiﬁed and so has level 1. The norm map 1 → UF1 is surjective (18.1 Lemma (2)), so χK = 1. NK/F : UK We ﬁrst prove (3) under the assumption that K/F is Galois. We can assume that ρ, ρ act on the same vector space V and, replacing ρ by a conjugate if necessary, that ρ(h) = ρ (h), h ∈ WK . For h ∈ WK and g ∈ WF , we have ρ(g)ρ(h)ρ(g)−1 = ρ(ghg −1 ) = ρ (ghg −1 ) = ρ (g)ρ (h)ρ (g)−1 = ρ (g)ρ(h)ρ (g)−1 . Thus ρ(g)−1 ρ (g) commutes with all the operators ρ(h). Schur’s Lemma implies that ρ(g)−1 ρ (g) = χ(g), for a scalar χ(g) ∈ C× . The map g → χ(g) is a character of WF trivial on WK , and ρ ∼ = χ ⊗ ρ. The character χ is tamely ramiﬁed, and uniquely determined modulo T(ρ). By hypothesis, T(ρ) contains no tamely ramiﬁed character other than 1, so χ is uniquely determined. This proves (3) for a Galois extension K/F . In the general case, let L/F be the normal closure of K/F . Surely ρL ∼ = ρL , ∼ so ρ = ρ ⊗ χ, for a character χ of WF trivial on WL . Therefore ρK ∼ = ρK ⊗ χK , and hence χK ∈ T(ρK ). The quadratic character χK is tamely ramiﬁed, therefore unramiﬁed. Since ρK is totally ramiﬁed, we have χK = 1. Viewing χ as a character of F × , this is equivalent to χ vanishing on norms from K. 42.2. We can now unpick the detailed structure of the 2-dimensional primitive representations of WF . Theorem. Let ρ ∈ G02 (F ) be primitive. There exists a cubic extension K/F such that ρK is imprimitive. Moreover: (1) If K/F is Galois (hence cyclic), the representation ρK is triply imprimitive.

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259

(2) Suppose K/F is not cyclic, let L/F be the normal closure of K/F and let E/F be the maximal unramiﬁed sub-extension of L/F . The representation ρK is simply imprimitive, ρL is triply imprimitive, and ρE is primitive. Proof. We start with a more general analysis. Lemma 1. Let ρ ∈ G02 (F ) be primitive, and let L/F be a ﬁnite, tamely ramiﬁed, Galois extension such that ρL is imprimitive. Then: (1) The Galois group Gal(L/F ) acts on T(ρL ) with only the trivial ﬁxed point. (2) The representation ρL is triply imprimitive. (3) If F ⊂ E ⊂ L and Gal(L/E) acts trivially on T(ρL ), the canonical map T(ρE ) → T(ρL ) is an isomorphism. Proof. Let E/F be the maximal unramiﬁed sub-extension of L/F . Suppose ξ ∈ T(ρL ) is ﬁxed by Gal(L/F ), ξ = 1. In particular, ξ is ﬁxed by Gal(L/E). Thus ξ = χL , for a character χ of E × . Since (χ ⊗ ρE )L ∼ = ρL , we have × θ ⊗ ρ , for a character θ of E trivial on norms from L× (42.1 χ ⊗ ρE ∼ = E −1 Proposition (3)). That is, we can replace χ by θ χ and assume χ ∈ T(ρE ). If σ generates Gal(E/F ), we have (χσ )L = χL , so χσ /χ is trivial on norms from L× . In particular, χσ /χ is trivial on UE1 . On the other hand, χσ ∈ T(ρE ), so T(ρE ) contains the tamely ramiﬁed character χσ /χ. Since ρE is totally ramiﬁed (42.1 Proposition (1)), this implies χσ = χ. We deduce that χ is of the form τE , for some character τ of F × . Going through the same argument, we get τ ∈ T(ρ), which is impossible. This proves (1). If ρL were simply imprimitive, the action of Gal(L/F ) on T(ρL ) would necessarily be trivial, contradicting (1) and proving (2). In (3), it is enough to treat the case where L/E is cyclic. A character χ ∈ T(ρL ) is of the form χ = ξL , for some character ξ of E × . Arguing as in the proof of (1), we may choose ξ to lie in T(ρE ). The map T(ρE ) → T(ρL ) is therefore surjective. It is injective by 42.1 Proposition (2). Next, we prove: Lemma 2. There is a ﬁnite, tamely ramiﬁed extension K/F such that the representation ρK is imprimitive. Proof. By 28.6 Proposition, we can assume that ρ is a representation of ΩF . We view ρ as a homomorphism ΩF → GL2 (C), and we write G = ρ(ΩF ). We can identify G with Gal(E/F ), for some ﬁnite Galois extension E/F . Let G denote the image of G in the projective linear group PGL2 (C) = GL2 (C)/C× .

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The group Z = G ∩ C× is ﬁnite cyclic, and central in G. We have an exact sequence: (42.2.1) 1 → Z −→ G −→ G → 1. Let H be a 2-Sylow subgroup of G, with inverse image H in G. The index (G : H) = (G : H) is odd, so the ﬁeld extension E H /F is tamely ramiﬁed. By 42.1 Proposition, the restriction ρ | H is irreducible. The group H is a cyclic central extension of a ﬁnite 2-group. Thus H is nilpotent and so any irreducible representation of H is induced from a one-dimensional representation. The desired ﬁeld K is then E H . So, by Lemma 2, there exists a ﬁnite, tamely ramiﬁed, Galois extension L/F such that ρL is imprimitive. Lemma 1 (2) implies that ρL is triply imprimitive. The action of Gal(L/F ) on T(ρL ) is given by a homomorphism Gal(L/F ) −→ Aut(T(ρL )) ∼ = GL2 (F2 ) ∼ = S3 ,

(42.2.2)

where S3 is the symmetric group on 3 letters. The image of the map (42.2.2) acts without ﬁxed point on the non-trivial elements of T(ρL ), so it is either S3 or the cyclic alternating group A3 . By Lemma 1 (3), we may choose L/F so that (42.2.2) is injective. That is, Gal(L/F ) is either S3 or A3 . If Gal(L/F ) ∼ = A3 , we are in the ﬁrst case of the theorem with K = L. Otherwise, a non-trivial involution σ ∈ Gal(L/F ) has a non-trivial ﬁxed point χ ∈ T(ρL ); if K is the ﬁxed ﬁeld of σ we have χ = ξL , for some ξ ∈ T(ρK ). The extension K/F is totally ramiﬁed and cubic, while the representation ρK is simply imprimitive (Lemma 1 (3)). We are in case (2) of the theorem, and the ﬁnal assertion follows from Lemma 1 (1). Remark. Suppose ρ ∈ G02 (F ) is primitive, and let K/F be a tamely ramiﬁed, Galois extension such that ρK is triply imprimitive. The group Gal(K/F ) permutes the non-trivial elements of T(ρK ) transitively, and so they all have the same level. Corollary 41.5 implies that this level is nK /3, where nK is the level of ρK . In particular, nK is divisible by 3. 42.3. We can work out the structure of the group G in (42.2.1). Take K as in the theorem, with L/F the normal closure of K/F . There are totally ramiﬁed quadratic extensions Ei /L, i = 1, 2, 3, such that the characters κEi /L are the non-trivial elements of T(ρL ). For each i, there is a character ξi of Ei× such that ρL = IndEi /L ξi . Let E/L be the composite of the extensions Ei /L; then E/L is totally ramiﬁed (42.2 Remark and 41.5 Corollary) and Gal(E/L) = V4 , the noncyclic group of order 4. The character ξ = ξi,E of E × is ﬁxed by Gal(E/F ) and ρ is eﬀectively a representation of a cyclic central extension of Gal(E/F ). We thus have G = Gal(E/F ).

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In the ﬁrst case of the theorem, we have an exact sequence 1 → V4 −→ G −→ A3 → 1, and A3 acts non-trivially on the subgroup V4 . Elementary group-theoretic arguments give G ∼ = A4 . In case (2), we have an exact sequence 1 → V4 −→ G −→ S3 → 1, giving G ∼ = S4 . Comment. It is a “well-known fact, equivalent to the classiﬁcation of the Platonic solids”, that the ﬁnite subgroups of PGL2 (C) are the cyclic groups Cn of order n, n 1, the dihedral groups Dn of order 2n, n 2, and the permutation groups A4 , S4 and A5 . The case G = Cn cannot arise (because ρ would be reducible), neither can G = Dn (since ρ would then be imprimitive). Since G is soluble, the case G = A5 cannot happen. This leaves the alternatives G = A4 or S4 , just as in the theorem. This background explains why the ﬁrst case is often called “tetrahedral” and the second “octahedral”. There is one aspect of this analysis to which we shall return. Proposition. Let ρ ∈ G02 (F ) be primitive. (1) There is a cubic extension K/F such that ρK is imprimitive, and this condition determines K uniquely, up to F -isomorphism. (2) Let E/F be a ﬁnite, tamely ramiﬁed ﬁeld extension. The representation ρE is imprimitive if and only if E contains some F -conjugate of K. Proof. The existence of K/F is given by 42.2 Theorem. Let L/F be the normal closure of K/F ; if K = L we set L0 = F while, if K = L, we let L0 /F be the maximal unramiﬁed sub-extension of L/F . It is enough to show that if E/F is a ﬁnite, tamely ramiﬁed Galois extension such that ρE is imprimitive, then E ⊃ L. We assume the contrary. It follows that E ∩ L = L0 or F . By Lemma 1 (2), the representations ρL , ρE are both triply imprimitive. The canonical maps T(ρL ) → T(ρEL ), T(ρE ) → T(ρEL ) are both isomorphisms. The group Gal(EL/E ∩ L) = Gal(L/E ∩ L) × Gal(E/E ∩ L) therefore acts trivially on T(ρEL ). Lemma 1 (3) now implies that ρE∩L is triply imprimitive. If E ∩ L = F , this contradicts the hypothesis. Otherwise, E ∩ L/F is unramiﬁed quadratic, and 42.2 Theorem (2) implies ρE∩L primitive. These contradictions prove the proposition. 42.4. Let K/F be a cubic extension, and let τ ∈ G02 (K). The question naturally arises as to whether τ is of the form ρK , for some ρ ∈ G02 (F ). Proposition. Let K/F be a cubic extension, and let τ ∈ G02 (K) be totally ramiﬁed.

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(1) Suppose K/F is cyclic. There exists ρ ∈ G02 (F ), such that τ ∼ = ρK , if and only if τ ∼ = τ σ , where σ generates Gal(K/F ). (2) Suppose K/F is not cyclic, and let L/F be the normal closure of K/F . There exists ρ ∈ G02 (F ) such that τ ∼ = ρK if and only if there exists 0 ∼ ρ ∈ G2 (F ) such that τL = ρL . (3) In the situation of (2), let ϑ ∈ G02 (L) be totally ramiﬁed. There exists ρ ∈ G02 (F ) such that ϑ ∼ = ϑ for all γ ∈ Gal(L/F ). = ρL if and only if ϑγ ∼ Proof. Part (1) is elementary. In part (2), the necessity of the condition is clear. As for suﬃciency, we observe that K/F is totally ramiﬁed while L/K is unramiﬁed and quadratic. We have τL ∼ = (ρK )L , so τ = φ ⊗ ρK , for an unram× 2 iﬁed character φ of K such that φ = 1 (42.1 Proposition (3)). Necessarily φ = χK , where χ is the unramiﬁed character of F × of the same order as φ. Thus τ = (χ ⊗ ρ )K , as required. In part (3), the necessity of the condition is clear. As for suﬃciency, let E/F be the maximal unramiﬁed sub-extension of L/F . Thus L/E is totally ramiﬁed and cubic. In particular, ϑ is of the form µL , for some µ ∈ G02 (E). Further, if µ ∈ G02 (E), then µL ∼ = ϑ if and only if µ ∼ = µ ⊗ χ, for some × character χ of E trivial on norms from L. These three representations µ of WE , which satisfy µL ∼ = ϑ, are mutually inequivalent as they have diﬀerent determinants. The group Gal(E/F ) acts on this set and it has a ﬁxed point. In other words, we may choose µ to satisfy µσ ∼ = µ, where σ generates Gal(E/F ). It follows that µ ∼ = ρE , for some 0 ρ ∈ G2 (F ).

43. A Converse Theorem In this section, we prove an analogue of the Converse Theorem (27.1) for imprimitive representations ρ ∈ G02 (F ). 43.1. We establish notation for a preliminary result. We let E/F be a separable, totally ramiﬁed quadratic extension. We put d(E|F ) = d and Gal(E/F ) = {1, σ}. We let ξ be a character of E × , which is minimal over F , of level n−d > d. The integer n is odd (41.4 Lemma (2)): we put n = 2m+1. The representation ρ = IndE/F ξ is irreducible, totally ramiﬁed, and of level n. For an integer j 0, we let Γj denote the group of characters of F × which are trivial on UFj . We put F (n) = {x ∈ F : υF (x) = −n}, E(n) = {y ∈ E : υE (y) = −n}.

43. A Converse Theorem

263

For x ∈ E(n), we have NE/F (x) ∈ F (n) and TrE/F (x) ∈ p−m+[d/2] ⊂ p−m . If u ∈ UEn−d+1 , then NE/F (xu) ≡ NE/F (x)

(mod UFm+1 ),

TrE/F (xu) ≡ TrE/F (x) (mod p), so we have a canonical map ΨE : Gal(E/F )\E(n)/UEn−d+1 −→ F (n)/UFm+1 × p−m /p, x −→ (NE/F (x), TrE/F (x)).

(43.1.1)

Lemma. The map ΨE is injective. Proof. Let γ1 , γ2 ∈ E(n), and suppose that ΨE (γ1 ) = ΨE (γ2 ). The relation NE/F (γ1−1 γ2 ) ∈ UF1 implies γ1 = γ2 (1+x), for some x ∈ pE . Since NE/F (1+x) ∈ UFm+1 ⊂ UFd+1 , we have 1+x ∈ Ker NE/F · UEd+1 . Replacing γ2 by γ2σ if necessary, we can assume 1+x ∈ UEd+1 . Since TrE/F (γ1 ) ≡ TrE/F (γ2 ) (mod p), we have υF (TrE/F (γ2 x)) 1. Temporarily set k = υE (γ2 x) and suppose ﬁrst that k ≡ d (mod 2). Parts (2) and (3) of 41.1 Lemma show that υF (TrE/F (γ2 x)) = (k+d)/2. This implies υE (x) n−d+1, as required. Suppose therefore that k ≡ d (mod 2) or, equivalently, υE (x) ≡ d (mod 2). In the same way, υF (TrE/F (x)) = (υE (x)+d)/2. The conditions NE/F (γ1 γ2−1 ) ∈ UFm+1 , x ∈ pd+1 E , then translate into (υE (x)+d)/2 m+1. The desired relation υE (x) n+1−d again follows. Let χ range over Γm+1 and µ over the characters of F which are trivial on p. We deﬁne a function Υ(χ, µ) of such characters by 0 if µ | o = 1, Υ(χ, µ) = 1 ε(χE ξ, 2 , µE ) otherwise. If µ ∈ F has level one, the local constant ε(χE ξ, s, µE ), for χ ∈ Γm+1 , depends only on µ | p−m . We therefore regard Υ as a function of χ ∈ Γm+1 and µ ∈ (p−m /p) . We ﬁx ψ ∈ F of level one. The map c → cψ then gives an isomorphism m+1 → (p−m /p) , under which the characters non-trivial on o/p correo/p spond to units c ∈ UF . Let a ∈ F (n); then ε(χE ξ, 12 , cψE )χ(a) depends only on a modulo UFm+1 and the coset χΓ0 .

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10 Arithmetic of Dyadic Fields

For a ∈ F (n)/UFm+1 , b ∈ p−m /p, consider the function

ε(χE ξ, 12 , cψE ) χ(a) ψ(−cb) Φ(a, b) = χ∈Γm+1 /Γ0 , m+1 c∈UF /UF

=

(43.1.2)

Υ(χ, cψ) χ(a) ψ(−cb).

χ∈Γm+1 /Γ0 , c∈o/pm+1

Thus Φ is essentially the Fourier transform of Υ on the ﬁnite group Γm+1 /Γ0 × o/pm+1 . In particular, we recover Υ as Υ(χ, cψ) = (q−1)−1 q −(2m+1) Φ(a, b) χ(a)−1 ψ(cb), a,b

where a runs through F (n)/UFm+1 and b through p−m /p. ˇ On the other hand, for x ∈ E(n), the quantity ξ(x)+ ξˇσ (x) depends only on ΨE (x). Thus we can deﬁne a function Φ on F (n)/UFm+1 × p−m /p, with support in ΨE (E(n)), such that ˇ Φ NE/F (x), TrE/F (x) = ξ(x)+ ξˇσ (x), x ∈ E(n). Proposition. There is a constant k > 0, depending only on n, such that Φ = kΦ . Proof. By Fourier inversion, it is enough to prove Φ (a, b) χ(a)−1 µ(b), Υ(χ, µ) = k a,b

for all characters µ of p−m trivial on p and all χ ∈ Γm+1 , where k is a positive constant depending only on n. The right hand side is ˇ k ξ(x)+ ξˇσ (x) χE (x)−1 µE (x) n−d+1 x∈E(n)/UE

= k

ˇ ξ(x)+ ξˇσ (x) χE (x)−1 µE (x) dν ∗ (x),

E(n)

where ν ∗ is the Haar measure on E × for which ν ∗ (UEn−d+1 ) = 1. If µ has level 0, then µE has level −d and the integral vanishes (as follows easily from 23.5 Lemma). If µ has level 1, then µE has level 1−d and, as in the proof of 23.5 Theorem, the integral reduces to k ε(χE ξ, 12 , µE ), for some k > 0 depending only on n.

43. A Converse Theorem

265

Let us re-state the proposition in more transparent form: Corollary. There exists a constant k > 0, depending only on n, such that ˇ ξ(x)+ ξˇσ (x) = k ε(χE ξ, 12 , cψ) χ(NE/F (x)) ψ(−cTrE/F (x)), χ∈Γm+1 /Γ0 , m+1 c∈UF /UF

for all x ∈ E(n). 43.2. The result we seek is: Galois Converse Theorem. Let ρ1 , ρ2 ∈ G02 (F ) be imprimitive and suppose that ρ1 is totally ramiﬁed. Suppose also that ε(χ ⊗ ρ1 , s, ψ) = ε(χ ⊗ ρ2 , s, ψ), for all characters χ of F × and some ψ ∈ F , ψ = 1. We then have ρ1 ∼ = ρ2 . We ﬁrst note that, using the Stability Theorem (29.4 Proposition (4)), the hypothesis implies det ρ1 = det ρ2 . The hypothesis is therefore independent of ψ. We may therefore choose ψ of level one. Further, we can assume that the ρi are both minimal over F . 43.3. The proof of 43.2 Theorem proceeds in two steps. We ﬁrst treat the case where T(ρ1 ) ∩ T(ρ2 ) is non-trivial. Let κE/F ∈ T(ρ1 ) ∩ T(ρ2 ). The extension E/F is then totally ramiﬁed. We set d(E|F ) = d, and use the other notation of 43.1. The integer n = n(ρ1 , ψ) = n(ρ2 , ψ) is then > 2d and so ρ2 is also totally ramiﬁed (41.4 Proposition (1)). By 41.3 Lemma, we can choose characters ξ1 , ξ2 of E × such that ρi = IndE/F ξi , i = 1, 2. Our hypothesis implies ε(χE ξ1 , 12 , ψE ) = ε(χE ξ2 , 12 , ψE ), for all characters χ of F × . Corollary 43.1 yields ξˇ1 (x) + ξˇ1σ (x) = ξˇ2 (x) + ξˇ2σ (x),

(43.3.1)

for all x ∈ E(n). The relation det ρ1 = det ρ2 implies ξ1 | F × = ξ2 | F × , so the two sides of (43.3.1) agree on the set F × E(n) = {y ∈ E × : υE (y) ≡ 1 (mod 2)}. Thus, if φ denotes the unramiﬁed character of E × of order 2, we have (ξ1 −φξ1 ) + (ξ1σ −φξ1σ ) = (ξ2 −φξ2 ) + (ξ2σ −φξ2σ ), as functions on E × . Since ρi is totally ramiﬁed, we have ξi = φξiσ . Linear independence of characters of E × now implies that ξ1 = ξ2 or ξ2σ . Therefore ρ1 ∼ = ρ2 , as desired.

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10 Arithmetic of Dyadic Fields

43.4. We prove 43.2 Theorem in the general case. Let κ = κK/F ∈ T(ρ2 ). For any character χ of F × , we have ε(χκ ⊗ ρ1 , s, ψ) = ε(χκ ⊗ ρ2 , s, ψ) = ε(χ ⊗ ρ2 , s, ψ) = ε(χ ⊗ ρ1 , s, ψ). Since ρ1 is totally ramiﬁed and T(ρ1 ) = T(κ ⊗ ρ1 ), we can apply the special case of 43.3 to deduce κ ⊗ ρ1 ∼ = ρ1 . Thus T(ρ1 ) ∩ T(ρ2 ) = {1}, and so ρ1 ∼ = ρ2 by the ﬁrst case. Remark. The theorem also holds if both ρi ∈ Gnr 2 (F ), because of the uniqueness of the tame Langlands correspondence (34.4). Further reading. Primitive 2-dimensional representations of WF are classiﬁed in Weil [88], using a somewhat diﬀerent approach from that here. A full classiﬁcation of primitive representations of WF , in arbitrary dimension and residual characteristic, is given by Koch [49]. The idea of writing the inducing character of an imprimitive representation in terms of local constants seems to have ﬁrst been employed in a series of papers of G´erardin and Li, for example [34].

11 Ordinary Representations

44. Ordinary representations and strata 45. Exceptional representations and strata We return to the partial correspondence 0 π : Gim 2 (F ) −→ A2 (F ),

deﬁned on the set Gim 2 (F ) of equivalence classes of irreducible imprimitive two-dimensional representations of WF , as in 40.1. The image π(Gim 2 (F )) comprises the cuspidal Weil representations of §39. The mode of construction of the map π is no longer relevant: we rely solely on its formal properties. The object of this chapter is to give an independent characterization of the representations π ∈ π(Gim 2 (F )), in terms of the classiﬁcation theory of Chapter IV. We know from the Tame Parametrization Theorem (34.4) that the set Anr 2 (F ) of equivalence classes of unramiﬁed cuspidal representations is conim 0 tained in π(Gim 2 (F )) and also, when p = 2, that G2 (F ) = G2 (F ) and π is bijective. So, we assume henceforth that the residual characteristic p of F is 2, and concentrate on totally ramiﬁed representations. (That said, the argument works unchanged when p = 2, provided we stick to totally ramiﬁed representations. Some variation is needed to deal with representations ρ ∈ Gnr 2 (F ) using this approach.)

44. Ordinary Representations and Strata Let π ∈ A02 (F ) satisfy (π) (χπ), for all characters χ of F × : we shall now say that such representations are minimal. Suppose also that π is totally ramiﬁed. There is a ramiﬁed simple stratum (A, n, α) in A = M2 (F ) such that π [n/2]+1 . As we shall discover in this section, it contains the character ψα of UA

268

11 Ordinary Representations

is the stratum (A, n, α) which determines whether or not π lies in the image of the partial correspondence π | Gim 2 (F ). 44.1. We gather together those properties of the partial correspondence on which we shall rely. Imprimitive Langlands correspondence. Let ψ ∈ F, ψ = 1. There is a unique map 0 π = πF : Gim 2 (F ) −→ A2 (F ) such that ε(χπ(ρ), s, ψ) = ε(χ ⊗ ρ, s, ψ)

(44.1.1)

×

for all characters χ of F . The map π has the further properties: (1) The relation (44.1.1) holds for all ψ ∈ F, ψ = 1. (2) The map π is injective. × (3) For any ρ ∈ Gim 2 (F ) and any character χ of F , we have π(χ ⊗ ρ) ∼ = χπ(ρ). (4) If ρ ∈ G02 (F ) and π = π(ρ), then ωπ = det ρ. Proof. The existence and uniqueness of π are taken from 40.1 Theorem, as are (1) and (3). Property (4) is 39.2 Proposition (3), 29.2 Proposition. Injectivity (2) is implied by the Galois Converse Theorem of 43.2. Deﬁnition. Let π ∈ A02 (F ); say π is ordinary if π = π(ρ), for some ρ ∈ Gim 2 (F ). Otherwise, call π exceptional. Observe that unramiﬁed cuspidal representations π are invariably ordinary. 44.2 We make a brief return to matrix calculations. Let (A, n, α) be a ramiﬁed simple stratum with n = 2m+1 1. We can replace A by a G-conjugate, when convenient, and identify it with the standard chain order I: oo A=I= . po We choose a prime element of F . The element α is then conjugate (in G or KA ) to the matrix 0 −ζ −(m+1) , −m t where ζ ∈ UF (so det α = ζ −n ) and t = tr α ∈ p−m (cf. 27.5). Lemma. Let (A, n, α), (A, n, α ) be ramiﬁed simple strata with n = 2m+1 1. Let ψ ∈ F have level one. The following conditions are equivalent:

44. Ordinary Representations and Strata

(1) (2) (3) (4)

the the the the

269

characters ψα , ψα of UAm+1 intertwine in G; characters ψα , ψα of UAm+1 are conjugate in KA ; cosets αUAm+1 , α UAm+1 are KA -conjugate; elements α, α satisfy det α ≡ det α tr α ≡ tr α

[m/2]+1

(mod UF (mod p

−[m/2]

),

).

Proof. The equivalence of (1) and (2) is the Conjugacy Theorem 15.2, and (3) is the dual version of (2). The equivalence of (3) and (4) follows from the “normal form” above. 44.3. We ﬁx a character ψ ∈ F of level one, to use the classiﬁcation theory of Chapter IV, and standardize our notation for the rest of the section. Let (A, n, α), n 1, be a ramiﬁed simple stratum in A = M2 (F ). By deﬁnition, the integer n is odd, so we write n = 2m+1. We let A02 (F ; α) denote the set of equivalence classes of representations π ∈ A02 (F ) which contain the character ψα : 1+x → ψA (αx) = ψ(tr(αx)) of UAm+1 . Note 1. The Classiﬁcation Theorem (15.5) and 15.6 Proposition 1 imply that we have a bijection C(ψα , A) −→ A02 (F ; α), Λ −→ c-IndG J Λ, where J = F [α]× UAm+1 . Note 2. Every π ∈ A02 (F ; α) has (π) = n/2. If (A , n , α ) is some other ramiﬁed simple stratum, with n 1, the Conjugacy Theorem (15.2) implies that either A02 (F ; α ) = A02 (F ; α) or A02 (F ; α ) ∩ A02 (F ; α) = ∅. If L/F is a ﬁnite ﬁeld extension, L(n) will denote the set of x ∈ L for which υL (x) = −n. If E/F is a separable, totally ramiﬁed quadratic extension, we set dE/F = pd+1 and d = d(E|F ). The unique non-trivial character of F × trivial on NE/F (E × ) will be denoted κE/F . We choose δE/F ∈ p−d so that κE/F (1+x) = ψ(xδE/F ), x ∈ p1+[d/2] . The section is devoted to proving the following result. Theorem. Let (A, n, α), n 1, be a ramiﬁed simple stratum, and write n = 2m+1. The following are equivalent: (1) The set A02 (F ; α) contains an ordinary representation. (2) All representations π ∈ A02 (F ; α) are ordinary. (3) There exist a totally ramiﬁed, separable quadratic extension E/F and an element β ∈ E(n) such that

270

11 Ordinary Representations

(a) n 3d(E|F ), and (b) the following congruences are satisﬁed: det α ≡ NE/F (β)

[m/2]+1

(mod UF

tr α ≡ δE/F + TrE/F (β)

),

(mod p−[m/2] ).

(44.3.1)

The proof occupies the rest of the section. Remark. If we regard E/F and β as given, the congruences (44.3.1) determine the KA -conjugacy class of the character ψα | UAm+1 completely (44.2). If, however, we start with ψα , they do not determine E/F or β with any precision: we have to analyze this phenomenon below. Note also that the conditions in the theorem are independent of the choice of ψ ∈ F of level one. 44.4. We start the proof of 44.3 Theorem with a calculation. Let ρ ∈ Gim 2 (F ). We assume that ρ is totally ramiﬁed and minimal, in the sense that n(ρ, ψ) n(χ ⊗ ρ, ψ), for all characters χ of F × (cf. 41.5). Thus there is a separable quadratic extension E/F , and a character ξ of E × , such that ρ ∼ = IndE/F ξ. By 41.5 Theorem, we may choose E so that n = n(ρ, ψ) 3d(E|F ).

(44.4.1)

The character ξ is minimal, of level n−d, d = d(E|F ). The integer n is invariably odd (41.5 Lemma), so we write n = 2m+1. The character ψE = ψ ◦ TrE/F has level 1−d (as follows from 41.2 Proposition (1)), so there exists β ∈ E(n) such that ξ(1+x) = ψE (βx),

υE (x) 1+[(n−d)/2].

(44.4.2)

[(n−d+1)/2]

. This condition determines the coset βUE The representation π = π(ρ) is minimal of normalized level (π) = n/2, and n = n(π, ψ) = n(ρ, ψ). Therefore there is a ramiﬁed simple stratum (A, n, α) such that π ∈ A02 (F ; α). Our calculation will show: Lemma. The elements α, β satisfy the congruences (44.3.1). Proof. We ﬁrst exploit the relation ωπ = det ρ = κE/F ⊗ ξ | F × . Let 1+x ∈ UAm+1 ∩ F × = UF . Thus ωπ (1+x) = ψ(x tr(α)). On the other hand, the hypothesis n 3d implies [m/2] [d/2], so det ρ(1+x) = ψ (δE/F +TrE/F (β))x , [m/2]+1

whence tr α ≡ δE/F +TrE/F (β)

(mod p−[m/2] ).

(44.4.3)

44. Ordinary Representations and Strata

271

44.5. We describe π in terms of the classiﬁcation theory of §15. The representation π contains a character Λ of the group J = F [α]× UAm+1 such that Λ | UAm+1 = ψα . Thus Λ ∈ C(ψα , A) and (A, J, Λ) is a cuspidal type in π. The character Λ induces an irreducible representation Ξ of KA : the pair (A, Ξ) is a cuspidal inducing datum in π. As before, we let Γr denote the group of characters of F × trivial on UFr , r 0. Let K(n) denote the set of g ∈ KA with υF (det g) = −n. From (27.7.1), we have the formula ˇ ε(χπ, 12 , cψ) χ(det g) ψ(−c tr(g)) tr Ξ(g) =k χ∈Γm+1 /Γ0 , m+1 c∈UF /UF

=k

ε(χπ, 12 , ψ) χ2 ωπ (c) χ(det g) ψ(−c tr(g)),

χ∈Γm+1 /Γ0 , m+1 c∈UF /UF

for g ∈ K(n), where k is a positive constant depending only on n. Lemma. Let du be the Haar measure on UAm+1 such that

m+1 UA

du = 1.

(1) Let γ ∈ K(n). The function of g ∈ K(n), deﬁned by ˇ ∗ ψγ−1 (g) = g− → tr Ξ

m+1 UA

ˇ tr Ξ(gu) ψγ (u) du,

is identically zero unless the cosets γUAm+1 , αUAm+1 are KA -conjugate. (2) For g ∈ J ∩ K(n), we have ˇ tr Ξˇ ∗ ψα−1 (g) = Λ(g). Proof. Set P = rad A. By deﬁnition, the representation Ξ | UAm+1 is the direct sum of the characters ψγ , γ ∈ K(n), such that the coset γ+P−m = γUAm+1 is KA -conjugate to αUAm+1 , whence follows (1). The natural representation of J on the ψα -isotypic space in Ξ is the character Λ, and this implies (2). In particular, tr Ξˇ ∗ ψ −1 (α) = 0. (44.5.1) α

The relation π = π(ρ) gives ˇ tr Ξ(g) =k

χ∈Γm+1 /Γ0 , m+1 c∈UF /UF

ε(χ ⊗ ρ, 12 , ψ) χ2 det ρ(c) χ(det g) ψ(−c tr(g)),

272

11 Ordinary Representations

for g ∈ K(n). Taking u ∈ UAm+1 , we have χ(det αu) = χ(det α)χ(det u) and ψ(−c tr(αu)) = ψ(−c tr(α)) ψ−cα (u). Therefore tr Ξˇ ∗ ψα−1 (α) =k

ε(χ ⊗ ρ, 12 , ψ) χ2 det ρ(c) χ(det α) ψ(−c tr(α))

χ, c

m+1 UA

χ(det u) ψ−cα (u) ψα (u) du.

We choose δχ ∈ p−m so that χ(1+x) = ψ(δχ x), x ∈ p[m/2]+1 . Thus χ(det u) = ψδχ (u), u ∈ UAm+1 . So, writing P = rad A, the inner integral is non-zero if and only if (44.5.2) δχ + (1−c)α ≡ 0 (mod P−m ). (When the integral is non-zero, its value is 1.) We could replace P by pF [α] here. In the ﬁeld F [α], the element δχ has even valuation while (c−1)α has odd valuation (or is zero). The congruence (44.5.2) is therefore equivalent to (mod P−m ).

δχ ≡ (1−c)α ≡ 0

A character χ thus contributes to the sum only if δχ ∈ P−m ∩ F = p−[m/2] : that is, χ has level [m/2]. A unit c ∈ UF contributes to the sum only if [m/2]+1 . For such χ and c we have χ(c) = 1. Therefore, c ∈ UAm+1 ∩ F = UF invoking (44.4.3), tr Ξˇ ∗ ψα−1 (α)

= k

ε(χ ⊗ ρ, 12 , ψ) det ρ(c) χ(det α) ψ(−c tr(α))

χ∈Γ[m/2]+1 /Γ0 , [m/2]+1

c∈UF

= k

m+1 /UF

ε(χ ⊗ ρ, 12 , ψ) χ(det α) ψ(−tr(α)),

χ∈Γ[m/2]+1 /Γ0

for positive constants k , k . For χ ∈ Γ[m/2]+1 , we have ε(χ ⊗ ρ, 12 , ψ) = λE/F (ψ) ε(χE ξ, 12 , ψE ). Writing ϕ = ϕE/F (cf. 44.2), the character χE has level ϕ([m/2]) < (n−d)/2. We can therefore apply 23.8 to get ε(χ ⊗ ρ, 12 , ψ) = χE (β)−1 ε(ρ, 12 , ψ), We achieve ˇ ∗ ψα−1 (α) = k ε(ρ, 1 , ψ) ψ(−tr(α)) tr Ξ 2

χ(det α/NE/F (β)),

χ

with χ ranging over Γ[m/2]+1 /Γ0 . This sum is non-zero if and only if det α ≡ NE/F (β)

(mod U [m/2]+1 ),

so completing the proof of 44.4 Lemma.

(44.5.3)

44. Ordinary Representations and Strata

273

44.6. We now prove (1) ⇔ (3) in 44.3 Theorem. Let (A, n, α), n = 2m+1 1, be a ramiﬁed simple stratum. Let π ∈ A02 (F ; α) be ordinary. Let ρ ∈ Gim 2 (F ) satisfy π = π(ρ). We write ρ = IndE/F ξ as in (44.4.2). We have n = n(ρ, ψ) and ρ is minimal. The congruences (44.3.1) are given by 44.4 Lemma. Conversely, let (A, n, α) satisfy (3), relative to an extension E/F and an element β ∈ E(n). We choose a character ξ of E × satisfying (44.4.2). Since n is odd, the character ξ is minimal over F , the representation ρ = IndE/F ξ is irreducible, and 44.4 Lemma shows that π(ρ) ∈ A02 (F ; α). Thus (A, n, α) satisﬁes (1). 44.7. At this point, it becomes convenient to introduce some new vocabulary and summarize the position. Let F /F be the separable algebraic closure of F used to deﬁne WF . Deﬁnition 1. An admissible wild triple (E/F, n, β) consists of (a) a totally ramiﬁed quadratic extension E/F , E ⊂ F , (b) an odd integer n = 2m + 1 1, and (c) an element β ∈ E(n), satisfying the condition n 3d(E|F ). Deﬁnition 2. Let (E/F, n, β) be an admissible wild triple. Deﬁne G02 (F ; β) to be the set of classes of representations ρ ∈ G02 (F ) of the form ρ = IndE/F ξ, where ξ(1+x) = ψE (βx),

1+[(n−d)/2]

x ∈ pE

,

(44.7.1)

and d = d(E|F ). Observe here that ξ has level n−d and n(ρ, ψ) = n. Moreover, ρ and ξ are minimal, so π(ρ) ∈ A02 (F ; α), for some ramiﬁed simple stratum (A, n, α). Lemma 44.4 shows that the stratum must satisfy the conditions (3) of 44.3 Theorem. To summarize: Proposition. Let (E/F, n, β) be an admissible wild triple, n = 2m+1. There exists a ramiﬁed simple stratum (A, n, α) satisfying det α ≡ NE/F (β) tr α ≡ δE/F + TrE/F (β)

[m/2]+1

(mod UF

−[m/2]

(mod p

),

).

These conditions determine the G-conjugacy class of the character ψα | UAm+1 uniquely, and π(G02 (F ; β)) ⊂ A02 (F ; α).

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44.8. To prove (1) ⇒ (2) in 44.3 Theorem, we rely on a counting argument. This will occupy the rest of the section. The group Γ1 acts on the set A02 (F ; α) by (χ, π) → χπ. Lemma 1. Let (A, n, α) be a ramiﬁed simple stratum, n = 2m+1 1; then

Γ1 \A02 (F ; α) = q m . Proof. Set J = F [α]× UAm+1 . The elements π of A02 (F ; α) are the induced representations πΛ = c-IndG J Λ, where Λ ranges over the set C(ψα , A) of characters of J satisfying Λ | UAm+1 = ψα . Indeed, Λ → πΛ is a bijection between C(ψα , A) and A02 (F ; α) (15.6 Proposition 1). Thus two representations πΛi lie in the same Γ1 -orbit if and only the characters Λi agree on J ∩ UA1 = UF1 [α] UAm+1 . Therefore

Γ1 \A02 (F ; α) = J ∩ UA1 : U m+1 = U 1 : U m+1 = q m , F [α] A F [α] as required. We need a corresponding result on the Galois side. If (E/F, n, β) is an χ ⊗ ρ. admissible wild triple, then Γ1 acts on G02 (F ; β) by (χ, ρ) → Lemma 2. Let (E/F, n, β) be an admissible wild triple. With d = d(E|F ), we have

q [(n−d)/2] if n > 3d,

Γ1 \G02 (F ; β) = 1 [(n−d)/2] if n = 3d. 2q Proof. Let σ ∈ Gal(E/F ), σ = 1. There are precisely q [(n−d)/2] characters ξ of UE1 satisfying (44.7.1). For any such ξ, the representation IndE/F ξ is irreducible. These characters ξ are Galois-conjugate in pairs if the characters 1+[(n−d)/2] are the same. η : 1+x → ψE (βx), η σ : 1+x → ψE (β σ x) of UE Otherwise, they give rise to distinct elements of Γ1 \G02 (F ; β). 1+[(n−d)/2] are the same if and only if The characters η, η σ of UE [(n−d+1)/2]

β/β σ ∈ UE

.

(44.8.1)

We have υE (β/β σ − 1) = d (41.1 Lemma), so (44.8.1) holds if and only if [(n−d+1)/2] d. This translates as n 3d, that is, n = 3d and the lemma follows. 44.9. In this paragraph, we analyze the set of admissible wild triples giving rise to the same simple stratum in 44.7. To do this, we ﬁx an admissible wild triple (E/F, n, β), n = 2m+1.

44. Ordinary Representations and Strata

275

Proposition. Let E /F be a totally ramiﬁed, separable quadratic extension, and let β ∈ E (n). Suppose that NE /F (β ) ≡ NE/F (β)

TrE /F (β ) ≡ TrE/F (β)

[m/2]+1

(mod UF (mod p

−[m/2]

),

).

(44.9.1)

We then have d(E |F ) = d(E|F ) and δE /F ≡ δE/F (mod p−[m/2] ). In particular, (E /F, n, β ) is an admissible wild triple. Proof. Let us abbreviate d = d(E|F ), d = d(E |F ). By 41.1 Lemma, υF (TrE/F (β)) −m + [d/2], with equality if and only if d is odd, i.e., d < υF (4). This translates into d = min {2υF (TrE/F (β))+2m+1, υF (4)}. Likewise, d = min {2υF (TrE /F (β ))+2m+1, υF (4)}. The assumption n 3d implies −m+[d/2] < −[m/2], so the congruence of traces implies υF (TrE/F (β)) = υF (TrE /F (β )). We deduce that d = d n/3, whence (E /F, n, β ) is admissible. The second assertion of the proposition is equivalent to demanding that [m/2]+1 . These the characters κ = κE/F , κ = κE /F agree on the group UF 2 2 characters both have level d and they satisfy κ = κ = 1. It is therefore [m/2]+1 which is a norm enough to show that κ vanishes on any element of UF from E. If [m/2] d, there is nothing to do, since both κ and κ are trivial [m/2]+1 ⊂ UFd+1 ⊂ Ker κ ∩ Ker κ . We therefore assume [m/2] < d. on UF Any element of E = F [β] can be written uniquely in the form x+yβ, x, y ∈ F . The element x+yβ lies in UE1 if and only if x ∈ UF1 and yβ ∈ pE . 2 That is, UE1 = UF1 (1+pm+1 β). Since κ = 1, this reduces us to showing that [m/2]+1 κ (NE/F (1+xβ)) = 1 whenever NE/F (1+xβ) ∈ UF . [m/2]+1 if and only if 1+xβ ∈ Let x ∈ pm+1 ; then NE/F (1+xβ) ∈ UF [m/2]+1 . Equivalently, 2υF (x) n+[m/2]+1. This condition is unchanged UE on replacing β by β . For such x, we have NE/F (1+xβ) = 1 + xTrE/F (β) + x2 NE/F (β) ≡ 1 + xTrE /F (β ) + x2 NE /F (β ) (mod pd+1 ), whence κ (NE/F (1+xβ)) = κ (NE /F (1+xβ )) = 1, as required. Consequently: Corollary. Let (E/F, n, β) be an admissible wild triple. Let (A, n, α) be a ramiﬁed simple stratum such that π(G02 (F ; β)) ⊂ A02 (F ; α). If (E , n, β ) is a triple satisfying (44.9.1), then π(G02 (F ; β )) ⊂ A02 (F ; α).

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11 Ordinary Representations

44.10. We let (E/F, n, β), (E /F, n, β ) be admissible wild triples satisfying the conditions (44.9.1), namely NE /F (β ) ≡ NE/F (β)

[m/2]+1

(mod UF

TrE /F (β ) ≡ TrE/F (β)

(mod p

−[m/2]

),

),

where n = 2m+1. We set d = d(E|F ) = d(E |F ). Proposition. Suppose, with the hypotheses above, that G02 (F ; β)∩G02 (F ; β ) = ∅. We then have NE /F (β ) ≡ NE/F (β)

TrE /F (β ) ≡ TrE/F (β)

[(1+n−d)/2]

(mod NE/F (UE (mod

)),

−n+[(1+n−d)/2] TrE/F (pE )).

(44.10.1)

In this case, G02 (F ; β) = G02 (F ; β ) and E = E . 1+[(n−d)/2]

Proof. Denote by ψE,β the character 1+x → ψE (βx) of UE , and similarly deﬁne ψE ,β . We take ρ ∈ G02 (F ; β) ∩ G02 (F ; β ) and accordingly write ρ = IndE/F ξ = × IndE /F ξ , for characters ξ of E × , ξ of E such that 1+[(n−d)/2]

ξ | UE

= ψE,β ,

ξ | UE

1+[(n−d)/2]

= ψE,β .

Consider ﬁrst the possibility E = E . We then have ξ = ξ τ , for some τ ∈ Gal(E/F ) (41.3 Lemma). Consequently, β ≡ βτ

[(1+n−d)/2]

(mod UE

).

(44.10.2)

Directly from the deﬁnition, therefore, G02 (F ; β ) = G02 (F ; β) and the congruence of norms holds. In additive notation, the congruence (44.10.2) says β ≡ βτ

−n+[(1+n−d)/2]

(mod pE

).

The congruence of traces is satisﬁed and the result therefore holds in the case E = E. If n > 3d, 41.5 Corollary implies that E = E . If n = 3d and ρ is triply imprimitive, there are three ﬁelds Ei /F from which ρ is induced. However, no two of the elements δEi /F are congruent modulo p1−d loc. cit. Since n = 3d, we have 1−d −[m/2], so 44.9 Proposition again implies E = E . 44.11. Let (E/F, n, β) be an admissible wild triple and (A, n, α) a simple stratum such that π(G02 (F ; β)) ⊂ A02 (F ; α). The triple determines the charac[n/2]+1 rather than the ﬁeld F [α]. However, one can extract some ter ψα of UA useful information.

44. Ordinary Representations and Strata

277

Proposition. Let (A, n, α) be a ramiﬁed simple stratum and let (E/F, n, β) be a wild admissible triple. Suppose that π(G02 (F ; β)) ⊂ A02 (F ; α). Then n > 3d(E|F ) ⇐⇒ n > 3d(F [α]|F ) n = 3d(E|F ) ⇐⇒ n 3d(F [α]|F ). When the ﬁrst alternative holds, we have d(F [α]|F ) = d(E|F ). Proof. We abbreviate d = d(E|F ), dα = d(F [α]|F ) and work case by case. Assume to start with that n > 3d and d is odd. We have υF (TrE/F β) = −m + (d−1)/2 < −d, since 3d < n. For the same reason, this valuation is < −[m/2], so υF (tr α) = υF (TrE/F (β) + δE/F ) = −m + (d−1)/2. If dα is odd, we get dα = d < n/3, as required. This valuation is < −m+υF (2), so the case of dα = υF (4) cannot arise. We take the case n > 3d, d = υF (4). We then have υF (TrE/F (β) + δE/F ) min {−m+υF (2), −υF (4)}. If dα were odd, so dα < d = υF (4), we would have −m + (dα −1)/2 min {−m+υF (2), −υF (4)}, which is impossible. Thus dα = d = υF (4) and n > 3dα . We therefore assume n = 3d. In particular, d < υF (4), d is odd and n−d is even. The element TrE/F (β) has valuation −m + (d−1)/2 = −d < −[m/2]. Thus υF (tr α) −d. If dα = υF (4), then surely n < 3dα , so we assume dα is odd. Therefore υF (tr α) = −m + (dα −1)/2 −d = −n/3. This yields dα n/3, as desired. Remark. Suppose we are in the case n 3d(F [α]|F ). If A02 (F ; α) has an ordinary element, there is a wild admissible triple (E/F, n, β) satisfying the congruences (44.3.1). The proposition gives n = 3d(E|F ) whence n ≡ 0 (mod 3). Put another way, if (A, n, α) is a ramiﬁed simple stratum with n 3d(F [α]|F ) and n ≡ ±1 (mod 3), every representation π ∈ A02 (F ; α) is exceptional.

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11 Ordinary Representations

44.12. We now prove the implication (1) ⇒ (2) in 44.3 Theorem. By hypothesis, there is an admissible wild triple (E/F, n, β) with π(G02 (F ; β)) contained in A02 (F ; α). In particular, the elements α, β satisfy the congruences (44.3.1). Set [m/2]+1 [(n−d+1)/2] −[m/2] −n+[(n−d+1)/2] : NE/F (UE ) p : TrE/F (pE ) . r = UF Proposition 44.10 gives admissible wild triples (E/F, n, βi ), 1 i r, satisfying π(G02 (F ; βi )) ⊂ A02 (F ; α), G02 (F ; βi )

∩

G02 (F ; βj )

1 i r, = ∅,

i = j.

Each set Γ1 \G02 (F ; βi ) has q [(n−d)/2] elements if n > 3d, or q [(n−d)/2] /2 elements if n = 3d (44.8 Lemma 2). The partial correspondence π is injective (44.1 Theorem (2)), so Lemma 1 of 44.8 reduces us to proving if n > 3d, q [(n−d)/2] q m /r = [(n−d)/2] /2 if n = 3d. q This follows from a routine calculation, and we have completed the proof of 44.3 Theorem. Remark. The list (Ei /F, n, βi ), 1 i r, in the last proof does not exhaust the set of admissible wild triples (E /F, n, β ) for which π(G02 (F ; β )) ⊂ A02 (F ; α): it counts only those satisfying (44.9.1) relative to a ﬁxed triple (E/F, n, β). These are enough. We have no further use for this fact, but it is worthy of a formal record: Corollary. Let (A, n, α), n = 2m+1, be a ramiﬁed simple stratum and let (E/F, n, β) be an admissible wild triple satisfying det α ≡ NE/F (β)

[m/2]+1

(mod UF

),

(mod p−[m/2] ).

tr α ≡ δE/F + TrE/F (β)

Let π ∈ A02 (F ; α). There exists an admissible wild triple (E /F, n, β ), satisfying [m/2]+1 ), NE /F (β ) ≡ NE/F (β) (mod UF TrE /F (β ) ≡ TrE/F (β)

(mod p−[m/2] ),

such that π = π(ρ), for some ρ ∈ G02 (F ; β ).

45. Exceptional Representations and Strata

279

45. Exceptional Representations and Strata Let (A, n, α), n 1, be a ramiﬁed simple stratum. We shall say that it is ordinary if A02 (F ; α) contains an ordinary representation. By 44.3 Theorem, this is equivalent to A02 (F ; α) ⊂ π(Gim 2 (F )). The purpose of this section is to give a simple criterion for a stratum to be ordinary. 45.1. We start with an elementary observation: Lemma. There exists ψ ∈ F such that (a) ψ has level one, and (b) ψ(x+x2 ) = 1, x ∈ o. A character ψ ∈ F has properties (a) and (b) if and only if ψ = uψ, for some u ∈ UF1 . Proof. Let η be the non-trivial character of F2 , and set ψ0 = η ◦ Trk/F2 . The Frobenius substitution x → x2 generates Gal(k/F2 ), so ψ0 (y 2 ) = ψ0 (y), y ∈ K, whence ψ0 (y+y 2 ) = 1. Moreover, ψ0 is the unique non-trivial character of k with this property. We take ψ ∈ F so that ψ | o is the inﬂation of ψ0 : this character has the desired properties (a), (b), and its uniqueness follows from that of ψ0 . We call special a character ψ ∈ F satisfying the conditions of the lemma. 45.2. We prove: Theorem. Let (A, n, α), n = 2m+1 1, be a ramiﬁed simple stratum. The following are equivalent: (1) The stratum (A, n, α) is ordinary. (2) Either (a) n > 3d(F [α]|F ), or (b) the polynomial Cα (X) = X 3 − tr(α)X 2 + det(α) has a root in F . Remark. The reducibility properties of the polynomial Cα (X) depend only on [n/2]+1 , not on the stratum and the choice of character the character ψα of UA −1 , and Cuα (X) ψ. For, if we replace ψ by u ψ, with u ∈ UF , we have ψα = ψuα splits over F if and only if Cα (X) does. This enables us to make convenient choices of ψ.

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11 Ordinary Representations

45.3. Suppose that (A, n, α) is ordinary, and choose an admissible wild triple (E/F, n, β) which satisﬁes the congruences (44.3.1) relative to α. Set d = d(E|F ), dα = d(F [α]|F ). Suppose that n 3dα , so that n = 3d, d = d(E|F ) (44.11 Proposition). It follows that d is odd and TrE/F (β) has valuation −d. Thus TrE/F (β)2 NE/F (β)−1 has valuation d. It will now be convenient to assume that ψ is special, as in 45.1. We take y ∈ o and set x = yTrE/F (β)/NE/F (β). Thus 1+xβ ∈ UEd and 1 = κE/F (NE/F (1+xβ)) = κE/F (1 + x TrE/F (β) + x2 NE/F (β)) = ψ δE/F (x TrE/F (β) + x2 NE/F (β) = ψ δE/F TrE/F (β)2 NE/F (β)−1 (y+y 2 ) . This holds for all y ∈ o. By 45.1 Lemma, we get δE/F TrE/F (β)2 NE/F (β)−1 ≡ 1 (mod p). Setting t = TrE/F (β), the congruences (44.3.1) imply Cα (t) = t3 − tr(α)t2 + det(α) ≡ 0 (mod p1−n ). We have υF (det α) = −n = −3d, so we write det α = ua−3 , for some a ∈ F × , u ∈ UF . Setting s = at, we have cα (s) = s3 − atr(α)s2 + u ≡ 0

(mod p).

The element tr(α) has valuation −d, so atr(α) ∈ o. The reduction modulo p of the polynomial cα (X) is separable, so Hensel’s Lemma applies to show that cα (X) has a root in F . Thus Cα has a root in F , and (1) ⇒ (2) in 45.2 Theorem. 45.4. We turn to the converse. Let (A, n, α) be a ramiﬁed simple stratum with n = 2m+1 1. Suppose ﬁrst that n 3dα , dα = d(F [α]|F ), and that Cα (X) has a root in F . If n ≡ 0 (mod 3), elementary changes of variable transform Cα (X) into an Eisenstein polynomial, so Cα is irreducible over F in this case. We deduce that n ≡ 0 (mod 3). Elementary Lemma. Let n be an integer satisfying n 1, n ≡ 0 (mod 3). Let a, b ∈ F satisfy υF (b) = −n, υF (a) −n/3. Any root γ ∈ F of the polynomial c(X) = X 3 + aX 2 + b then satisﬁes υF (γ) = −n/3. Proof. Extending the base ﬁeld, we can assume that c(X) splits completely in F , without changing the relative conditions on the coeﬃcients. If c(X) has a root of valuation < −n/3, the vanishing of the linear term in c(X) implies it is the only root with this property. This, however, is inconsistent with the condition on the coeﬃcient a.

45. Exceptional Representations and Strata

281

Since υF (tr α) −n/3, we deduce that any root γ ∈ F of Cα (X) has υF (γ) = −n/3. Lemma 1. Let ∈ F , υF () −m. There exists β = β ∈ KA such that det β = det α,

tr β = .

The algebra E = F [β] is a ﬁeld satisfying E × ⊂ KA . Proof. This is immediate, as a consequence of the normal form used in 44.2. We also need a reﬁnement of 45.1 Lemma. Lemma 2. Let j be an integer, 0 j [(υF (2)−1)/2]. There exists a character ψ ∈ F, of level one, such that ψ(y+y 2 ) = 1,

y ∈ p−j .

Proof. Consider the map ℘ : p−j /p → F/p given by y → y+y 2 . The condition on j implies that ℘ is additive, but its image does not contain o/p. We now assume ψ has been chosen to satisfy the conditions of Lemma 2 for the largest possible value of j. Set d = n/3 (which is odd) and let be a root of Cα (X) in F . We deﬁne β as in Lemma 1 for this and set E = F [β]. This gives d(E|F ) = d = n/3 and (E/F, n, β) is an admissible wild triple. We put δ = tr(α)−. Thus υF (δ) −d. We show that δ ≡ δE/F

(mod p−[m/2] ).

This will imply that (E/F, n, β) is an admissible wild triple satisfying (44.3.1) and that (A, n, α) is ordinary, as required. To do this, it is enough to show that the character κ : 1+x → ψ(δx) [m/2]+1 [m/2]+1 is trivial on norms from UE . This reduces to showing that of UF [m/2]+1 κ(NE/F (1+xβ)) = 1 when x ∈ F and xβ ∈ pE . For such x, we can write x = yTrE/F (β)/NE/F (β) = y/ det α, where 2υF (y) = 2υF (x) − 2n + 2d m 2 − n + 2d + 1 = m 2 − d + 1 1 − υF (2). We now repeat the calculation of 45.3: κ(NE/F (1+xβ)) = ψ(δ(x TrE/F (β) + x2 NE/F (β)) = ψ(δ TrE/F (β)2 NE/F (β)−1 (y+y 2 )).

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11 Ordinary Representations

Since δ TrE/F (β)2 NE/F (β)−1 = (tr α − ) 2 det α−1 = 1, this reduces to κ(NE/F (1+xβ)) = ψ(y+y 2 ) = 1, as required. 45.5. It remains to show that a ramiﬁed simple stratum (A, n, α), which satisﬁes n > 3d, d = d(F [α]|F ), is necessarily ordinary. The case n > 4d is trivial: (F [α]/F, n, α) is an admissible wild triple which satisﬁes the requirements of (44.3.1). We therefore assume 3d < n < 4d. This implies [m/2] < d. We consider an admissible wild triple (E/F, n, β), with E × ⊂ KA , in which d(E|F ) = d. Lemma. Let v be an integer satisfying d v > [m/2]. Let β ∈ KA satisfy det β = det β,

tr(β ) ≡ tr(β)

(mod p−v ).

Writing E = F [β ], we then have δE /F ≡ δE/F

(mod pn−2d−2v ).

Proof. The condition on the trace of β implies that d(E |F ) = d. Write κ = κE/F , κ = κE /F . The assertion of the lemma is equivalent to −1 κ κ being trivial on (1+2d+2v−n)-units in F . Since 1+2d+2v−n d, 2 and κ 2 = κ = 1, it is enough to show that κ, κ agree on norms of (1+2d+2v−n)-units of E of the form 1+xβ, with x ∈ F . The condition 1+xβ ∈ UE1+2d+2v−n is equivalent to 2υF (x) 1+2d+2v, or υF (x) 1+d+v. Writing out, NE/F (1+xβ) = 1 + x tr(β) + x2 det(β) ≡ 1 + x tr(β ) + x2 det(β )

(mod pd+1 ).

The character κ has level d, so κ (NE/F (1+xβ)) = κ (NE /F (1+xβ )) = 1, and symmetrically. The lemma is proved. We return to the stratum (A, n, α), with d = d(F [α]|F ), 3d < n < 4d. We construct a sequence of quadratic ﬁeld extensions Ei = F [βi ]/F , with δi = δEi /F , as follows.

45. Exceptional Representations and Strata

283

We put β0 = α. For i 0, we set det βi+1 = det α,

tr(βi+1 ) = tr(α) − δi .

In particular, tr(β1 ) ≡ tr(β0 ) (mod p−d ), so we can apply the lemma to get δ1 ≡ δ0

(mod pn−4d ).

Since n−4d > −d, this amounts to δ1 ≡ δ0 (mod p1−d ). Therefore tr(β2 ) = tr(α) − δ1 = tr(β1 ) − δ1 + δ0 ≡ tr(β1 ) (mod p1−d ). The condition 4d > n implies d−1 > [m/2], we can apply the lemma to get δ2 ≡ δ1

(mod pn−4d+2 ).

In particular, δ2 ≡ δ1 (mod p2−d ). Iterating, we get δi+1 ≡ δi

(mod pi+1−d )

provided i < d−[m/2]. Setting k = d−[m/2], (Ek /F, n, βk ) is an admissible wild triple satisfying the conditions (44.3.1): det βk = det α,

tr βk + δk ≡ tr α

(mod p−[m/2] ).

By 44.3 Theorem, the stratum (A, n, α) is ordinary, as required. 45.6. For ease of reference, we display the formal consequence of the theorem. We call the ramiﬁed simple stratum (A, n, α) exceptional if it is not ordinary. Equivalently, any π ∈ A02 (F ; α) is exceptional. Corollary. Let (A, n, α), n 1, be a ramiﬁed simple stratum. The following are equivalent: (1) the stratum (A, n, α) is exceptional; (2) n 3d(F [α]|F ) and the polynomial Cα (X) = X 3 − tr(α)X 2 + det(α) is irreducible over F . Remark. Let (A, n, α) be a ramiﬁed simple stratum, 1 n 3dα . We have already observed (45.3) that Cα (X) is irreducible over F in the case n ≡ 0 (mod 3). So, according to the theorem, (A, n, α) cannot be ordinary: cf. 44.11 Remark.

12 The Dyadic Langlands Correspondence

46. Tame lifting 47. Interior actions 48. The Langlands-Deligne local constant modulo roots of unity 49. The Godement-Jacquet local constant and lifting 50. The existence theorem 51. Some special cases 52. Octahedral representations

In the sole remaining case p = 2, the Langlands correspondence has only been deﬁned (44.1) on the set Gim 2 (F ) of classes of imprimitive representations ρ ∈ G02 (F ). We have to deﬁne πF (ρ) when ρ is primitive, and verify that the map πF : G02 (F ) → A02 (F ) so obtained satisﬁes the requirements of the statement in 33.1. Any primitive ρ ∈ G02 (F ) is totally ramiﬁed, that is, ρ ∈ Gwr 2 (F ) = 0 wr 0 nr (F ). We correspondingly put A (F ) = A (F ) A G2 (F ) Gnr 2 2 2 2 (F ). Let wr ρ ∈ G2 (F ); the deﬁnition of π = πF (ρ) is given in 50.3: after a little preparation, it is explicit and direct. At ﬁrst pass, however, it only gives the relation ε(ρ, s, ψ)3 = ε(π, s, ψ)3 . The process of extracting the cube root, to get the required equality of local constants, depends on a further analysis of these objects and their properties relative to certain sorts of base ﬁeld extension (§48, §49).

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12 The Dyadic Langlands Correspondence

wr That achieved, we have a map πF : Gwr 2 (F ) → A2 (F ) satisfying the deﬁning (and determining) property of the Langlands correspondence. It remains only to prove that it is bijective. This requires a diﬀerent family of methods. The strong uniqueness properties of the correspondence, expressed via the Converse Theorems, imply that it is compatible with automorphisms of the base ﬁeld F (§47). The argument revolves around the ability to recognize ﬁxed points for certain automorphisms of F . On the Galois side, this has already been clariﬁed in Chapter X. If K/F is a ﬁnite, cyclic, tamely ramiﬁed ﬁeld extension, the set of Galois-ﬁxed points Gal(K/F ) is the image of Gwr Gwr 2 (K) 2 (F ) under restriction of representations (from WF to WK ): wr ResK/F : Gwr 2 (F ) −→ G2 (K).

On the automorphic side, there is a similar map wr LftK/F : Awr 2 (F ) −→ A2 (K),

deﬁned for any ﬁnite, tamely ramiﬁed extension K/F , and called tame lifting. Its properties are exactly parallel to those of ResK/F and, in particular, it gives the same description of Galois-ﬁxed points. If F contains a primitive cube root of unity, then any primitive ρ ∈ G02 (F ) is tetrahedral, in the sense of 42.3 Comment. In this case, the ﬁxed-point properties of the tame lifting map imply the bijectivity of πF with little diﬃculty (51.3). The general case (§52) requires a more delicate analysis of Galois-ﬁxed points, and of the manner in which two representations are distinguished by the local constants of their twists. It is known that the operations of restriction and tame lifting are related via the Langlands correspondence: πK ◦ ResK/F = LftK/F ◦ πF , for any ﬁnite tamely ramiﬁed extension K/F . However, we have not used this fact: the only known proof relies on global methods and is beyond the scope of this book: we discuss the matter further at the end of the chapter. Except where the contrary is explicitly allowed, we assume that p = 2 throughout this chapter.

46. Tame Lifting 0 Recall that Anr 2 (F ) denotes the subset of A2 (F ) consisting of the classes of unramiﬁed representations (20.1). We set 0 nr Awr 2 (F ) = A2 (F ) A2 (F ).

We call representations π ∈ Awr 2 (F ) totally ramiﬁed.

46. Tame Lifting

287

This corresponds to another dissection of the set A02 (F ): a cuspidal representation π is unramiﬁed if and only if contains a cuspidal inducing datum of the form (M2 (o), Ξ) (20.3 Lemma). Thus π is totally ramiﬁed if and only if it contains a cuspidal inducing datum (A, Ξ) in which eA = 2. The object of this section is to deﬁne a canonical map wr LftK/F : Awr 2 (F ) −→ A2 (K),

for any ﬁnite, tamely ramiﬁed ﬁeld extension K/F . The deﬁnition is straightforward in principle, but it proceeds via the classiﬁcation theory of Chapter IV. Some eﬀort, therefore, is required to show it is independent of choices: this is a tortuous process extending from the statement of 46.3 Proposition to 46.5 Deﬁnition, and could be omitted at a ﬁrst reading. 46.1. We start by deﬁning a notion of lifting for ramiﬁed simple strata. Let (A, n, α), n = 2m+1 1, be a ramiﬁed simple stratum in A = M2 (F ). In particular, E = F [α] is a ﬁeld and A is the unique o-chain order in A with α ∈ KA (cf. 13.5 Proposition). Let K/F be a ﬁnite, tamely ramiﬁed ﬁeld extension and abbreviate eK = e(K|F ). (In particular, eK is odd.) We set AK = M2 (K) and GK = GL2 (K). We regard A as embedded in AK in the obvious way. If K/F happens to be Galois, we can make Gal(K/F ) act on AK coeﬃcient by coeﬃcient, and then A is the set of Gal(K/F )-ﬁxed points in AK . Similarly for G and GK . Since E/F is totally wildly ramiﬁed and K/F is tamely ramiﬁed, the algebra EK = E ⊗F K is a ﬁeld, naturally identiﬁed with the subﬁeld K[α] of AK . The extension EK/K is quadratic and totally ramiﬁed. The valuation υEK (α) = −eK n is odd, whence α is minimal over K. Proposition. Let (A, n, α) be a ramiﬁed simple stratum in A = M2 (F ). Let K/F be a ﬁnite, tamely ramiﬁed ﬁeld extension with eK = e(K|F ). (1) There is a unique oK -chain order AK in AK = M2 (K) such that α ∈ KAK . The triple (AK , nK , α) is a ramiﬁed simple stratum in AK , where nK = eK n. (2) Write rad A = P and rad AK = PK ; then 1+[j/eK ] , P1+j K ∩A=P

j ∈ Z,

and, in particular, A = AK ∩ A,

P = PK ∩ A.

Proof. Since α is minimal over K, part (1) follows from 13.5 Proposition.

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12 The Dyadic Langlands Correspondence

We may choose bases so that A is the standard chain order I and α is a matrix of the form ( ∗0 ∗∗ ). The order AK is then given by (1−e )/2 oK pK K . (46.1.1) AK = (1+e )/2 pK K oK Since K/F is tamely ramiﬁed, we have j+1 1+[j/eK ] TrK/F (pj+1 , K ) = pK ∩ F = p

(46.1.2)

for all j ∈ Z (18.1 Lemma). All assertions now follow from simple calculations. The group KA is generated by UA , F × , and the element α. The GK normalizer KAK of AK thus contains KA . The proposition shows: Gloss. Let A be a chain order in A with eA = 2. There is a unique oK -chain order AK in AK such that KA ⊂ KAK . If α ∈ KA is minimal over F , then AK is the unique chain order in AK such that α ∈ KAK . 46.2. With K/F as in 46.1, deﬁne T : AK = K ⊗F A −→ A, x ⊗ a −→ TrK/F (x)a. In matrix terms,

T :

ab cd

−→

TrK/F a TrK/F b TrK/F c TrK/F d

.

Let (A, n, α), n = 2m+1 1, be a ramiﬁed simple stratum in A, and deﬁne (AK , nK , α) as in 46.1. Writing nK = e(K|F ) n = 2mK +1, we have nK +1 T (PK ) = Pn+1 ,

K +1 T (Pm ) = Pm+1 . K

Thus there is a surjective group homomorphism K +1 T ∗ : UAmKK +1 /UAnK −→ UAm+1 /UAn+1 ,

1+x −→ 1+T (x). then has level Let ψ ∈ F have level one. The character ψK = ψ ◦ TrK/F ∈ K one. Consider the characters ψα : 1+x −→ ψ(trA (αx)),

1+x ∈ UAm+1 ,

ψαK : 1+y −→ ψK (trAK (αy)),

1+y ∈ UAmKK +1 .

(46.2.1)

Immediately: ψαK = ψα ◦ T ∗ .

(46.2.2)

This process preserves the intertwining (or conjugacy, cf. 15.2) structures in which we are interested:

46. Tame Lifting

289

m+1 Lemma. Let (B, n, β) be a ramiﬁed simple stratum in A; then ψβ | UB mK +1 m+1 K intertwines in G with ψα | UA if and only if ψβ | UBK intertwines with ψαK | UAmKK +1 in GK .

Proof. The G-conjugacy class of the character ψα is determined by the pair of cosets [m/2]+1 , trA (α)+p−[m/2] , det(α) UF (44.2 Lemma). Similarly for ψαK . We have [mK /2]+1

UK

[m/2]+1

∩ F = UF

,

−[mK /2]

∩ F = p−[m/2] ,

pK

whence the lemma follows.

The lemma says that the GK -conjugacy class of the character ψαK | UAmKK +1 and the G-conjugacy class of ψα | UAm+1 determine each other uniquely (cf. 15.2). 46.3. With (A, n, α), K/F and ψ as before, let E = F [α] and J = E × UAm+1 . Thus J is the set of elements of G which intertwine the character ψα of UAm+1 (15.1). Likewise, JK = EK × UAmKK +1 is the set of g ∈ GK which intertwine ψαK . Recall that C(ψα , A) is the set of characters Λ of J such that Λ | UAm+1 = ψα (15.3 Deﬁnition). The set C(ψαK , AK ) is deﬁned similarly. Consider the character κK/F = det IndK/F 1K , where 1K denotes the triv2 =1 ial character of WK . We view κK/F as a character of F × . It satisﬁes κK/F and it is tamely ramiﬁed. It is therefore unramiﬁed. We deﬁne

K/F = κK/F () = ±1,

(46.3.1)

for any prime element of F . Using this notation, we prove: Proposition. Let Λ ∈ C(ψα , A). There exists a unique character K ΛK = Lα K/F (Λ) ∈ C(ψα , AK )

such that υ

EK ΛK (y) = K/F

(y)

Λ(NEK/E (y)),

y ∈ EK × .

(46.3.2)

Further: (1) Let Λ, Λ ∈ C(ψα , A); then ΛK = ΛK if and only if Λ = χ ◦ det ⊗Λ, for a character χ of F × which is trivial on NK/F (K × ). (2) If L/K is a ﬁnite, tamely ramiﬁed ﬁeld extension, then α α Lα L/F = LL/K ◦ LK/F .

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12 The Dyadic Langlands Correspondence

(3) If (A, n, β) is a simple stratum such that ψβ = ψα on UAm+1 , then C(ψβ , A) = C(ψα , A), C(ψβK , AK ) = C(ψαK , AK ) and LβK/F = Lα K/F . Proof. We take T as in 46.2. For x ∈ EK, we have T (x) = TrEK/E (x). mK +1 , then ψαK (u) = ψα NEK/E (u) . The So, if u ∈ EK × ∩ UAmKK +1 = UEK formula (46.3.2) therefore speciﬁes a unique character ΛK of JK with ΛK ∈ C(ψαK , AK ). 1 → UE1 is surjective, since EK/E is In (1), the norm map NEK/E : UEK tamely ramiﬁed. Thus, if ΛK = ΛK , the characters Λ, Λ agree on UE1 UAm+1 . Therefore Λ = χ ◦ det ⊗Λ, for a tamely ramiﬁed character χ of F × . The character χEK of EK × is trivial; since EK/K is totally wildly ramiﬁed, this implies χK = 1. This proves assertion (1). Assertion (2) is immediate from the deﬁnition. In (3), the equality C(ψβK , AK ) = C(ψαK , AK ) follows from 46.2 Lemma and the Conjugacy Theorem 15.2. To prove the main assertion, we ﬁrst assume that the extension K/F is cyclic, of degree d say. We set Γ = Gal(K/F ) and choose a generator γ of Γ . We let Γ act, coeﬃcient by coeﬃcient, on AK . The group JK is then stable under Γ , and we can form the semi-direct product Γ JK . For x ∈ JK , the element Nγ (x ) = (γx )d surely lies in JK . Lemma. Let x ∈ UAmKK +1 ; then Nγ (x ) ∈ UAm+1 UAnKK+1 , and Nγ induces a surjective homomorphism K +1 Nγ : UAmKK +1 /UAnK −→ UAm+1 /UAn+1

satisfying ψαK (x) = ψα (Nγ (x)),

x ∈ UAmKK +1 .

K +1 ; then Proof. Write x = 1+a, a ∈ Pm K

Nγ (x ) ≡ 1 +

d−1

γ j (a)

(mod PnKK +1 ).

j =0

The right hand side is 1 + T (a) = T ∗ (x), in the notation of 46.2. As in 46.2, K +1 → UAm+1 /UAn+1 is surjective, and the lemma the map T ∗ : UAmKK +1 /UAnK follows.

46. Tame Lifting

291

Write X = Ker ψα , XK = Ker ψαK . It follows from the lemma that Nγ induces an isomorphism Nγ : UAmKK +1 /XK ∼ = UAm+1 /X. For β as in (3), we have J = F [β]× UAm+1 and JK = K[β]× UAmKK +1 . So, if x ∈ JK , we can write x = zu, z ∈ K[β]× , u ∈ UAmKK +1 . Since both γ and conjugation by elements of K[β]× ﬁx ψαK , we get Nγ (x ) ≡ NK[β]/F [β] (z ) Nγ (u)

(mod XK ).

Thus Nγ induces a group homomorphism Nγ : JK /XK −→ J/X, which is visibly independent of the choice of β. For y ∈ JK , we have υ (det y)

K ΛK (y) = K/F

Λ(Nγ (y)),

and we have proved part (3) of the proposition when K/F is cyclic. The transitivity property (2) implies (3) in the case where K/F is Galois. 46.4. To deal with the general case, we use a rather diﬀerent sort of method: Lemma. Let K/F be a ﬁnite, tamely ramiﬁed ﬁeld extension. Let Λ ∈ G GK C(ψα , A) and put ΛK = Lα K/F (Λ). Set π = c-IndJ Λ and = c-IndJK ΛK . We then have (46.4.1) ε(, s, ψK ) = K/F ε(π, s, ψ)[K:F ] . Proof. Combining formula (25.2.1) with 25.5 Corollary, we get ε(π, s, ψ) = q n( 2 −s) Λ(α)−1 ψ(trA α), 1

where, we recall, q = |k| is the size of the residue ﬁeld of F . Similarly, n ( 12 −s)

ε(, s, ψK ) = qKK

ΛK (α)−1 ψK (trAK α),

where qK = |kK | = q f (K|F ) and nK = e(K|F )n. Also ψK (trAK α) = ψK (trA α) = ψ(trA α)[K:F ] , so the result follows from the deﬁnition of ΛK . We ﬁnish the proof of 46.3 Proposition. The result for cyclic extensions and the transitivity properties reduce us to the case in which K/F is totally tamely ramiﬁed, and hence of odd degree.

292

12 The Dyadic Langlands Correspondence

Let C1 (ψα , A) denote the set of characters of J 1 = J ∩ UA1 = UE1 UAm+1 which extend ψα on UAm+1 . Use similar notation for other base ﬁelds. 1 1 For any tame extension K /F , the norm map NEK /E : UEK → UE is α surjective, since EK /E is tamely ramiﬁed. It follows that LK /F induces an

K injective map L1,α K /F : C1 (ψα , A) → C1 (ψα , AK ). 1 If L/F is the normal closure of K/F , then ΛK | JK is the unique element 1,α 1,α K 1 1 is of C1 (ψα , AK ) with image LL/F (Λ | JL ) under LL/K . Thus ΛK | JK determined independently of the choice of α. We have ΛK | K × = (Λ | F × ) ◦ NK/F . The character LβK/F (Λ) is therefore either ΛK or χ ◦ det ⊗ΛK , where χ is unramiﬁed of order 2. The second possibility is excluded by the lemma. We are therefore justiﬁed in henceforward writing LK/F = Lα K/F .

46.5. To be useful, the lifting operation Λ → ΛK must respect the relations of twisting and intertwining (or conjugacy). We check this. Lemma 1. Let (A, n, α), (B, n, β) be ramiﬁed simple strata in A, and set m+1 n = 2m+1 1. Suppose that the characters ψα | UAm+1 , ψβ | UB intertwine in G = GL2 (F ). Let Λ ∈ C(ψα , A), and let Λ be the unique element of C(ψβ , B) which intertwines with Λ in G. Let K/F be a ﬁnite, tamely ramiﬁed ﬁeld extension. The characters LK/F (Λ), LK/F (Λ ) then intertwine in GL2 (K). m+1 . The intertwining hypothesis Proof. Set J = F [α]× UAm+1 , J = F [β]× UB m+1 −1 , ψβ = ψαg = says there exists g ∈ G such that B = g Ag and, on UB −1 g ψg−1 αg . We then have J = g Jg, and Λ = Λ . Reverting brieﬂy to our earlier notation, the deﬁnition gives g −1 LgK/Fαg (Λg ) = Lα K/F (Λ) . −1

g αg , LβK/F on C(ψβ , B) are the same (46.3 Proposition (3)), so The maps LK/F the lemma is proven.

Directly from the original deﬁnition (46.3.2), we get: Lemma 2. Let (A, n, α) be a ramiﬁed simple stratum in A, n = 2m+1 1. Let χ be a character of F × of level l < n/2, and let c ∈ F satisfy χ(1+x) = ψ(cx), x ∈ p[l/2]+1 , or, if l = 0, set c = 0. Let Λ ∈ C(ψα , A). The character χΛ = (χ ◦ det) ⊗ Λ of F [α]× UAm+1 lies in C(ψα+c , A) and, if K/F is a ﬁnite, tamely ramiﬁed ﬁeld extension, then χΛ K = χK ΛK , where χK = χ ◦ NK/F .

46. Tame Lifting

293

We can ﬁnally make the desired: Deﬁnition. Let π ∈ Awr 2 (F ), and let K/F be a ﬁnite, tamely ramiﬁed ﬁeld extension. Deﬁne a representation πK = LftK/F (π) ∈ Awr 2 (K) as follows. (1) If π is minimal, choose a ramiﬁed simple stratum (A, n, α) in M2 (F ) such that π contains a character Λ ∈ C(ψα , A). The representation πK is then the unique element of Awr 2 (K) containing the character K ΛK ∈ C(ψα , AK ) of (46.3.2). (2) Otherwise, choose a decomposition π = χπ , where χ is a character of F × and π ∈ Awr 2 (F ) is minimal. Deﬁne πK by (1) and put πK = χK πK . The representation πK is called the K/F -lift of π. The lemmas show that the deﬁnition of πK is independent of the various choices. It is also independent of the choice of character ψ. We therefore have a well-deﬁned map wr LftK/F : Awr 2 (F ) −→ A2 (K),

π −→ πK ,

(46.5.1)

which we call tame lifting (relative to K/F ). Summarizing its basic properties: Proposition. Let K/F be a ﬁnite, tamely ramiﬁed ﬁeld extension and let π ∈ Awr 2 (F ). (1) The central characters ωπ , ωπK are related by ωπK = ωπ ◦ NK/F . (2) If χ is a character of F × , then (χπ)K = χK πK . (3) If L/K is a ﬁnite, tamely ramiﬁed ﬁeld extension, then LftL/F = LftL/K ◦ LftK/F . ∼ ∼ (4) If π, π ∈ Awr 2 (F ), then πK = πK if and only if π = χπ, for a character × × χ of F trivial on NK/F (K ).

Proof. Part (1) is immediate from the deﬁnition (46.3.2). Parts (2) and (3) have already been noted. Assertion (4) follows from the deﬁnition, 46.2 Lemma and 46.3 Proposition (1). As a direct consequence of 45.2 Theorem, we get: Corollary. Let π ∈ Awr 2 (F ); there exists a cubic extension K/F such that πK is ordinary.

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12 The Dyadic Langlands Correspondence

46.6. Suppose, for the moment, that K/F is a ﬁnite Galois extension. The natural action of Gal(K/F ) on GK = GL2 (K) induces an action on the sets γ A02 (K), Awr 2 (K), which we denote π → π , γ ∈ Gal(K/F ). Clearly, if K/F is also tamely ramiﬁed, then wr Gal(K/F ) LftK/F Awr . 2 (F ) ⊂ A2 (K) Proposition. Let K/F be a ﬁnite, tamely ramiﬁed, cyclic extension and set wr Σ Gal(K/F ) = Σ. If τ ∈ Awr 2 (K) , there exists π ∈ A2 (F ) such that τ = πK . The representation π is determined up to twisting with a character of F × trivial on NK/F (K × ). Proof. Suppose ﬁrst that τ is minimal. We choose a ramiﬁed simple stratum r+1 . (B, s, β) in AK , s = 2r+1, such that τ contains the character ψβK of UB K K σ K For σ ∈ Σ, the characters ψβ , (ψβ ) = ψβ σ intertwine in GK , since they both occur in τ = τ σ . We conclude: det β ≡ det β σ

[r/2]+1

(mod UK

),

tr β ≡ tr β σ

−[r/2]

(mod pK

).

1 The ﬁrst of these congruences implies that the coset det β UK has a Σ-ﬁxed point, whence e(K|F ) divides υK (det β) = −s. Set n = s/e(K|F ) = 2m+1. We then have [r/2]+1

UK

[m/2]+1

∩ F = UF

,

−[r/2]

pK

∩ F = p−[m/2] .

Since K/F is tamely ramiﬁed, it is easy to show that the cohomology groups [r/2]+1 −[r/2] H 1 (Σ, UK ) and H 1 (Σ, pK ) are both trivial. It follows that there exist × x ∈ F with υF (x) = −n and y ∈ p−m satisfying x ≡ det β

[r/2]+1

(mod UK

),

y ≡ tr β

−[r/2]

(mod pK

).

There is a ramiﬁed simple stratum (A, n, α) in A with det α = x, tr α = y. The r+1 characters ψβK | UB , ψαK | UAr+1 are then conjugate in GK (44.2 Lemma). K Consequently, we can replace (B, s, β) by (AK , nK , α) (in the notation of 46.1). The character ψαK of UAmKK +1 is ﬁxed by Σ, whence the same applies to the character Θ ∈ C(ψαK , AK ) which occurs in τ . Thus Θ | K[α]× is ﬁxed under the natural action of Σ on K[α] as Gal(K[α]/F [α]). Therefore Θ | K[α]× factors through NK[α]/F [α] and so Θ = ΛK , for a character Λ ∈ C(ψα , A). Setting π = c-IndG J Λ, we get τ = πK . In general, we write τ = χτ , for a character χ of K × and τ minimal. Let (B, s, β) be a ramiﬁed simple stratum such that τ ∈ A02 (K; β). This only determines the isomorphism class of the oK -chain order B. We may therefore take B = AK (notation of 46.1), for some oF -chain order A in A. That is, we may choose B to be Σ-stable. If χ has level t, then 2t > s, and the characters

47. Interior Actions

295

2t 2t χ ◦ det | UB , χσ ◦ det | UB intertwine in GK . It follows that χσ , χ, agree on t t UK , whence χ | UK factors through NK/F . In other words, we can reduce the level of τ by twisting with a character lifted from F . Iterating, this gets us back to the ﬁrst case. The ﬁnal assertion follows directly from 46.3 Proposition (1).

We shall mainly use the proposition in the following context: Corollary. Let K/F be a cubic extension, let L/F be the normal closure of wr Σ K/F and write Σ = Gal(L/F ). Let τ ∈ Awr 2 (L) . There exists π ∈ A2 (F ) such that πL = τ . The representation π is determined up to twisting with a character of F × trivial on NL/F (L× ). Proof. If K/F is cyclic, we are in a special case of the proposition. We assume, therefore, that K/F is not cyclic. Let E/F be the maximal unramiﬁed subextension of L/F . Thus [E:F ] = 2 while L/E is totally ramiﬁed and cyclic of degree 3. By the proposition, there exist precisely three representations ζ ∈ Awr 2 (E) such that ζL = τ . They form a single orbit under twisting by characters of E × /NL/E (L× ), and so are distinguished by their central characters. They are permuted by Gal(E/F ), with a single ﬁxed point ζ 0 , say. There are precisely 0 two representations π ∈ Awr 2 (F ) satisfying πE = ζ ; they are twists of each × × other by the non-trivial character of F /NE/F (E ). Since E/F is the maximal abelian sub-extension of L/F , we have NE/F (E × ) = NL/F (L× ) (cf. 29.1), and the result follows.

47. Interior Actions The strong uniqueness properties of the partial Langlands Correspondence of 44.1 translate into a selection of naturality properties with respect to isomorphisms of the base ﬁeld. We give an account of the matter now, since it will be useful in the sections to follow. The arguments centre on properties of local constants. Everything in this section applies in arbitrary residual characteristic. 47.1. Let φ : F → F be an isomorphism of local ﬁelds. It extends to an isomorphism φ : F → F of separable algebraic closures, yielding an isomorphism φ : ΩF → ΩF of absolute Galois groups, determined by the original map φ : F → F up to an inner automorphism of ΩF . Explicitly, for σ ∈ ΩF , the automorphism φσ of F is φ ◦ σ ◦ φ−1 . The map φ : ΩF → ΩF on Galois groups induces a topological isomorphism φ : WF → WF of Weil groups, determined up to inner automorphism of

296

12 The Dyadic Langlands Correspondence

ΩF . It gives a bijection G0 (F ) −→ G0 (F ), ρ −→ ρφ , where G0 (F ) denotes the set of equivalence classes of irreducible smooth representations of WF of arbitrary dimension. This map extends to a bijection Gss (F ) → Gss (F ) on the sets of equivalence classes of ﬁnite-dimensional semisimple smooth representations of WF . (See also 28.7 Remark.) In particular, the isomorphism φ : F ∼ = = F induces an isomorphism WFab ∼ ab WF which relates the Artin Reciprocity maps aF , aF (29.1) in the obvious way, namely φ ◦ aF = aF ◦ φ. 47.2. Likewise, φ induces a bijection A02 (F ) → A02 (F ) via the obvious isomorphism GL2 (F ) → GL2 (F ) given by φ. We denote this by π → π φ = π ◦ φ. φ φ Proposition. Let ρ ∈ Gim 2 (F ); then πF (ρ ) = πF (ρ) .

, Proof. Let E/F be a ﬁnite extension, E ⊂ F , and set E = φ(E). If η ∈ E φ and φ transports the self-dual Haar measure on η = 1, then η = η ◦ φ ∈ E × E (relative to η) to that on E relative to η φ . If χ is a character of E , then χφ = χ ◦ φ is a character of E × and, following through the deﬁnitions, we get ε(χφ , s, η φ ) = ε(χ, s, η). The maps on equivalence classes of representations of Weil groups, induced by φ, commute with induction. We deduce that ε(ρφ , s, η φ ) = ε(ρ, s, η) for any ρ ∈ Gss (E ). , ψ = 1, then ε(π φ , s, ψ φ ) = Similarly, if π ∈ A02 (F ) and if ψ ∈ F × φ φ φ ε(π, s, ψ). Surely (χπ) = χ π , for any character χ of F . im × Let ρ ∈ G2 (F ), and let χ be a character of F . We have ε(χπF (ρ), s, ψ) = ε(χφ πF (ρ)φ , s, ψ φ ), ε(χ ⊗ ρ, s, ψ) = ε(χφ ⊗ ρφ , s, ψ φ ), ε(χ ⊗ ρ, s, ψ) = ε(χπF (ρ), s, ψ),

whence ε(χφ πF (ρ)φ , s, ψ φ ) = ε(χφ ⊗ ρφ , s, ψ φ ), × , ψ = 1. The Converse Theorem 27.1 for all characters χ of F and all ψ ∈ F implies that πF (ρ)φ = πF (ρφ ), as required.

48. The Langlands-Deligne Local Constant modulo Roots of Unity

297

In particular: Corollary. Let K/F be a ﬁnite Galois extension. If ρ ∈ Gim 2 (K) and γ ∈ Gal(K/F ), then πK (ργ ) = πK (ρ)γ . 47.3. The operation of tame lifting likewise behaves properly with regard to isomorphisms of ﬁelds. Suppose we have ﬁnite, tamely ramiﬁed extensions K/F , K /F , together with an isomorphism φ : K → K such that φ(F ) = F . The original deﬁnitions (46.3.2), 46.5 yield straightaway: φ φ Proposition. Let π ∈ Awr 2 (F ); then LftK/F (π ) = LftK /F (π) .

48. The Langlands-Deligne Local Constant modulo Roots of Unity We have to make connections between the operation of tame lifting and the process of restriction of representations of Weil groups. We do this via two parallel series of properties of the local constants of Langlands-Deligne and of Godement-Jacquet. In both cases, we get a stability theorem in a direction opposite to that of our earlier ones 25.7, 29.4 Proposition (4). This leads to very useful descriptions of the local constants via multiplicative congruences. In this section, we deal with the Langlands-Deligne constant. As in §30, we have to work in arbitrary dimension and use the machinery of induction constants. 48.1. The ﬁrst of these results concerns the eﬀect on the local constant of twisting with a tamely ramiﬁed character of F × . It holds for arbitrary residual characteristic p. Recall that PF denotes the wild inertia subgroup of WF . Theorem. Let ρ ∈ Gss (F ), and suppose that ρ | PF does not contain the trivial character. Let ψ ∈ F, ψ = 1. (1) There exists cρ = c(ρ, ψ) ∈ F × such that ε(χ ⊗ ρ, s, ψ) = χ(cρ )−1 ε(ρ, s, ψ),

(48.1.1)

for all tamely ramiﬁed characters χ of F × . This property determines the coset cρ UF1 uniquely. (2) If φ is a tamely ramiﬁed character of WF , then c(φ ⊗ ρ, ψ) = c(ρ, ψ). (3) Let K/F be a ﬁnite, tamely ramiﬁed ﬁeld extension, and set ρK = ρ | WK , ψK = ψ ◦ TrK/F . We have c(ρK , ψK ) ≡ c(ρ, ψ)

1 (mod UK ).

298

12 The Dyadic Langlands Correspondence

Proof. We ﬁrst observe that if the result holds for ρ1 , ρ2 ∈ Gss (F ), then it holds for ρ1 ⊕ ρ2 with c(ρ1 ⊕ ρ2 , ψ) = c(ρ1 , ψ) c(ρ2 , ψ). We may as well, therefore, start with the assumption that ρ is irreducible. Twisting ρ with an unramiﬁed character of WF amounts to a translation of the variable s and so changes nothing of relevance here. We therefore assume that ρ is an irreducible representation of Γ = Gal(E/F ), for some ﬁnite Galois extension E/F . Let F1 /F be the maximal tamely ramiﬁed sub-extension of E/F . By hypothesis, ρ | Gal(E/F1 ) does not contain the trivial character. The ﬁnite p-group Gal(E/F1 ) has non-trivial centre, Z say. The group Z is a normal subgroup of Γ and the restriction of ρ to Z is a sum of characters, all conjugate under Γ . We can assume that none of these characters is trivial: otherwise, they all are and we can replace E by E Z . Thus ρ | Z is a sum of characters φγ , where φ is non-trivial and γ ∈ Γ . We need a variant of the Brauer induction theorem: Lemma. There is an expression ρ=

r

ni IndΓ∆i ξi ,

(48.1.2)

i=1

where ∆i is a subgroup of Γ containing Z, ξi is a character of ∆i such that ξi | Z = φ, and ni ∈ Z. Proof. Let Γ0 be the Γ -centralizer of φ, and let ρ0 be the natural representation of Γ0 on the φ-isotypic subspace of ρ. The representation ρ0 is then irreducible and ρ = IndΓΓ0 ρ0 . It is therefore enough to prove the lemma under the assumption that Γ = Γ0 . We can further factor out the kernel of φ, and assume that φ is a faithful character of the central subgroup Z. We use the standard Brauer induction theorem to write ρ=

r

ni IndΓ∆i ξi ,

i=1

where ξi is a linear character of the subgroup ∆i . The φ-isotypic component of the term IndΓ∆i ξi is zero unless ξi | ∆i ∩ Z = φ | ∆i ∩ Z. If this condition is satisﬁed, we extend ξi to a character ξ˜i of ∆i Z so that ξ˜i | Z = φ. We so get the required expression ρ= IndΓ∆i Z ξ˜i , i

including only those terms for which ξi | ∆i ∩ Z = φ | ∆i ∩ Z.

48. The Langlands-Deligne Local Constant modulo Roots of Unity

299

Consider a typical term θ = IndΓ∆ ξ in the expression (48.1.2). Let L = E ∆ and view ξ as a character of L× via class ﬁeld theory. Let k be the level of ξ; then k 1 and there exists α ∈ L× such that ξ(1+x) = ψL (αx), x ∈ pkL . Assertion (1) then holds for θ with c(θ, ψ) = NL/F (α) (cf. 23.6 Proposition). It therefore holds for ρ. Explicitly, if Li = E ∆i and αi is deﬁned by analogy with α, (48.1.2) gives c(ρ, ψ) =

r

NLi /F (αi )ni .

i=1

The uniqueness assertion is immediate, and this proves part (1). Abbreviating cρ = c(ρ, ψ), in part (2) we have ε(χφ ⊗ ρ, s, ψ) = χφ(cρ ) ε(ρ, s, ψ) = χ(cρ ) ε(φ ⊗ ρ, s, ψ), whence χ(cρ ) = χ(cφ⊗ρ ), for all tamely ramiﬁed characters χ of F × . Thus c(φ ⊗ ρ, ψ) ≡ c(ρ, ψ) (mod UF1 ), as required. To prove (3), let K/F be a ﬁnite, tamely ramiﬁed extension. Enlarging E if necessary, we can assume that K ⊂ E. Set Gal(E/K) = Σ. The expression (48.1.2) yields r ni IndΓ∆i ξi | Σ. ρK = i=1

Taking a typical term as before, θ | Σ = IndΓ∆ ξ | Σ =

γ IndΣ ∆γ ∩Σ ξ .

γ∈∆\Γ/Σ

Translating this in terms of ﬁelds θ|Σ= IndLγ K/K (ξ γ ◦ NLγ K/Lγ ). γ

The extension Lγ K/Lγ is tamely ramiﬁed, so we get

NLγ K/K (αγ ) = NL/F (α). c(θ | Σ, ψK ) = γ∈∆\Γ/Σ

Assertion (3) therefore holds for θ, so it holds for ρ. 48.2. We need a mild extension of the deﬁning property (48.1.1) of cρ = c(ρ, ψ). Corollary. Let ρ ∈ G0n (F ); suppose that ρ | PF is irreducible and non-trivial. Let θ ∈ Gss (F ) be trivial on PF . We then have ε(θ ⊗ ρ, s, ψ) = det θ(cρ )−1 ε(ρ, s, ψ)dim θ .

300

12 The Dyadic Langlands Correspondence

Proof. We use the language and techniques of §30. We can assume that θ is a representation of G = Gal(K/F ), for a ﬁnite, tamely ramiﬁed Galois extension K/F . Consider the set Γ(G) of pairs (E, χ), where E is a ﬁeld, F ⊂ E ⊂ K, and χ is a character of Gal(K/E). (We regard χ as a character of E × , of level zero, via local class ﬁeld theory.) In the language of §30, the map (E, χ) −→

ε(χ ⊗ ρE , s, ψE ) = χ(cρ )−1 , ε(ρE , s, ψE )

where cρ = c(ρ, ψ), is a division on G. The maps ε(χ ⊗ ρE , s, ψE ), (E, χ) −→ (E, χ) ∈ Γ(G), ε(ρE , s, ψE ), are pre-inductive divisions on G, being the boundaries, respectively, of the induction constants ε(θ ⊗ ρE , s, ψE ) 0 G. (E, θ) −→ (E, θ) ∈ K ε(ρE , s, ψE )dim θ . On the other hand, the division (E, χ) → χ(cρ )−1 is the boundary of the induction constant (E, θ) → det θ(cρ )−1 . An induction constant is determined by its boundary (30.1 Lemma 1). The corollary now follows. Remark. 48.1 Theorem and 48.2 Corollary apply, in particular, to representations ρ ∈ Gwr 2 (F ). 48.3. For the remainder of this section, our blanket hypothesis p = 2 is essential. Let K/F be a ﬁnite, tamely ramiﬁed extension. We set (as before) RK/F = IndK/F 1K and κK/F = det RK/F : thus κK/F is unramiﬁed of order at most 2. Let λK/F (ψ) be the Langlands constant, as in 30.4. We need the following particular consequence of 48.2 Corollary. Proposition. Let ρ ∈ Gwr 2 (F ) and write cρ = c(ρ, ψ). If K/F is a ﬁnite, tamely ramiﬁed extension, then ε(ρK , s, ψK ) = λK/F (ψ)−2 κK/F (cρ )−1 ε(ρ, s, ψ)[K:F ] .

(48.3.1)

Proof. We have IndK/F (ρK ) = ρ ⊗ RK/F ; the inductive property of the local constant and 48.2 Corollary give λK/F (ψ)2 ε(ρK , s, ψK ) = ε(ρ ⊗ RK/F , s, ψ) = κK/F (cρ )−1 ε(ρ, s, ψ)[K:F ] , as required.

48. The Langlands-Deligne Local Constant modulo Roots of Unity

301

48.4. The invariant c(ρ, ψ) has a more subtle rˆole. Let µC (2∞ ) denote the group of roots of unity in C of order a power of 2. Theorem. Let ρ ∈ Gwr 2 (F ). Let ψ ∈ F , ψ = 1, and write cρ = c(ρ, ψ). Then: ε(ρ, 12 , ψ) ≡ det ρ(cρ )−1/2

(mod µC (2∞ )).

Proof. Suppose ﬁrst that ρ is imprimitive. We can write ρ = IndE/F ξ, for a ramiﬁed quadratic extension E/F and a character ξ of E × of level k 1, say. We then have cρ = NE/F (α), for any α ∈ E × such that ξ(1+x) = ψE (αx), x ∈ pkE . Lemma. We have ε(ξ, 12 , ψE ) ≡ ξ(α)−1

(mod µC (2∞ )).

Let us assume the lemma for the moment: we prove it in the next paragraph. We have ε(ρ, 12 , ψ)/ε(ξ, 12 , ψE ) = λE/F (ψ), which is a 4-th root of unity (30.4.3). Also, det ρ = κE/F ⊗ ξ | F × , and κE/F takes values ±1. Thus det ρ(cρ ) ≡ ξ(NE/F (α))

(mod µC (2∞ )).

If σ ∈ Gal(E/F ), σ = 1, we have ασ ≡ α (mod UE1 ), with the result that ξ(NE/F (α)) ≡ ξ(α)2

(mod µC (2∞ )),

implying the theorem in this case. We therefore assume that ρ is primitive, and use the analysis of such representations in §42. We can assume that ρ is a representation of Γ = ), for some ﬁnite extension E/F . The centre Z of Γ is cyclic and Γ/Z Gal(E/F Z is A4 or S4 . Write E = E and let F1 /F be the maximal tamely ramiﬁed subextension of E/F . Thus Gal(F1 /F ) is cyclic of order 3 or dihedral of order 6, and F1 /F has a cubic sub-extension K/F such that ρK is imprimitive. The extension F1 /K is either trivial or unramiﬁed quadratic. The extension E/F1 is totally wildly ramiﬁed with non-cyclic Galois group of order 4. It is the composite of three quadratic extensions E (j) /F1 and the exponents d(E (j) |F1 ) = d, say, are all the same (cf. 42.2 Remark). There is a ramiﬁed quadratic extension K1 /K such that ρK = IndK1 /K φ, for some character φ of K1× . The ﬁeld K1 F1 is one of the E (j) , with the result that d(K1 |K) = d. Moreover, the character φ has level l(K1 ) 2d. If we write l(K ) φ(1+x) = ψK1 (αx), x ∈ pK1 1 , then c(ρK , ψK ) ≡ c(ρ, ψ) ≡ NK1 /K (α) as in 48.1.

1 (mod UK ), 1

302

12 The Dyadic Langlands Correspondence

The restriction ρ | Z is of the form χ ⊕ χ, for some character χ of E × . Clearly χ = φ ◦ NE/K1 , and χ has level l(E) 2d. Indeed, χ(1+x) = ψE (αx), l(E) x ∈ pE . We now use 48.1 Lemma to write ni IndLi /F ξi , ρ= i

for various ﬁelds Li ⊂ E. Each character ξi satisﬁes ξi ◦ NE/Li = χ. If li is the level of ξi and we choose αi ∈ Li such that ξi (1+x) = ψLi (αi x), x ∈ plLii , we have αi ≡ α (mod UE1 ). Using ∼ to denote congruence modulo µC (2∞ ), it follows that

ε(IndLi /F ξi , 12 , ψLi )ni , ε(ρ, 12 , ψ) ∼ i

since the Langlands constants λLi /F (ψ) are 4-th roots of unity (30.4 Remark). The lemma now gives:

ξi−ni (αi ). ε(ρ, 12 , ψ) ∼ i

We extend each ξi , somehow, to a smooth character of the group generated by α and UE1 . Writing cρ = c(ρ, ψ), we get

ξini (NK1 /K (α)) det ρ(cρ ) ∼ i

∼

ξi2ni (α) ∼

i

ξi2ni (αi )

i

∼ ε(ρ, 12 , ψ)−2 , whence the theorem follows.

48.5. We have to prove 48.4 Lemma. Re-normalizing our notation, this amounts to1 : Lemma. Let ψ ∈ F, ψ = 1. Let χ be a character of F × of level l 1. Let δ ∈ F × satisfy χ(1+x) = ψ(δx), x ∈ pl . The quantity χ(δ) ε(χ, 12 , ψ) is then a root of unity of order a power of 2. Proof. The assertion is independent of the choice of ψ, so it will be convenient to assume ψ has level one. Thus ε(χ, 12 , ψ) = τ (χ, ψ)/q (l+1)/2 . 1

The global argument used in the proof to follow is a variation of part of the standard proof of the unit theorem for global ﬁelds: see [54] or [31], for instance.

48. The Langlands-Deligne Local Constant modulo Roots of Unity

303

The condition imposed on δ determines only the coset δUF1 , but χ(UF1 ) ⊂ µC (2∞ ), so the assertion is independent of the choice of δ. We have χ(δy)−1 ψ(δy), τ (χ, ψ) = q 1 /U l y∈UF F

so χ(δ) τ (χ, ψ) lies in the ﬁeld F = Q[ζ] generated by a 2j -th root of unity ζ, for some j ∈ Z. We take j suﬃciently large to ensure that all values of χ | UF1 and all values of ψ | p−l lie in F. As we may take j 3, we can further assume that F contains a square root of q. Using z → z¯ to denote complex conjugation, a simple manipulation gives τ (χ, ψ) = χ(−1) |χ(δ)|−2 τ (χ, ˇ ψ). The relation (23.6.2) and the functional equation (23.4.2) yield τ (χ, ψ) τ (χ, ˇ ψ) = χ(−1) q l+1 ,

(48.5.1)

implying that |χ(δ) τ (χ, ψ)| = q (l+1)/2 . Let us now write ξ(χ, ψ) = χ(δ) τ (χ, ψ), and take σ ∈ Gal(F/Q). Thus ψ σ | p−l is the restriction of a character of F of level one. Since χ(UFl ) ⊂ {±1}, we also have χσ (1+x) = ψ σ (δx) for x ∈ pl . We conclude that ξ(χ, ψ)σ ≡ ξ(χσ , ψ σ ) (mod µC (2∞ )), and hence that |ξ(χ, ψ)σ | = q (l+1)/2 ,

σ ∈ Gal(F/Q).

Put another way, q −(l+1)/2 ξ(χ, ψ)v = 1, for all Archimedean places v of the ﬁeld F. Let v2 be the unique non-Archimedean place of F lying over 2. Let v be a non-Archimedean place of F, v = v2 . The quantity ξ(χ, ψ)/q (l+1)/2 is then integral at v and the functional equation (48.5.1) implies that ξ(χ, ψ)/q (l+1)/2 v = 1 However, the product formula for normalized valuations of F,

xv = 1, x ∈ F× , v

304

12 The Dyadic Langlands Correspondence

now implies that ξ(χ, ψ)/q (l+1)/2 v = 1 for all v. In particular, ξ(χ, ψ)/q (l+1)/2 is a unit in F. Set

Fv ∼ F∞ = = C[F:Q]/2 , v|∞

where v ranges over all Archimedean places of F. The canonical map F → × F∞ identiﬁes o× F with a discrete subgroup of F∞ . The multiplicative group generated by ξ(χ, ψ)/q (l+1)/2 is therefore discrete, but it is contained in the compact subgroup of vectors (xv )v|∞ such that |xv | = 1 for all v|∞. It is therefore ﬁnite. We deduce that ξ(χ, ψ)/q (l+1)/2 is a root of unity in F, and so it lies in µC (2∞ ), as required.

49. The Godement-Jacquet Local Constant and Lifting We seek some properties of the Godement-Jacquet local constant, analogous to those of §48. In this section, it will be enough to treat the case where ψ ∈ F has level one, and is the character used to relate simple strata and characters. 49.1. The ﬁrst step is extremely simple: Lemma. Let π ∈ Awr 2 (F ). (1) There exists cπ = c(π, ψ) such that ε(χπ, s, ψ) = χ(cπ )−1 ε(π, s, ψ)

(49.1.1)

for all tamely ramiﬁed characters χ of F × . This condition determines the coset cπ UF1 uniquely. (2) Let K/F be a ﬁnite, tamely ramiﬁed ﬁeld extension; then c(πK , ψK ) ≡ c(π, ψ)

1 (mod UK ).

(49.1.2)

Proof. Suppose ﬁrst that π is minimal: there exists a ramiﬁed simple stratum (A, n, α) in A such that π ∈ A02 (F ; α) (notation of 44.3). The Gauss sum formula (25.5 Corollary) for the local constant then gives (49.1.1) with cπ = det α. The second assertion is immediate and part (2) follows from the deﬁnition of tame lifting. Otherwise, we write π = ξπ0 , where ξ is a character of F × and (π0 ) < (π). Let ξ have level k 1, and choose δξ ∈ F so that ξ(1+x) = ψ(δξ x), x ∈ pk . The same Gauss sum formula then yields the result with cπ = δξ2 .

49. The Godement-Jacquet Local Constant and Lifting

305

It will be useful to record the speciﬁc values given by the proof of the lemma: Gloss. (1) Suppose there exists a ramiﬁed simple stratum (A, n, α) such that π ∈ A02 (F ; α); then c(π, ψ) = det α. (2) Suppose there exists a character ξ of F × , of level l 1, such that (ξ −1 π) < (π); then c(π, ψ) = δ 2 , where δ ∈ p−l satisﬁes ξ(1+x) = ψ(δx), x ∈ pl . 49.2. We can now work out the behaviour of the local constant under tame lifting. We only need a very special case: Proposition. Let π ∈ Awr 2 (F ), and let K/F be a cubic extension; then ε(πK , s, ψK ) = κK/F (cπ )−1 ε(π, s, ψ)3 ,

(49.2.1)

where cπ = c(π, ψ) and λK/F = λK/F (ψ). Proof. Suppose ﬁrst that π ∈ A02 (F ; α), for some ramiﬁed simple stratum (A, n, α). Immediately, 2(π) 3 2(π ) q = qK K , so in this case the Gauss sum formula and the deﬁnition of πK give ε(πK , s, ψK ) = K/F ε(π, s, ψ)3 . We have κK/F (det α) = K/F , and the result follows in this case. The next step is to prove: λK/F (ψ)2 = 1.

(49.2.2)

If K/F is unramiﬁed, 23.5 Proposition gives λK/F (ψ) = 1. Suppose next that K/F is totally ramiﬁed and cyclic. Let φ be a non-trivial character of F × vanishing on norms from K. By (23.4.2), λK/F (ψ) = ε(φ, s, ψ) ε(φ−1 s, ψ) = φ(−1). Since φ has order 3, we have φ(−1) = 1 = λK/F (ψ). This leaves only the case where K/F is non-cyclic, hence totally ramiﬁed. Let E/F be unramiﬁed quadratic. Thus EK/E is cyclic and cubic. We have λEK/F (ψ) = λEK/K (ψK ) λK/F (ψ)2 = λEK/E (ψE ) λE/F (ψ)3 .

306

12 The Dyadic Langlands Correspondence

By the cyclic cubic case, λEK/E (ψE ) = 1 while, for the quadratic unramiﬁed extensions E/F , EK/K, 23.5 Proposition gives λEK/K (ψK ) = λE/F (ψ) = −1. Thus λK/F (ψ)2 = 1, as desired. We return to the proof of the Proposition. If π is not minimal, there is a ramiﬁed simple stratum (A, n, α), a character χ of F × of level l > n/2, and a representation π ∈ A02 (F ; α) such that π = χπ . The representation π contains a character Λ ∈ C(ψα , A). Here, (π) = l and we have q 3(π) = (π ) qK K , so it is again enough to check (49.2.1) at s = 12 . We choose δ ∈ p−l such that χ(1+x) = ψ(δx), x ∈ p[l/2]+1 . This gives χΛ ((α+δ)x)−1 ψA ((α+δ)x), ε(π, 12 , ψ) = q −1 x

where ψA = ψ ◦ trA and the sum is taken over x ∈ UAl /UAl+1 (25.5 Corollary). For such x, we have Λ (αx)−1 ψA (αx) = Λ (α)−1 ψA (α), so this expression reduces to ε(π, 12 , ψ) = q −1 ψA (α) Λ (α+δ)−1 χ(det(1+αδ −1 ))−1 χ(det δx)−1 ψA (δx). x

Using the notation of 25.6, the identity (25.6.1) yields χ(det δx)−1 ψA (δx) = q −l τ (χ, ψ)2 , x

where τ (χ, ψ) =

χ(δy)−1 ψ(δy).

l+1 y∈UF /UF

That is, ε(π, 12 , ψ) = ψ(trA α) Λ (α+δ)−1 χ(det(1+αδ −1 ))−1 τ (χ, ψ)2 /q (l+1) . The corresponding expression for πK is eK l+1 ε(πK , 12 , ψK ) = ψK (trAK α) ΛK (α+δ)−1 χK (det(1+αδ −1 ))−1 τ (χK , ψK )2 /qK ,

where eK = e(K|F ). The deﬁnition of ΛK gives ΛK (α+δ) = Λ (α+δ)3 . We have cπ = δ 2 , so we are reduced to proving eK l+1 τ (χK , ψK )2 /qK = τ (χ, ψ)6 /q 3(l+1) .

(49.2.3)

50. The Existence Theorem

307

To this end we use the identities τ (χ, ψ) = q (l+1)/2 ε(χ, 12 , ψ), (e(K|F )l+1)/2

τ (χK , ψK ) = qK

ε(χK , 12 , ψK ).

The inductive property of the local constant gives ε(χK , 12 , ψK ) = ε(χ ⊗ RK/F , 12 , ψ) λK/F (ψ)−1 . A trivial instance of 48.1 gives cχ = δ and ε(χ ⊗ RK/F , 12 , ψ) = κK/F (δ)−1 ε(χ, 12 , ψ)3 . 2 Since κK/F = 1, the desired identity (49.2.3) now follows from (49.2.2).

50. The Existence Theorem In this section, we construct a map πF : G02 (F ) → A02 (F ) satisfying the local constant relation (33.4.1). That relation determines the map uniquely, so πF is the Langlands correspondence. Until further notice, ψ ∈ F is of level one. We use it to make the connection between simple strata and cuspidal types, as in Chapter IV. Our blanket 0 nr hypothesis p = 2 remains in force. We recall that Gwr 2 (F ) = G2 (F ) G2 (F ) 0 is the set of totally ramiﬁed ρ ∈ G2 (F ). 50.1. We extend some of the elementary observations of §41 to the class of primitive representations. Let ρ ∈ G02 (F ); 29.4 Proposition (1) gives an integer n(ρ, ψ) such that 1 ε(ρ, s, ψ) = q ( 2 −s)n(ρ,ψ) ε(ρ, 12 , ψ). We call n(ρ, ψ) the level of ρ. We say that ρ is minimal if n(ρ, ψ) n(χ⊗ρ, ψ) for all characters χ of F × . Proposition. Let ρ ∈ Gwr 2 (F ). (1) The representation ρ is minimal if and only if n(ρ, ψ) is odd. (2) Suppose that ρ is minimal and let χ be a character of F × of level l 0. The representation χ ⊗ ρ is minimal if and only if 2l < n(ρ, ψ). If 2l > n(ρ, ψ), then n(χ ⊗ ρ, ψ) = 2l. Proof. In the case where ρ is imprimitive, all assertions follow easily from the discussions in 41.4, 41.5. Suppose therefore that ρ is primitive, and choose a cubic extension K/F such that ρK is imprimitive (42.2). We have n(ρK , ψK ) = e(K|F ) n(ρ, ψ).

308

12 The Dyadic Langlands Correspondence

So, if n(ρ, ψ) is odd, then n(ρK , ψK ) is odd and ρK is consequently minimal. Thus ρ is minimal. Conversely, suppose that n = n(ρ, ψ) is even, and let L/F be the normal closure of K/F : in particular, L/K is unramiﬁed of degree 2. The representation ρL has level nL = e(K|F )n and so is not minimal over L. We write ρL = χ⊗τ , for a character χ of L× and a representation τ with n(τ, ψL ) < nL . The character χ thus has level nL /2. For γ ∈ Gal(L/F ), we have ργL ∼ = ρL , so τ ∼ = χ−1 χγ ⊗ τ γ . Thus χ−1 χγ has level < nL /2, and the restriction of χ to n /2 UL L factors through NL/F . In other words, we could have taken χ = φL , for some character φ of F × . This character satisﬁes n(φ−1 ⊗ ρ, ψ) < n(ρ, ψ), and ρ is not minimal. This completes the proof of (1) and (2) now follows readily. 50.2. Let ρ ∈ Gwr 2 (F ) be minimal. If ρ is imprimitive, then π = πF (ρ) has been deﬁned. The representation π is minimal, and 44.7 Proposition speciﬁes a simple stratum (A, n, α) such that π ∈ A02 (F ; α). We generalize this construction to the case of ρ primitive, by an indirect route. Proposition. Let ρ ∈ Gwr 2 (F ) be minimal over F . (1) There is a ramiﬁed simple stratum (A, n, α) in M2 (F ), n = 2m+1, with the following property: if K/F is a ﬁnite, tamely ramiﬁed ﬁeld extension such that ρK is imprimitive, then πK (ρK ) ∈ A02 (K; α). (2) Condition (1) determines the G-conjugacy class of the character ψα | UAm+1 , and hence the set A02 (F ; α), uniquely. Proof. In the case where ρ is imprimitive, all assertions in (1) are immediate consequences of 44.7. We therefore assume that ρ is primitive, and choose a cubic extension K/F such that ρK is imprimitive. We apply the ﬁrst case to ρK to get a simple stratum (A , n , α ) in AK = M2 (K) with the property (1) relative to the base ﬁeld K. Let L/F be the normal closure of K/F and consider the representation ζ = πL (ρL ). This lies in A02 (L; α ) and is ﬁxed by Gal(L/F ). By 46.6 Corollary, there is π ∈ Awr 2 (F ) such that ζ = πL . The representation π satisﬁes (π) = / Z, so π is minimal. There exists a simple stratum (A, n, α) in A e(L|F )−1 (ζ) ∈ such that π ∈ A02 (F ; α), whence ζ ∈ A02 (L; α). That is, A02 (L; α) = A02 (L; α ). Lemma 46.2 now implies A02 (K; α) = A02 (K; α ), so we could have taken (A , n , α ) = (AK , nK , α) (in the notation of 46.1)). If E/F is a ﬁnite tame extension such that ρE is imprimitive, then E contains some F -conjugate of K (42.3 Proposition) and the ﬁrst case implies πE (ρE ) ∈ A02 (E; α). Property (2) is implied by 46.2 Lemma.

50. The Existence Theorem

309

50.3. We come to the central result of the section, which establishes the existence of the Langlands correspondence. We continue with our ﬁxed ψ ∈ F of level one. Existence Theorem. Let ρ ∈ G02 (F ). There exists a unique representation π = πF (ρ) ∈ A02 (F ) such that ε(χπ, s, ψ) = ε(χ ⊗ ρ, s, ψ),

(50.3.1)

for all characters χ of F × . Further: (1) ωπF (ρ) = det ρ, for all ρ ∈ G02 (F ); (2) πF (χ ⊗ ρ) = χ πF (ρ), for all ρ ∈ G02 (F ) and all characters χ of F × ; (3) the relation (50.3.1) holds for all ψ ∈ F, ψ = 1. Proof. The uniqueness assertion is a direct consequence of (50.3.1) and the Converse Theorem 27.1. Parts (1) and (2) will be seen as consequences of the construction (50.6.2), (50.6.3). Given these, part (3) follows from 24.3 Proposition and 29.4 Proposition (2). If ρ is imprimitive, we set π = πF (ρ), as in 44.1: this has all the required properties. We assume henceforth that ρ is primitive. To start with, we also assume ρ is minimal. We choose a cubic extension K/F such that ρK is imprimitive. We take a ramiﬁed simple stratum (A, n, α) in M2 (F ), as in 50.2 Proposition: thus ζ = πK (ρK ) ∈ A02 (K; α). In particular, there is a unique character Λζ ∈ C(ψαK , AK ) which occurs in ζ. Let ν be the unramiﬁed character of K[α]× of the same order as κK/F , and write n = 2m+1. Lemma 1. There exists a unique character Λ ∈ C(ψα , A) such that Λ | F × = det ρ and

Λ3 = ν ⊗ Λζ F [α]× U m+1 . A

Proof. The restriction of Λζ to UAm+1 is simply ψ3α . The restriction of Λζ to UF1 [α] has values in the group µC (2∞ ) of roots of unity in C of 2-power order. Thus Λζ | UF1 [α] UAm+1 is of the form λ3 , for a unique character λ of UF1 [α] UAm+1 such that λ | UAm+1 = ψα . We also have Λζ | F × = (det ρ)3 , so we extend λ to a character Λ0 of F × UF1 [α] UAm+1 by deeming that Λ0 | F × = det ρ. The group F × UF1 [α] UAm+1 has index 2 in F [α]× UAm+1 , so there is a unique character Λ, extending Λ0 and with the required properties. × m+1 . We Taking Λ as in Lemma 1, we set π = c-IndG J Λ, where J = F [α] UA deﬁne

πF (ρ) = π.

(50.3.2)

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12 The Dyadic Langlands Correspondence

By construction, we have ωπ = det ρ,

(50.3.3)

as required for statement (1) of the theorem. Lemma 2. Let ρ ∈ G02 (F ) be primitive and minimal over F , and deﬁne π = πF (ρ), as in (50.3.2). Let χ be a character of F × such that χ ⊗ ρ is minimal over F . We then have πF (χ ⊗ ρ) = χπ. Proof. This is immediate from 15.9 and the construction of π. We highlight a feature of the construction: Proposition. Let ρ ∈ Gwr 2 (F ) be minimal. (1) There is a ramiﬁed simple stratum (A, n, α) in A such that πF (ρ) ∈ A02 (F ; α). (2) If E/F is a ﬁnite, tamely ramiﬁed extension, then πE (ρE ) ∈ A02 (E; α). Proof. If ρ is imprimitive, all assertions are given by 50.2 Proposition. Suppose, therefore, that ρ is primitive, and choose K/F cubic such that ρK is imprimitive. By construction, we have a ramiﬁed simple stratum (A, n, α) such that πF (ρ) ∈ A02 (F ; α) and πK (ρK ) ∈ A02 (K; α). Proposition 50.2 gives (2) in the case where ρE is imprimitive. Otherwise, we use 50.2 to get a ramiﬁed simple stratum (B, r, β) in AE such that πL (ρL ) ∈ A02 (L; β) for any ﬁnite tame extension L/E with ρL imprimitive. However, we can assume K ⊂ L and then 50.2 Proposition (2) gives A02 (L; β) = A02 (L; α). It follows from 46.2 Lemma that A02 (E; β) = A02 (E; α), as required for (2). Remark. We do not assert that πK (ρK ) is the K/F -lift πK of π (although, in fact, it is: see 52.9 below). 50.4. We start the process of verifying the local constant relation (50.3.1) for the representation π = πF (ρ). Proposition. Let ρ ∈ G02 (F ) be primitive and minimal over F , and deﬁne π = πF (ρ) as in (50.3.2). Then ε(π, s, ψ) = ε(ρ, s, ψ). Proof. We use the notation of the construction of π, as in 50.3. By deﬁnition, ε(ζ, s, ψK ) = ε(ρK , s, ψ). In particular, e(K|F ) n(π, ψ) = n(ζ, ψK ) = n(ρK , ψK ) = e(K|F ) n(ρ, ψ), so n(π, ψ) = n(ρ, ψ). We need only, therefore, check the desired relation at the point s = 12 .

50. The Existence Theorem

311

If φ is a tamely ramiﬁed character of K × , then ε(φ ⊗ ρK , s, ψK ) = φ(c(ρ, ψ))−1 ε(ρK , s, ψK ), ε(φζ, s, ψK ) = φ(det α)−1 ε(ζ, s, ψK ), by, respectively, 48.1 Theorem (3) and 49.1 Gloss. The deﬁning relation (48.1.1), with χ unramiﬁed, gives υF (c(ρ, ψ)) = n(ρ, ψ), and this is odd (50.1 Proposition). From 48.3 Proposition and (49.2.2), we therefore get ε(ρK , s, ψK ) = K/F ε(ρ, s, ψ)3 , while, by deﬁnition, ε(ρK , s, ψK ) = ε(ζ, s, ψK ). The Gauss sum formula 25.5 Corollary and the deﬁnition of π give ε(π, 12 , ψ)3 = K/F ε(ζ, 12 , ψK ), with the result that (50.4.2) ε(π, 12 , ψ)3 = ε(ρ, 12 , ψ)3 . We need to extract the cube root. Certainly ε(π, 12 , ψ) ≡ Λ(α)−1 ≡ ωπ (det α)−1/2

(mod µC (2∞ )).

(50.4.3)

Invoking 48.4 Theorem, we get ε(π, 12 , ψ)/ε(ρ, 12 , ψ) ∈ µC (2∞ ), so (50.4.2) implies ε(π, 12 , ψ) = ε(ρ, 12 , ψ), as required. Combining the proposition with 50.3 Lemma 2, we get: Corollary. The identity (50.3.1) holds for any χ such that χ ⊗ ρ is minimal over F . 50.5. We now have to check (50.3.1) for characters χ such that χ ⊗ ρ is not minimal over F . Thus χ has level l > n/2, n = n(π, ψ) = n(ρ, ψ). We choose δ ∈ p−l such that χ(1+x) = ψ(δx), x ∈ p[l/2]+1 . Straightforward manipulations and 48.1 Theorem (3) give cρ ≡ cπ ≡ δ 2

(mod UF1 ).

The standard Gauss sum formula and 50.1 Proposition (2) give n(χ ⊗ ρ, ψ) = n(χπ, ψ) = 2l. From 48.3 Proposition and (49.2.2), we get ε(χK ⊗ ρK , s, ψK ) = ε(χ ⊗ ρ, s, ψ)3 , while 49.2 Proposition and (49.2.2) give ε(χK πK , s, ψK ) = ε(χπ, s, ψ)3 .

(50.5.1)

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12 The Dyadic Langlands Correspondence

In all, this yields ε(χ ⊗ ρ, 12 , ψ)3 = ε(χπ, 12 , ψ)3 , whence det(χ ⊗ ρ)(δ)3 ε(χ ⊗ ρ, 12 , ψ)3 = ωχπ (δ)3 ε(χπ, 12 , ψ)3 . The left hand side lies in µC (2∞ ). The result is therefore implied by the following analogue of 48.4 Theorem: Proposition. The quantity ωχπ (δ) ε(χπ, 12 , ψ) is a 2-power root of unity. Proof. The Gauss sum formula 25.5 Corollary gives ε(χπ, 12 , ψ) = q −1 χΛ((α+δ)x)−1 ψA ((α+δ)x), x

where ψA = ψ ◦ trA and the sum is taken over x ∈ UAl /UAl+1 . For such x, we have Λ(αx)−1 ψA (αx) = Λ(α)−1 ψA (α), so this expression reduces to ε(χπ, 12 , ψ) = q −1 ψA (α) Λ(α+δ)−1 χ(det(1+αδ −1 ))−1

χ(det δx)−1 ψA (δx).

x

Using the notation of 25.6, the identity (25.6.1) yields χ(det δx)−1 ψA (δx) = q −l τ (χ, ψ)2 , x

where τ (χ, ψ) =

χ(δy)−1 ψ(δy).

l+1 y∈UF /UF

That is, ε(χπ, 12 , ψ) = ψA (α) Λ(α+δ)−1 χ(det(1+αδ −1 ))−1 τ (χ, ψ)2 /q (l+1) . In this formula, surely ψA (α) ∈ µC (2∞ ). We have δ −1 α ∈ pF [α] , therefore ωπ (δ) ε(χπ, 12 , ψ) ≡ τ (χ, ψ)2 /q l+1

(mod µC (2∞ )),

or, equivalently, ωχπ (δ) ε(χπ, 12 , ψ) ≡ χ(δ)2 τ (χ, ψ)2 /q l+1

(mod µC (2∞ )).

However, the quantity χ(δ) τ (χ, ψ)/q (l+1)/2 is a root of unity, of order a power of 2, by 48.5 Lemma. This completes the proof of the proposition. We have also completed the proof of the Existence Theorem 50.3 when the representation ρ is minimal.

51. Some Special Cases

313

50.6. It remains only to treat the case where ρ is primitive but not minimal. We choose a character φ of F × such that ρ = φ−1 ⊗ ρ is minimal. We set πF (ρ) = φ πF (ρ ).

(50.6.1)

This is independent of the choice of φ, by 50.3 Lemma 2. This deﬁnition gives (cf. (50.3.3)) (50.6.2) ωπF (ρ) = det ρ, and also πF (χ ⊗ ρ) = χ πF (ρ),

ρ ∈ G02 (F ),

(50.6.3)

for all characters χ of F × . The local constant relation (50.3.1) follows from (50.6.3) and the case already done. The proof of the Existence Theorem is complete.

51. Some Special Cases In preparation for the ﬁnal result in the next section, we examine further the Langlands correspondence πF of 50.3 in some special cases. 51.1. The map πF of 50.3 is the only map G02 (F ) → A02 (F ) with the property (50.3.1). From an argument identical to the proof of 47.2 Proposition, we obtain: Proposition. Let φ : F → F be an isomorphism of ﬁelds; we then have πF (ρφ ) = πF (ρ)φ , for all ρ ∈ G02 (F ). 51.2. In one case, it is easy to establish a connection between restriction and tame lifting. Proposition. Let ρ ∈ Gwr 2 (F ), let K/F be a cyclic cubic extension, and suppose that ρK is imprimitive. We then have πK (ρK ) = πF (ρ)K . Proof. Write Σ = Gal(K/F ) and ζ = πK (ρK ). Thus ζ ∈ A02 (K)Σ , so ζ = πK , for some π ∈ Awr 2 (F ) (46.6 Proposition). We have ωζ = det ρK , so there is a unique choice of π such that ωπ = det ρ. The character κK/F is trivial, so 49.2 Proposition gives us ε(χK ζ, s, ψK ) = ε(χπ, s, ψ)3 ,

314

12 The Dyadic Langlands Correspondence

for all characters χ of F × . On the other hand, ε(χK ζ, s, ψK ) = ε(χK ⊗ ρK , s, ψK ) = ε(χ ⊗ ρ, ψ)3 by (48.3.1), (49.2.2). So, if κ = πF (ρ), we have ε(χκ, s, ψ)3 = ε(χπ, s, ψ)3 , for all characters χ of F × . Further, ωκ = ωπ while cχπ ≡ cχκ ≡ cχ⊗ρ (mod UF1 ). We next observe: Lemma. The quantities ωχπ (cχπ )1/2 ε(χπ, 12 , ψ), ωχκ (cχκ )1/2 ε(χκ, 12 , ψ) are 2-power roots of unity. Proof. The two cases are the same, so we deal only with π. If the representation χπ is not minimal, the assertion is given by 50.5 Proposition. Otherwise, it follows directly from the formula in 25.5 Corollary. So, ε(χκ, 12 , ψ) = ε(χπ, 12 , ψ), for all χ. Surely n(π, ψ) = n(κ, ψ); it follows that ε(χκ, s, ψ) = ε(χπ, s, ψ), and hence π ∼ = κ, as required. 51.3. In many cases, we can now complete the picture: Proposition. Suppose that the ﬁeld F satisﬁes the condition q ≡ 1 (mod 3). The map πF : G02 (F ) → A02 (F ) is then bijective. Proof. The eﬀect of the hypothesis is that any cubic extension K/F is cyclic. We ﬁrst show that πF is injective. Let ρ, ρ ∈ G02 (F ) satisfy πF (ρ) = πF (ρ ). If ρ and ρ are both imprimitive, 44.1 gives ρ = ρ . Suppose next that ρ, ρ are both primitive. Set π = πF (ρ) = πF (ρ ). In particular, we have det ρ = det ρ = ωπ . Take cubic extensions K/F , K /F such that ρK and ρK are both imprimitive. If K = K , 51.2 Proposition gives πK (ρK ) = πK = πK (ρK ). Therefore ρK ∼ = ρK , whence ρ ∼ = χ ⊗ ρ, for some χ with χ3 = 1 (42.1 Proposition (3)). The relation det ρ = det ρ implies χ = 1. If K = K , set L = KK , so that L/F is abelian of degree 9. Applying 51.2 Proposition twice, we get πL (ρL ) = πL = πL (ρL ), and so ρL ∼ = ρL . Surely det ρK = det ρK , so we deduce ρK ∼ = ρK and then ∼ ρ = ρ , using 42.1 again. If ρ is primitive while ρ is imprimitive, the same argument again leads to the conclusion ρ = ρ . Thus πF is injective. We show that πF is surjective. Let π ∈ A02 (F ). If π is ordinary then, by deﬁnition, it lies in πF (Gim 2 (F )). We therefore assume π is exceptional.

51. Some Special Cases

315

By 45.2 Theorem, there is a cubic extension K/F such that πK is ordinary: we have πK = πK (τ ), for some τ ∈ Gwr 2 (K). The representation τ is ﬁxed by Gal(K/F ), hence of the form τ = ρK , for some ρ ∈ Gwr 2 (F ). We may choose ρ such that det ρ = ωπ . Proposition 51.2 now implies π = πF (ρ). 51.4. We return to the general case q ≡ ±1 (mod 3). Let (A, n, α) be a ramiﬁed simple stratum in A = M2 (F ); we deﬁne G02 (F )α = {ρ ∈ G02 (F ) : πF (ρ) ∈ A02 (F ; α)}. Proposition. Let (A, n, α) be an ordinary, ramiﬁed simple stratum in A = M2 (F ). The set G02 (F )α contains no primitive representation. In particular, the map πF induces a bijection ≈

G02 (F )α −→ A02 (F ; α).

(51.4.1)

0 Proof. The map πF induces a bijection G02 (F )α ∩Gim 2 (F ) → A2 (F ; α), by 44.3 Theorem and 43.2 Theorem. So, πF : G02 (F )α → A02 (F ; α) is bijective if and only if G02 (F )α contains no primitive representation. If q ≡ 1 (mod 3), the bijectivity property is given by 51.3 Proposition, so we take q ≡ −1 (mod 3). Let E/F be the unramiﬁed quadratic extension and Σ = Gal(E/F ). Thus qE = q 2 ≡ 1 (mod 3). If ρ ∈ G02 (F )α is primitive, then ρE is primitive (42.2 Theorem (2)). It follows from 50.2 Proposition and 46.2 Lemma that ρE ∈ G02 (E)α , contradicting what we have just said.

51.5. We highlight a consequence of 51.4. Let π ∈ A02 (F ); by analogy with the group T(ρ) deﬁned in 41.3, let T(π) denote the group of characters φ of F × such that φπ ∼ = π. Proposition. (1) Let ρ ∈ G02 (F ) be imprimitive and set π = πF (ρ). We then have T(π) = T(ρ). (2) Let (A, n, α) be an exceptional, ramiﬁed simple stratum in M2 (F ). Every π ∈ A02 (F ; α) then satisﬁes T(π) = 1. Proof. For any ρ ∈ G02 (F ), part (2) of the Existence Theorem (50.3) implies that T(ρ) ⊂ T(πF (ρ)). The injectivity of πF on imprimitive representations yields (1). Likewise, if q ≡ 1 (mod 3), the bijectivity of πF (51.3) gives T(πF (ρ)) = T(ρ) in all cases. In particular, π ∈ A02 (F ) is ordinary if and only if T(π) = 1. We therefore pass to the case q ≡ −1 (mod 3), and take π ∈ A02 (F ; α). Suppose, for a contradiction, that χ ∈ T(π), χ = 1. The representation π is totally ramiﬁed, so χ is not unramiﬁed. Let E/F be the unramiﬁed quadratic extension. Thus χE = 1 and χE ∈ T(πE ). However, πE ∈ A02 (E; α) and

316

12 The Dyadic Langlands Correspondence

the polynomial Cα (X) (as in 45.2) remains irreducible over E. Thus πE is exceptional, implying χE = 1 by the ﬁrst case. This contradiction completes the proof of (2). We can re-phrase the proposition: Corollary. An irreducible cuspidal representation of G is ordinary if and only if T(π) = 1. Or, in the language of §39, π is a Weil representation πE/F (Θ) if and only if κE/F ∈ T(π). 51.6. Although we will not use it in the proofs, there is one further special case which is worthy of mention. Let (A, n, α), n = 2m+1 1, be a ramiﬁed simple stratum in A = M2 (F ). Write d(F [α]|F ) = dα . Proposition. Suppose that (A, n, α) is exceptional and n = 3dα . The splitting ﬁeld of Cα (X) is then unramiﬁed over F . Proof. The hypothesis n = 3dα implies that tr(α) = TrF [α]/F (α) has valuation −dα . Let be a prime element of F . The polynomial cα (Y ) = 3dα Cα (−dα Y ) = Y 3 − aY 2 + b has coeﬃcients a, b ∈ UF and is irreducible over F . Its image in k[Y ] is also irreducible, by Hensel’s Lemma, whence the result follows.

52. Octahedral Representations In this section, we prove: 52.1 Bijectivity Theorem. The Langlands correspondence πF : G02 (F ) −→ A02 (F ) is bijective. After the discussion of imprimitive representations and the special cases of §51, all the remaining diﬃculties centre on primitive representations ρ ∈ G02 (F ) which become imprimitive only over a non-cyclic cubic extension K/F . In the terminology of 42.3 Comment, these are the octahedral representations of WF .

52. Octahedral Representations

317

52.2. Before embarking on the proof, we need to investigate some rather delicate properties of wildly ramiﬁed cuspidal types. We take a ramiﬁed simple stratum (A, n, α) in A, with n = 2m+1 1, and impose a “metric” on the set A02 (F ; α). Set J = F [α]× UAm+1 . Let π1 , π2 ∈ A02 (F ; α), so that πi = c-IndG J Λi , for a 1 unique Λi ∈ C(ψα , A). Let k denote the level of the character Λ−1 1 Λ2 | UF [α] . We put π1 −π2 = k. Thus 0 π1 −π2 m. The condition π1 −π2 = 0 is equivalent to the existence of a tamely ramiﬁed character χ of F × such that π2 = χπ1 . One can determine this “distance” between representations in terms of local constants: Lemma. Let (A, n, α), n 1, be a ramiﬁed simple stratum. Let π, π ∈ A02 (F ; α) satisfy ωπ = ωπ and π−π > 0. The integer ν = π−π is then odd, and (n−ν)/2 is the minimum of the levels of characters χ of F × such that ε(π, 12 , ψ)/ε(χπ, 12 , ψ) = ε(π , 12 , ψ)/ε(χπ , 12 , ψ).

(52.2.1)

Proof. For any j 1, we have UF2j[α] = UFj UF2j+1 [α] ; the condition ωπ = ωπ so implies that ν is odd. Write n = 2m+1, J = F [α]× UAm+1 . Let Λ, Λ ∈ C(ψα , A) satisfy π = G −k satisfy c-IndG J Λ, π = c-IndJ Λ . Take χ of level k m, and let δχ ∈ p m+1 χ(det 1+x) = ψδχ (1+x), 1+x ∈ UA . We then have (25.5 Corollary)

ε(π, 12 , ψ)/ε(χπ, 12 , ψ) = χ(det(α+δχ )) Λ(1+α−1 δχ ) ψA (−δχ ), and similarly for π in place of π. The inequality (52.2.1) is therefore equivalent to Λ(1+α−1 δχ ) = Λ (1+α−1 δχ ), and the result follows.

52.3. There is a complementary property, concerning the stability of a set A02 (F ; α) under twisting by characters of comparatively small level. This gives an alternative description of the elements of A02 (F ; α) suﬃciently close to a ﬁxed representation π. Proposition. Let (A, n, α), n = 2m+1 1, be a ramiﬁed simple stratum. Put dα = d(F [α]|F ), suppose that n 3dα , and write c = [n/3]. Let π ∈ A02 (F ; α); then χπ ∈ A02 (F ; α) for every character χ of F × of level c.

318

12 The Dyadic Langlands Correspondence

Proof. Set rad A = P. Let χ have level k c; if k = 0, there is nothing to prove, so we assume k 1 (and hence n 3). We choose δ ∈ p−k such that χ(det 1+x) = ψδ (x), x ∈ Pm+1 . Let Λ ∈ C(ψα , A) occur in π. We consider the character χΛ : x −→ χ(det x) Λ(x), x ∈ J. The character χΛ lies in C(ψα+δ , A) (15.9) and occurs in χπ. Lemma. The cosets αUAm+1 , (α+δ)UAm+1 are conjugate in KA . Proof. By 44.2 Lemma, we have to show that det 1+α−1 δ ∈ UF υF (2δ) −[m/2]. Starting with the trace condition,

[m/2]+1

υF (2δ)

and

dα n n n dα −k − − . 2 2 3 6 3

This is −[m/2] in all cases except n = 3. If n = 3, we have k 1 and υF (2δ) 0 = −[m/2]. On the other hand, 1+α−1 δ ∈ UFn−2k [α] . We have n−2k n/3 and n/3 dα ; therefore [n/3] det 1+α−1 δ = NF [α]/F (1+α−1 δ) ∈ UF . The lemma then follows, provided [n/3] [m/2]+1. This holds for all odd integers n 3 except n = 5. When n = 5, both n−2k and dα are 2, and [m/2]+1 , as required. NF [α]/F (UF2 [α] ) ⊂ UF2 = UF So, we choose g ∈ KA such that gαg −1 ≡ α+δ (mod UAm+1 ): in terms of g . The KA -normalizer of ψα+δ | characters of UAm+1 , this means ψα = ψα+δ m+1 is again J, so g normalizes J. It follows that the character Λ = (χΛ)g UA of J lies in C(ψα , A). It is contained in χπ, so χπ ∈ A02 (F ; α), as required. Continuing in the same vein, we now sharpen our hypotheses and prove: Corollary. Let (A, n, α) be exceptional. Let π1 , π2 ∈ A02 (F ; α) satisfy π1 −π2 c = [n/3]. There then exists a character χ of F × , of level c, such that π2 = χπ1 . Proof. Since (A, n, α) is exceptional, we have n 3dα . We carry on from the end of the proof of the proposition, and push the calculation a little further. The element g ∈ KA from the proof of the proposition, satisfying gαg −1 ≡ α+δ (mod P−m ), is only determined modulo F [α]× . Since KA = F [α]× UA ,

52. Octahedral Representations

319

we may as well take g ∈ UA . It satisﬁes gαg −1 ≡ α α

−1

gα ≡ g

(mod P−2k ), n−2k

(mod P

or

).

By 16.2 Lemma, we can take g = 1+x, x ∈ Pn−2k . We consider the restriction of the character Λ = (χΛ)1+x to the group c+1 UF [α] . Taking y ∈ pc+1 F [α] , we get (1+x)(1+y)(1+x)−1 ≡ 1+y+xy−yx (mod Pn+1 ), giving (1+x)(1+y)(1+x)−1 (1+y)−1 ≡ 1 + xy−yx (mod Pn+1 ). Certainly xy−yx ∈ Pm+1 so, in the obvious commutator notation, χΛ[1+x, 1+y] = ψα+δ (1+xy−yx) = 1. Since k c dα , the character χ ◦ det | UF1 [α] has level k. Therefore c+1 c+1 1+x Λ | UFc+1 | UFc+1 [α] = (χΛ) [α] = χΛ | UF [α] = Λ | UF [α] .

Setting π = χπ, we therefore have π−π c. Let χ range over the characters of F × of level c; since (A, n, α) is exceptional, the representations χπ are distinct (51.5 Proposition). Counting, the proposition shows that they account for all π ∈ A02 (F ; α) with π −π c. 52.4. We return to the Bijectivity Theorem to make a preliminary reduction. We know (Tame Langlands Correspondence, 34.4) that πF induces a bijection nr Gnr 2 (F ) → A2 (F ). We must therefore show that πF restricts to a bijection wr wr G2 (F ) → A2 (F ). wr Let k 0 be an integer; deﬁne Gwr 2 (F )k to be the set of classes ρ ∈ G2 (F ) wr for which n(ρ, ψ) k, where ψ∈F one. Deﬁne A2 (F )k in exactly has level wr ⊂ A (F ) (F ) the same way. Thus πF Gwr k k . We write πF,k for the map 2 2 wr (F ) → A (F ) induced by π . Gwr k k F 2 2 Proposition. The following are equivalent: (1) the (2) the (3) the (4) the

map map map map

wr πF : Gwr 2 (F ) → A2 (F ) is bijective; πF,k is bijective for all k 0; πF,k is injective for all k 0; wr πF : Gwr 2 (F ) → A2 (F ) is injective.

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12 The Dyadic Langlands Correspondence

Proof. Clearly (1) ⇔ (2) ⇒ (3) ⇔ (4). We show that (3) ⇒ (2). We also observe that all assertions hold when q ≡ 1 (mod 3), by 51.3 Proposition. In general, the set Γ1 \Awr 2 (F )k is ﬁnite where, we recall, Γ1 denotes the × group of characters of F /UF1 . It follows that, when q ≡ 1 (mod 3), we have

wr

Γ1 \Gwr

2 (F )k = Γ1 \A2 (F )k < ∞. Suppose therefore that q ≡ −1 (mod 3). Let E/F be the unramiﬁed quadratic extension and put Σ = Gal(E/F ). The canonical maps wr Gwr 2 (F ) −→ G2 (E), ρ −→ ρE ,

wr Awr 2 (F ) −→ A2 (E), π −→ πE ,

wr wr wr take Gwr 2 (F )k to G2 (E)k and A2 (F )k to A2 (E)k . Thus, writing Γ1 (E) for × 1 the group of characters of E /UE , we have canonical maps

Σ wr , Γ1 \Gwr 2 (F )k −→ Γ1 (E)\G2 (E)k Γ1 \Awr 2 (F )k

−→

Σ Γ1 (E)\Awr . 2 (E)k

(52.4.1)

Lemma. The maps (52.4.1) are both bijective. Proof. In the ﬁrst case, representations ρ, ρ ∈ Gwr 2 (F ) lie in the same Γ1 -orbit ∼ if and only if ρ | PF = ρ | PF . The same applies over E and, since E/F is unramiﬁed, we have PE = PF . The ﬁrst map is therefore injective. Let σ ∈ Σ, σ = 1. If τ ∈ Gwr 2 (E) deﬁnes an orbit Σ Γ1 (E) ⊗ τ ∈ Γ1 (E)\Gwr , 2 (E) ∼ χ ⊗ τ , for some χ ∈ Γ1 (E). It follows that χχσ = 1. This implies then τ σ = χ | µE to be of the form λ−1 λσ , for some character λ ∈ Γ1 (E). Replacing τ by λ−1 ⊗τ , we can assume that χ is unramiﬁed. It follows that χ2 = 1. Suppose for a contradiction that χ = 1. Thus χ = φE , where φ is an unramiﬁed character of F × of order 4. The representation ρ = IndE/F τ is irreducible and satisﬁes ρ∼ = φ ⊗ ρ. Thus ρ is induced from the quartic unramiﬁed extension of F and Σ τ cannot be totally ramiﬁed. Therefore χ = 1 and τ ∈ Gwr 2 (E) . Therefore wr τ = ρE , for some ρ ∈ G2 (F ) and the ﬁrst map in (52.4.1), we deduce, is surjective. Let π, π ∈ Awr 2 (F ) satisfy πE = χπE , for some χ ∈ Γ1 (E). This implies σ χ πE = χπE and, since πE is totally ramiﬁed, it follows that χ = χσ . Thus χ = φE , for some φ ∈ Γ1 . From 46.6 Proposition, we deduce that π = φ π, for some φ ∈ Γ1 . The second map of (52.4.1) is therefore injective. σ = χτ , for some χ ∈ Γ1 (E). We deduce Now let τ ∈ Awr 2 (E) satisfy τ σ that χχ = 1. Exactly as in the preceding paragraph, we reduce to the case

52. Octahedral Representations

321

where χ is unramiﬁed and χ2 = 1. An argument identical to the proof of 46.6 Proposition gives a representation π ∈ Awr 2 (F ) such that πE = φτ , where φ is unramiﬁed and φ2 = 1. Therefore the orbit Γ1 (E) · τ meets the image of Awr 2 (F ), as required to show that the second map (52.4.1) is surjective. We thus have canonical bijections wr Σ Γ1 \Gwr 2 (F )k ←→ Γ1 (E)\G2 (E)k , wr Σ Γ1 \Awr 2 (F )k ←→ Γ1 (E)\A2 (E)k ,

while πE,k induces a bijection wr Σ Σ Γ1 (E)\Gwr 2 (E)k ←→ Γ1 (E)\A2 (E)k . wr If πF,k is injective, it induces an injection Γ1 \Gwr 2 (F )k → Γ1 \A2 (F )k . This is necessarily bijective, whence πF,k is bijective.

52.5. We now prove the Bijectivity Theorem 52.1. We take ρ, ρ ∈ Gwr 2 (F ), and assume that πF (ρ) = πF (ρ ). We show that ρ = ρ : in the light of 52.4 Proposition, this will be enough. Our assumption that πF (ρ) = πF (ρ ) carries the implication ε(χ ⊗ ρ, s, ψ) = ε(χ ⊗ ρ , s, ψ),

(52.5.1)

for all χ. We deduce that, in particular, det ρ = det ρ

and n(ρ, ψ) = n(ρ , ψ).

We can further assume that ρ, ρ are both minimal. The minimality hypothesis implies that there is a ramiﬁed simple stratum (A, n, α) such that πF (ρ) = πF (ρ ) ∈ A02 (F ; α). If either of ρ, ρ is imprimitive, the stratum (A, n, α) is ordinary and 51.4 Proposition implies both imprimitive. It follows that ρ = ρ in this case. We therefore assume that ρ, ρ are both primitive. The stratum (A, n, α) is therefore exceptional. Let K/F be a cubic extension such that ρK is imprimitive. If ρK = ρK , then ρ ∼ = χ⊗ρ, for a tame character χ of F × (42.1); our assumption πF (ρ ) = πF (ρ) implies χ = 1 (51.5 Proposition) and ρ = ρ . We therefore assume ρK = ρK . However, by Propositions 50.2 and 50.3, we have πK (ρK ), πK (ρK ) ∈ A02 (K; α). As ρK is imprimitive, the stratum (AK , e(K|F )n, α) in M2 (K) is ordinary, and so ρK is imprimitive (51.4 Proposition). If K/F is cyclic, 51.2 Proposition implies πK (ρK ) = πF (ρ)K = πF (ρ )K = πK (ρK ), whence ρK = ρK , contrary to hypothesis.

322

12 The Dyadic Langlands Correspondence

52.6. We are left, therefore, with the case where K/F is not cyclic. Since the stratum (A, n, α) is exceptional, we have n 3d(F [α]|F ), by 45.2 Theorem. Set n = 2m+1. Let L/F be the normal closure of K/F , and put Σ = Gal(L/F ). Set τ = πL (ρL ), τ = πL (ρL ). The stratum (AL , 3n, α) in M2 (L) is ordinary and τ, τ ∈ A02 (L; α)Σ . Moreover, ωτ = ωτ . Deﬁne νL = τ −τ . This satisﬁes 0 νL 3m+2. If νL = 0, there is a tamely ramiﬁed character φ of L× such that τ = φτ . We deduce that ρL ∼ = φ ⊗ ρL . The relation 2 det ρL = det ρL implies φ = 1 and hence φ is unramiﬁed. Therefore ρ ∼ = χ⊗ρ, for a character χ of F × of the form χ = χ χ , where χ is unramiﬁed (with χL = φ) and χ is trivial on norms from L. In particular, χ is unramiﬁed; thus χ is also unramiﬁed and the relation πF (ρ ) = πF (ρ) implies χ = 1. We therefore assume νL > 0. There exist representations π, π ∈ A02 (F ; α) (46.6 Corollary). Set νF = π−π ; we have such that τ = πL , τ = πL νL = 3νF

(52.6.1)

from the deﬁnition of tame lifting. If νL n, then νF [n/3], so there is a character χ of F × such that π = χπ (52.3 Corollary). This implies τ = χL τ and, since (AL , 3n, α) is ordinary, ρL = χL ⊗ ρL by 51.4 Proposition. Thus there is a character χ of F × such that ρ = χ ⊗ ρ (42.1 Proposition (3)). The relation πF (ρ ) = χ πF (ρ) = πF (ρ) yields χ = 1 and ρ = ρ. This leaves only the possibility νL > n. We consider the imprimitive representations ρK , ρK . There are admissible wild triples (E/K, 3n, β), (E /K, 3n, β ) such that ρK ∈ G02 (K; β) and ρK ∈ G02 (K; β ). By deﬁnition, we have υE (β) = −3n while n = d(E|K) = d(EL|L) (44.11 Proposition). Likewise for β . By deﬁnition, there is a character ξ of E × , of level 2n, such that ρK = IndE/K ξ. Thus ρL = IndEL/L ξL , where ξL = ξ ◦ NEL/E . By 52.2 Lemma, there is a character χ of L× , of level l = (3n−νL )/2 < n such that ε(τ, 12 , ψL )/ε(χτ, 12 , ψL ) = ε(τ , 12 , ψL )/ε(χτ , 12 , ψL ). The level of χ is l < n = d(EL|L), so χEL has level l. Therefore ε(τ, 12 , ψL ) ε(ρL , 12 , ψL ) ε(ξL , 12 , ψEL ) = = 1 1 ε(χτ, 2 , ψL ) ε(χ ⊗ ρL , 2 , ψL ) ε(χEL ξL , 12 , ψEL ) = χEL (β) = χ(NE/K (β)),

52. Octahedral Representations

323

using 23.6 Proposition. Likewise, ε(ρL , 12 , ψL ) = χ(NE /K (β )). ε(χ ⊗ ρL , 12 , ψL ) Thus, by 52.2 Lemma, 3n−νL = υK NE/K (β)/NE /K (β ) − 1 . 2

(52.6.2)

52.7. We turn to the representations ζ = πK (ρK ) and ζ = πK (ρK ). We can do the same calculation over K, using the same elements β, β . We conclude that ζ−ζ = νL = 3νF . Let Υ, Υ ∈ C(ψαK , AK ) be such that ζ = c-Ind Υ , ζ = c-Ind Υ . The character 1 has level νL . The norm NK[α]/F [α] induces an isomorphism Υ −1 Υ of UK[α] νL 1+νL F /UK[α] −→ UFνF[α] /UF1+ν UK[α] [α] ,

which restricts to the isomorphism νF 1+νF F UFνF[α] /UF1+ν [α] −→ UF [α] /UF [α] ,

x −→ x3 . It follows that Υ −1 Υ | UF1 [α] = 1. Looking back at the construction in 50.3 (especially Lemma 1), the hypothesis πF (ρ) = πF (ρ ) would imply Υ −1 Υ | UF1 [α] trivial. This case also cannot arise therefore, and we have completed the proof of the Bijectivity Theorem 52.1. We have completed the proof of 33.1 Theorem and our discussion of the Langlands Correspondence for GL(2). 52.8. The Langlands correspondence gives a bijection between imprimitive representations of WF and ordinary representations of GL2 (F ). It therefore gives a bijection between primitive representations of WF and exceptional representations of GL2 (F ). It is easy to produce exceptional representations of GL2 (F ), using 45.2 Theorem: if (A, n, α) is a ramiﬁed simple stratum with 1 n 3d(F [α]|F ) and n ≡ 0 (mod 3), any π ∈ A02 (F ; α) is exceptional (45.6 Remark). The cubic extension K/F which renders such representations π ordinary is totally ramiﬁed. On the other hand, it is easy to produce exceptional strata (A, n, α) with n = 3d(F [α]|F ); in this case, K/F is unramiﬁed (51.6). In particular, while all primitive representations ρ ∈ G02 (F ) are tetrahedral in the case q ≡ 1 (mod 3), both tetrahedral and octahedral cases arise when q ≡ −1 (mod 3).

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12 The Dyadic Langlands Correspondence

52.9 Comment. The Langlands correspondence is known to satisfy πE (ρE ) = πF (ρ)E ,

ρ ∈ Gwr 2 (F ),

(52.9.1)

for any ﬁnite, tamely ramiﬁed ﬁeld extension E/F . However, no proof of this fact, using purely local methods, is known to the authors. The central case is that where ρ is imprimitive and E/F is quadratic unramiﬁed. If we assume this fact, we get a quicker construction of the Langlands correspondence on primitive representations and an easier proof that it is bijective. To sketch the argument, let ρ ∈ G02 (F ) be primitive. Choose K/F cubic so that ρK is imprimitive. Let L/F be the normal closure of K/F , and put Σ = Gal(L/F ). Set τ = πK (ρK ). By (52.9.1), we have τL = πL (ρL ), Σ so τL ∈ Awr 2 (L) . Thus (as follows readily from 46.6 Proposition) there ex0 ists π ∈ A2 (F ) such that τL = πL . There is a unique choice of π for which ωπ = det ρ. We set πF (ρ) = π. The local constant relations are established much as before. Bijectivity is easily proved, in the manner of 51.3 Proposition. Exercise. Assume that (52.9.1) holds for E/F quadratic unramiﬁed and ρ imprimitive. Show that it holds for all ρ ∈ Gwr 2 (F ) and all tame extensions E/F . Further reading The notion of “tame lifting” has a quite general version [13], [14]. Likewise the twisting and congruence properties of the LanglandsDeligne local constant [28], [39]. Kutzko’s original proof [52] (plus [53], covered here in Chapter XI) uses a diﬀerent local constant technology, and derives (52.9.1) (for ρ imprimitive and E/F unramiﬁed quadratic) from Langlands’ base change [57], via the preliminary summary [33]. From that point, Kutzko’s argument is as summarized in 52.9 above. The argument, both here and in Chapter XI, relies heavily on counting. This has been a persistent feature of attempts to prove the Langlands conjecture. Both approaches to general dimension in characteristic zero ([38], or [37] plus [43]), the proof in positive characteristic [58], and the attempt at an “explicit” version [14], all depend on the general counting arguments of [41].

13 The Jacquet-Langlands Correspondence

53. 54. 55. 56.

Structure of division algebras Representations Functional equation Jacquet-Langlands correspondence

We return to a general non-Archimedean local ﬁeld F , for which the residual characteristic p is arbitrary. The ﬁeld F possesses a central division algebra D of dimension 4 and, up to isomorphism, only one. The group D× is locally proﬁnite and is compact modulo its centre F × . We consider the irreducible smooth representations of D× . One can classify these by a method parallel to that of Chapter IV, the principal series being eﬀectively absent for D× . There is a straightforward theory of L-functions and local constants attached to irreducible representations of D× , rather similar to that of §23 for characters of F × . Let A1 (D) denote the set of equivalence classes of irreducible smooth representations of D× . On the other hand, let A♦ 2 (F ) denote the set of equivalence classes of irreducible smooth representations of G = GL2 (F ) which are essentially square-integrable modulo centre, as in 17.4. This chapter is concerned with a canonical bijection, between A♦ 2 (F ) and A1 (D), called the Jacquet-Langlands correspondence. We specify it in terms of L-functions and local constants. It relates the explicit classiﬁcation of representations on either side. We give virtually no proofs in this chapter. Once the basic structure of D has been described, all arguments are simpler versions of those we have already seen. The reader is invited to explore the topic as a sequence of exercises.

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13 The Jacquet-Langlands Correspondence

53. Division Algebras Let D be an F -algebra which is a division ring, with centre F and dimension 4. We review1 the basic structure of D as an F -algebra, and say something of the structure of the locally proﬁnite group D× of non-zero elements of D. 53.1. The relevant elements of the structure of D as F -algebra can be summed up as follows. Proposition. Let E/F be a separable quadratic ﬁeld extension. (1) There exists an embedding E → D of F -algebras. Any two such embeddings are conjugate by an element of D× . (2) The E-algebra B = E ⊗F D is isomorphic to M2 (E). (3) Let a ∈ D; then trB/E (1 ⊗ a) ∈ F and, if a = 0, then detB (1 ⊗ a) ∈ F × . In the context of the proposition, the quantities trB (1 ⊗ a), detB (1 ⊗ a) actually depend only on a, not on E. One denotes them TrdD a, NrdD a respectively. The reduced trace TrdD is a F -linear map D → F and the reduced norm NrdD is a homomorphism D× → F × . Let a ∈ D, a ∈ / F . The F -algebra F [a] is then a ﬁeld; since D is an F [a]-vector space, we must have [F [a]:F ] = 2. Further, TrdD F [a] = TrF [a]/F , NrdD F [a]× = NF [a]/F . The polynomial X 2 − TrdD (a)X + NrdD (a) ∈ F [X] is called the reduced characteristic polynomial of a. Remark. Suppose, for the moment, that F is an arbitrary ﬁeld. If D is a central F -division algebra of dimension 4, there exists a separable quadratic extension E/F such that E admits an F -embedding in D. The proposition then holds relative to that ﬁeld E. From this general point of view, nonArchimedean local ﬁelds exhibit two idiosyncrasies. First, the algebra D is unique up to isomorphism, although that will not intervene in what we do. More importantly, any quadratic ﬁeld extension E/F can be embedded in D: this has a profound eﬀect. 53.2. The map υD : a −→ υF (NrdD a),

a ∈ D,

is a valuation on D, in that it satisﬁes υD (ab) = υD (a)+υD (b) 1

For a full discussion of division algebras over local ﬁelds, similar to the approach we take here, see [74].

53. Division Algebras

327

and the ultrametric inequality υD (a+b) min {υD (a), υD (b)}. We set |a|D = NrdD a = q −υD (a) . The metric space topology on D induced by | · |D coincides with the natural topology of D as F -vector space. 53.3. The set O = {x ∈ D : υD (x) 0} is thus an open subring of D, and q = {x ∈ D : υD (x) 1} is an open, two-sided ideal of O: it is the unique maximal left (or right) ideal of O. If D ∈ D satisﬁes υD (D ) = 1, we have q = D O = OD ; one calls D a prime element of D. Also n D O = qn = {x ∈ D : υD (x) n},

n ∈ Z.

All the sets qn are stable under conjugation by D× . In particular, D× acts by conjugation on the ring kD = O/q. Proposition. We have: (1) pO = q2 ; (2) the ring kD = O/q is a ﬁeld of q 2 elements; (3) if D is a prime element of D, then −1 D xD = xq ,

x ∈ kD .

For n 1, the ring O/qn has q 2n elements. As in the commutative case of the ﬁeld F (1.2), we deduce that O is the projective limit of the system of ﬁnite rings O/qn , n 1, and is compact. It follows that the groups 0 UD = UD = O×

and

n UD = 1+qn ,

n 1,

are compact open normal subgroups of D× . They provide a fundamental system of open neighbourhoods of the identity in D× . The subgroup F × UD has index 2 in D× ; we deduce that D× is compact modulo its centre F × .

328

13 The Jacquet-Langlands Correspondence

of characters of D. We deﬁne the level of 53.4. We consider the group D η ∈ D to be the least integer k such that qk ⊂ Ker η (with the understanding that the trivial character has level −∞). and a ∈ D, the map Lemma 1. For η ∈ D aη : x −→ η(ax) is a character of D. If η = 1, the map a → aη is an isomorphism D → D. For ψ ∈ F, we set ψD = ψ ◦ TrdD ∈ D. Lemma 2. If ψ ∈ F has level one, the character ψD of D has level one. Let us now ﬁx ψ ∈ F of level one. For a ∈ D, we form the function ψaD : x −→ ψD (a(x−1)),

x ∈ D.

Proposition. For integers m, n such that 0 [n/2] m < n, the map a → m+1 n+1 /UD . ψaD is an isomorphism of q−n /q−m with the character group of UD 53.5. The group of characters of D× takes a particularly simple form. If χ is a character of D× , we deﬁne the level of χ to be the least integer n 0 such n+1 ⊂ Ker χ. that UD Proposition. The map φ → φD = φ ◦ NrdD is an isomorphism of the character group of F × with the character group of D× . If φ has level j, then φD has level 2j.

54. Representations We can give a complete classiﬁcation of the irreducible smooth representations of D× , following parts of Chapter IV quite closely. For the purposes of this section, it is convenient to ﬁx a character ψ of F of level one. 54.1. We have noted already (53.3) that the group D× /F × is compact. So, if π is an irreducible smooth representation of D× , then dim π is ﬁnite (2.7) and Ker π contains an open normal subgroup of D× . In particular, π is trivial m , for some m 0. We deﬁne the level D (π) of π to be on a unit group UD n+1 ⊂ Ker π. the least integer n 0 such that UD We write A1 (D) for the set of equivalence classes of irreducible smooth representations of D× . If χ is a character of F × and π ∈ A1 (D), we set χπ = χD ⊗ π, where, as in 53.5, χD = χ◦NrdD . We say that π is minimal if D (π) D (χπ), for all characters χ of F × .

54. Representations

329

54.2. From 53.5, we know that the one-dimensional smooth representations of D× are the characters χD , where χ ranges over the characters of F × . We let A01 (D) denote the set of classes of representation π ∈ A1 (D) such that dim π = 1. By way of an example, let us consider the elements π of A01 (D) of level zero. We use the notion of admissible pair, as in Chapter 5. Let (E/F, θ) be an admissible pair in which θ has level zero. In particular, E/F is unramiﬁed. We identify E with a subﬁeld of D, and extend θ, by 1 . We form the smooth triviality, to a character Θ of the group D J = E × UD representation × Θ. πθD = c-IndD DJ Proposition. (1) The representation πθD is irreducible and of level 0. (2) The map (E/F, θ) → πθD is a bijection between the set of isomorphism classes of admissible pairs of level zero and the set of π ∈ A01 (D) of level zero. 54.3. Let α ∈ D, α ∈ / F . The algebra F [α] is then a quadratic ﬁeld extension. As we noted in 13.4, the notion of α being minimal over F can be formulated purely in terms of the extension F [α]/F . Adapting 13.4 Deﬁnition to the context of α ∈ D, we get: Deﬁnition. Let α ∈ D× , and set n = −υD (α). The element α is minimal over F if either (1) n is odd, or (2) n is even and, for a prime element of F , the reduced characteristic polynomial of n/2 α over F is irreducible modulo p. Observe that the ﬁeld extension F [α]/F is totally ramiﬁed in case (1), unramiﬁed in case (2). Just as in 13.4, we have: Proposition. Let α ∈ D× , with n = −υD (α). (1) The element α is minimal over F if and only if α+q1−n ∩ F = ∅. 1 contains only minimal (2) If α is minimal over F , the coset α+q1−n = αUD elements. 54.4. Let α ∈ D be minimal over F , and such that n = −υD (α) 1. We can [n/2]+1 (53.4). We get the same then form the character ψαD of the group UD intertwining properties as in G = GL2 (F ): Intertwining Theorem. Let α ∈ D be minimal over F , with n = −υD (α) 1. Write E = F [α] and let x ∈ D× . The following are equivalent: [n/2]+1

(1) x intertwines the character ψαD of UD [(n+1)/2] (2) x ∈ E × UD .

.

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13 The Jacquet-Langlands Correspondence

m Since each UD is a normal subgroup of D× , the element x intertwines [n/2]+1 | UD if and only if it normalizes it. The analogue of the Conjugacy Theorem (15.2), for characters ψαD , is thus trivial: if β is another minimal [n/2]+1 intertwine element of D with υD (β) = −n, the characters ψαD , ψβD of UD in D× if and only if they are conjugate. In the situation of the Intertwining Theorem, we write

ψαD

DJ α

= E × UD

[(n+1)/2]

.

(54.4.1)

Immediately: [n/2]+1

Lemma. Let Λ be an irreducible representation of D J α such that Λ | UD [n/2]+1 contains the character ψαD ; then Λ | UD is a multiple of ψαD .

We write C(ψαD ) for the set of equivalence classes of irreducible represen[n/2]+1 is a multiple of ψαD . tations of D J α such that Λ | UD Proposition. (1) Let Λ ∈ C(ψαD ). The representation ×

πΛ = c-IndD Jα Λ is irreducible. (2) Representations Λ1 , Λ2 ∈ C(ψαD ) intertwine in D× if and only if they are equivalent. In particular, πΛ1 ∼ = πΛ2 if and only if Λ1 ∼ = Λ2 . 54.6. Let π ∈ A01 (D) be minimal of level n 1. Consider the restriction of π n . This is a sum of characters ψβD , where the elements β are all minimal to UD [n/2]+1

, with α minimal over F . Thus π contains a character ψαD of the group UD over F . The conjugacy class of this character is determined uniquely by π. Taking this further, π must contain a unique Λ ∈ C(ψαD ). This reduces the classiﬁcation of the minimal elements of A01 (F ) to the description of the elements of the sets C(ψαD ). 54.7. Let α ∈ D be minimal over F , with n = −υD (α) 1. Write E = F [α]. Thus e(E|F ) = 2/gcd(2, n). Lemma 1. (1) If n is odd or if n ≡ 0 (mod 4), then D J α = E × UD . × [n/2]+1 (2) If n ≡ 2 (mod 4), then D J α contains E UD with index q 2 . [n/2]+1

In case (1) of Lemma 1, therefore, the set C(ψαD ) consists of the characters [n/2]+1 [n/2]+1 such that Λ | UD = ψαD . Λ of E × UD

55. Functional Equation

331

Lemma 2. Suppose that n ≡ 2 (mod 4). Let θ be a character of the group [n/2]+1 extending ψαD . UE1 UD

1 DHα

=

(1) There exists a unique irreducible representation ηθ of the group D J 1α = [(n+1)/2] such that η | D H 1α contains θ. UE1 UD (2) The representation ηθ satisﬁes dim ηθ = q, and ηθ | D H 1α is a multiple of θ. (3) An irreducible representation Υ of D J α lies in C(ψαD ) if and only if [n/2]+1 = ψαD . Υ | D J 1α = ηθ , for some character θ of D H 1α such that θ | UD 54.8. There is an entertaining assymmetry. Let (A, n, α), n 1, be an unramiﬁed simple stratum in A = M2 (F ). In particular, α is minimal over F . The elements of C(ψα , A) have dimension 1 if n is odd, and dimension q if n is even. Now choose an F -embedding F [α] → D, and view α as an element of D× . Thus υD (α) = −2n. The elements of C(ψαD ) have dimension q if n is odd, and dimension 1 if n is even.

55. Functional Equation We review the theory of functional equations for irreducible smooth representations of D× . While the formalism resembles that of §24, the underlying methods are much closer to those of §23. 55.1. Let (π, V ) be an irreducible smooth representation of D× and, as before, let C(π) denote the space of coeﬃcients of π (deﬁned exactly as in 10.1). We consider integrals of the form Φ(x) f (x) NrdD xs dµ∗ (x), ζ(Φ, f, s) = D×

Cc∞ (D),

f ∈ C(π) and a Haar measure µ∗ on D× . This for functions Φ ∈ integral can be re-written as a formal Laurent series in q −s . We set Z(π) = {ζ(Φ, f, s+ 12 ) : Φ ∈ Cc∞ (D), f ∈ C(π)}. Proposition. There is a unique polynomial Pπ (T ) ∈ C[T ] such that Pπ (0) = 1 and Z(π) = Pπ (q −s )−1 C[q s , q −s ]. We deﬁne L(π, s) = Pπ (q −s )−1 . Theorem. Let (π, V ) be an irreducible smooth representation of D× . We have L(π, s) = 1 except when π = χD , for an unramiﬁed character χ of F × . In this exceptional case, we have L(χD , s) = L(χ, s+ 12 ).

332

13 The Jacquet-Langlands Correspondence

55.2. We ﬁx a character ψ ∈ deﬁne an operation of Fourier Haar measure on D, exactly as

F, ψ = 1. Using the character ψD , we can ˆ on Cc∞ (D), and self-dual transform Φ → Φ for F (23.1) or A (24.1).

Theorem. Let ψ ∈ F, ψ = 1. There is a unique rational function γ(π, s, ψ) ∈ C(q −s ) such that ˆ fˇ, 3 −s) = γ(π, s, ψ) ζ(Φ, f, s+ 1 ), ζ(Φ, 2 2 for all Φ ∈ Cc∞ (D), f ∈ C(π). The function ε(π, s, ψ) = γ(π, s, ψ)

L(π, s) L(ˇ π , 1−s)

is of the form aq bs , for some a ∈ C× and b ∈ Z. Warning. It is conventional to insert a minus sign in the deﬁnition of ε (or γ) in this context. We feel the picture is clearer without it, so we consciously and deliberately omit it. 55.3. The local constant ε(π, s, ψ), for π ∈ A1 (D), has properties analogous to that for representations of G. In particular, ε(π, s, aψ) = ωπ (a) a2s−1 ε(π, s, ψ),

a ∈ F ×,

(55.3.1)

where ωπ denotes the central character of π. So, when calculating local constants, we can always reduce to the convenient case where ψ has level one. Since the irreducible representation (π, V ) has ﬁnite dimension, there is no diﬃculty in working with operator-valued integrals ζ(Φ, π, s) = Φ(x) π(x) NrdD xs dµ∗ (x), D×

for Φ ∈ Cc∞ (D). Using this formalism, the proofs of the results 55.1, 55.2 are virtually identical to the corresponding arguments in §23. For n ∈ Z, write D(n) = {x ∈ D : υD (x) = −n}. Theorem. Let (π, V ) ∈ A01 (D) have level n, and let ψ ∈ F have level one. The integral 3 π ˇ (x) NrdD x 2 −s dµ∗ (x) (55.3.2) D(n) n+1 is a scalar operator on Vˇ with eigenvalue is µ∗ (UD ) ε(π, s, ψ)/q (2n+1) .

The integral (55.3.2) reduces to a non-abelian Gauss sum exactly like (25.1.1). If π is written in the form c-Ind Λ, as in §55, one can equally express this Gauss sum in terms of Λ, just as in 25.5.

55. Functional Equation

333

55.4. For the one-dimensional representations, we have: Proposition. Let χ be a character of F × and ψ ∈ F, ψ = 1. If χ is unramiﬁed, then (55.4.1) ε(χD , s, ψ) = ε(χ, s, ψ), while otherwise, ε(χD , s, ψ) = −ε(χ, s− 12 , ψ) ε(χ, s+ 12 ).

(55.4.2)

55.5. The proof of (55.4.1) is straighforward, but that of (55.4.2) relies one an elementary calculation, which is also crucial at later stages of the section. Let χ be a character of F × of level k 1. Let c ∈ p−k satisfy χ(1+x) = ψ(cx), x ∈ p[k/2]+1 . We have various neo-classical species of Gauss sum. Taking a chain order A in A = M2 (F ) with e = eA (and putting χA = χ ◦ det): χ(cx)−1 ψ(cx) ; τ (χ, ψ) = k+1 x∈UF /UF

τA (χ, ψ) =

χA (cx)−1 ψA (cx) ;

ek+1 x∈UA /UA

τD (χ, ψ) =

χD (cx)−1 ψD (cx) .

2k+1 x∈UD /UD

Lemma. There are positive constants cA , cD , depending only on k and eA , such that τ (χ, ψ)2 = cA τA (χ, ψ) = −cD τD (χ, ψ). (The ﬁrst equality is (25.6.1).) 55.6. Following through the development for GL2 (F ), the next step is to prove: Converse Theorem. Let π1 , π2 be irreducible smooth representations of D× . Suppose that L(χπ1 , s) = L(χπ2 , s), ε(χπ1 , s, ψ) = ε(χπ2 , s, ψ), for all characters χ of F × and all ψ ∈ F, ψ = 1. We then have π1 ∼ = π2 . The L-function L(χπ, s) is non-trivial, for some χ, if and only if dim π = 1. It follows that, if one πi is one-dimensional, then so is the other and the result is a straightforward consequence of 55.1 Theorem. Assume therefore that dim πi > 1, i = 1, 2. One reduces, as in §27, to the case where ψ has level one. The key step in the proof is again:

334

13 The Jacquet-Langlands Correspondence

Inversion Formula. Let π ∈ A01 (D) satisfy n = (π) (χπ), for all characters χ of F × . Let ψ ∈ F have level one, and set r = [n/2]. There is a constant kD , depending only on n, such that ε(χπ, 12 , cψ) χ(NrdD g) cψ(−TrdD g), tr π ˇ (g) = kD χ∈Γr+1 /Γ0 , r+1 c∈UF /UF

for all g ∈ D(n) which are minimal over F .

56. Jacquet-Langlands Correspondence We come to the main point. 56.1. Let A♦ 2 (F ) denote the set of equivalence classes of irreducible smooth representations of G = GL2 (F ) which are essentially square-integrable (see 17.4). Jacquet-Langlands Correspondence. Let D be a central F -division algebra of dimension 4. Let ψ ∈ F, ψ = 1. There is a unique map A♦ 2 (F ) −→ A1 (D), π −→ πD ,

(56.1.1)

such that L(χπD , s) = L(χπ, s),

(56.1.2)

ε(χπD , s, ψ) = −ε(χπ, s, ψ),

(56.1.3)

A♦ 2 (F )

×

and all characters χ of F . for all π ∈ The map π → πD is bijective and (56.1.3) holds for all ψ ∈ F, ψ = 1. It further satisﬁes (χπ)D = χπD , ωπD = ωπ ,

(56.1.4) ∨

(ˇ π )D = (πD ) . The relation (56.1.3) implies (πD ) = 2(π), π ∈ A02 (F ). In the remainder of the section, we sketch a proof of the result. 0 56.2. Let π ∈ A♦ 2 (F ) A2 (F ). Therefore (17.5 Theorem) π is a special representation φ·StG , for some character φ of F × . In this case, we set πD = φD . This gives a bijection 0 0 A♦ 2 (F ) A2 (F ) −→ A1 (D) A1 (D),

φ · StG −→ φD .

(56.2.1)

Our calculations of L-functions in 55.1, 26.1 Theorem show that this is the unique map satisfying (56.1.2). The ε-calculations in 26.1 Theorem and 55.4 show that it also satisﬁes (56.1.3).

56. Jacquet-Langlands Correspondence

335

56.3. A representation π ∈ A2 (F ) lies in π ∈ A02 (F ) if and only if L(χπ, s) = 1, for all characters χ of F × (27.2 Proposition). The analogous property holds in A1 (D). So, we have to produce a bijection A02 (F ) → A01 (D) satisfying (56.1.3). We need only do this for one character ψ, arguing exactly as in 27.4. We may therefore assume ψ has level one. We ﬁrst deﬁne πD when π ∈ A02 (F ) is minimal. We proceed by cases. 56.4. Let π ∈ A02 (F ) have level zero. Thus there is an admissible pair (E/F, θ), in which θ has level zero, such that π ∼ = πθ , in the notation of 19.1. Using the notation of 54.2, we set πD = (πθ )D = πθD . The relation ε(π, s, ψ) = −ε(πD , s, ψ) follows from (25.4.1) and an easy calculation in D. 56.5. Let π ∈ A02 (F ) be minimal, with (π) > 0. There is a simple stratum [n/2]+1 . More precisely, (A, n, α) in A such that π contains the character ψα of UA π contains a representation Λ ∈ C(ψα , A), and so π = c-IndG J Λ, where J = [(n+1)/2] , E = F [α]. E × UA Suppose, in this paragraph, that eA = 2. Thus dim Λ = 1. We choose an F -embedding E → D, and henceforth regard α equally as [n/2]+1 and deﬁne a character an element of D. We form the group D J = E × UD ΛD of D J by [n/2]+1 u ∈ UD , ΛD (u) = ψαD (u), ΛD (x) = (−1)υE (x) Λ(x),

x ∈ E×.

We let πD be the representation of D× induced by ΛD . Remark. Various choices were made in the deﬁnition of πD . We shall see (56.8 below) that the deﬁnition is independent of these, so the matter need not deﬂect us now. 56.6. In the same initial situation, we assume that eA = 1 and that n is odd. [n/2]+1 , Λ ∈ C(ψα , A). Let θ denote Again, π contains a character Λ of E × UA 1 [n/2]+1 the restriction Λ | UE UA and set ωΛ = Λ | F × . n+1 by Now view E as embedded in D. We get a character θD of UE1 UD θD | UE1 = θ | UE1 ,

n+1 θD | UD = ψαD .

n such that There is a unique irreducible representation ηD of the group UE1 UD 1 n+1 ηD | UE UD is a multiple of θD (54.7 Lemma 2). We deﬁne an irreducible n by (cf. 19.4) representation ΛD of D J = E × UD n+1 = ηD ; ΛD | UE1 UD

ΛD | F × = a multiple of ωΛ ; tr ΛD (ζ) = −Λ(ζ),

ζ ∈ µE µF .

We let πD be the representation of D× induced by ΛD .

336

13 The Jacquet-Langlands Correspondence

56.7. Finally, we have the case eA = 1 and n even. Here, it is easiest to take an admissible pair (E/F, χ), with E/F unramiﬁed and χ of level n, such that π = πχ , in the notation of (19.4.2). We deﬁne the character ΛD of × n+1 by D J = E UD ΛD | E × = χ,

n+1 ΛD | UD = ψαD .

Again, πD is deﬁned to be the representation of D× induced by ΛD . 56.8. Taking π ∈ A02 (F ) of minimal normalized level, the next step is to use these deﬁnitions to verify the relation ε(χπD , s, ψ) = −ε(χπ, s, ψ), for all characters χ of F × and the ﬁxed ψ ∈ F. This done, the Converse Theorem (55.6) for D shows that the isomorphism class of πD is determined by that of π, so πD is independent of the various choices made in its deﬁnition. We now extend the deﬁnition of πD to all π ∈ A02 (F ) using the relation (χπ)D = χπD . The equation (56.1.3) is then satisﬁed and it is clear from the classiﬁcation in §55 that the map π → πD gives a surjection A02 (F ) → A01 (D). It is injective, because of the Converse Theorem (27.1) for GL2 (F ). 56.9. Comment. Let g ∈ G = GL2 (F ); one says that g is regular semisimple if its characteristic polynomial has distinct roots (in a splitting ﬁeld), elliptic if its characteristic polynomial is irreducible, elliptic regular if the characteristic polynomial is also separable. For an elliptic element g ∈ G, there is an element gD ∈ D whose minimal polynomial over F coincides with that of g: this condition determines gD up to conjugacy in D× . The process g → gD gives a bijection between the set of elliptic conjugacy classes in G and the set of D× -conjugacy classes in D× F × . By way of an example, consider a representation π ∈ A02 (F ) of minimal normalized level = (π) ∈ 12 Z. Let (A, Ξ) be a cuspidal inducing datum in π. Let g ∈ KA be minimal, with υF (det g) = −n. Taking more care with positive constants than hitherto, the two inversion formulas and the local constant relation (56.1.3) together yield tr Ξ(g) = −tr πD (gD ).

(56.9.1)

Taking account of the relation ωπ = ωπD , the equation (56.9.1) is valid for all minimal elements g ∈ G such that υF (det g) ≡ 2 It reﬂects a deeper reality.

(mod 2).

(56.9.2)

56. Jacquet-Langlands Correspondence

337

One knows that there is a function θπ , deﬁned (at least) on the open dense set Gss reg of regular semisimple elements of G and locally constant there, such that θπ (g)f (g) dg, f ∈ H(G). tr π(f ) = G

(Observe that, since π is admissible, the operator π(f ) has ﬁnite-dimensional image and its trace is therefore deﬁned.) It is more suggestive to write θπ = tr π. For minimal elements g satisfying (56.9.2), one can show that tr π(g) = tr Ξ(g). Thus we have tr π(g) = −tr πD (gD ). The Jacquet-Langlands correspondence is usually stated in the form: There is a unique bijection A♦ 2 (F ) −→ A1 (D),

(56.9.3)

π −→ πD , such that tr π(g) = −tr πD (gD ), for all regular elliptic elements g ∈ G. Our characterization (56.1.2), (56.1.3) is thus weaker than the usual one, but nontheless adequate. Further reading For a central F -division algebra H of arbitrary ﬁnite dimension, the representations of H × have been described by Zink [91] and, in a version closely parallel to [19], by Broussous [9]. In characteristic zero, the Jacquet-Langlands correspondence was proved in dimension 4 in [46] and in arbitrary dimension by Rogawski [72]. The case of positive characteristic and arbitrary dimension is due to Badulescu [3]. The compatibility of the classiﬁcation schemes of [9] and [19] via the Jacquet-Langlands correspondence has not been completely written down, but see [16] for a particularly hard case. The functional equation is in [46], but is a special instance of [35], which treats GLm (H), for any m 1. An introductory account of the basic theory of characters can be found in an appendix to [13]. The domain of deﬁnition of a character is a subtle matter, for which one can consult [59].

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Index

A2 (F ), 212 A02 (F ), 129 A02 (F ; α), 269 Anr 2 (F ), 129 Awr 2 (F ), 285 Additive duality, 11 Admissible pair, 124 — , minimal, 124 Admissible wild triple, 273 Artin reciprocity map, 187 Bijectivity Theorem, 316 Bruhat decomposition, 44 Boundary, 191 Cα (X), 279 Cc∞ (G), 25 C(π), 70 C(ψα , A), 106 Cartan decomposition, 51 Central character, 21 Chain order, 88 Character, 11 — , unitary, 11 Classiﬁcation Theorem (principal series), 69 — (cuspidal), 108 Coeﬃcient, 70 Conjugacy Theorem, 105 Containment (in representation), 77 Contragredient, 23 Converse Theorem, 170 Convolution, 33 Cuspidal inducing datum, 110

Cuspidal representation, 47, 62, 71 — , minimal, 267 — , totally ramiﬁed, 129, 287 — , unramiﬁed, 129 Cuspidal type, 107 d(E|F ), 252 Deligne representation, 200 — , semisimple, 200 Division, 191 — pre-inductive, 191 Dual, smooth, 23 Duality Theorem, 32, 56 eK , 34 K/F , 289 Exceptional representation, 268 — stratum, 283 Exhaustion Theorem, 104 Existence Theorem, 309 F, 11 Formal degree, 75 Fourier inversion formula, 139, 147 Fourier transform, 138, 147 Frobenius element, 181 Frobenius Reciprocity, 18, 20 Functional equation, Godement-Jacquet, 149 — , Tate, 143 Gn (F ), 200 G0n (F ), 185 Gim 2 (F ), 249 Gnr 2 (F ), 214 Gss n (F ), 185 Gwr 2 (F ), 287

346

Index

Galois Converse Theorem, 265 γ(∗, s, ψ), 141, 148 Gauss sum, 145, 146 — , non-abelian, 156 Group, locally proﬁnite, 8 — , proﬁnite, 8 — , unimodular, 27 H(G), 33 H(G, K), 35 Haar integral, 26 Haar measure, 27 — , self-dual, 139, 148 Hecke algebra, 33 — , ρ-spherical, 79 Herbrand function, 252 I, 87 IF , 181 Imprimitive representation, 249 —simply, 255 — triply, 255 Induction, smooth, 18 — , compact, 19 — , normalized, 69 — , unitary, 69 Induction constant, 191 Inertia group, 181 — wild, 182 Intertwining, 77 Intertwining Theorem, 105 Irreducibility Criterion, 65 Isotypic component, 15 Iwahori decomposition, 52 Iwahori subgroup, 51 Iwasawa decomposition, 50 Jacquet functor, 61 — module, 61 KA , 89 κE/F , 188, 236, 254 Kirillov model, 228, 231 L-function, Artin, 189 — , Godement-Jacquet, 149 — , Tate, 143 -adic representation, Φ-semisimple, 208 Langlands constant, 194 Langlands correspondence, 212 — , -adic, 223 — , tame, 219 Lattice, 10 — o-lattice, 10

Lattice chain, 86 Level, 11, 12, 256 — , normalized, 91 LftK/F , 293 Lift, 293 Local constant, Godement-Jacquet, 161 — , Langlands-Deligne, 190 — , Tate, 143 M, 87 Module, 29, 54 Minimal character, 124, 285 Minimal element, 98 Minimal representation, 256, 267 Occurrence (in representation), 77 ΩF , 180 Ordinary representation, 268 — stratum, 279 PF , 182 Primitive representation, 255 Projectivity Theorem, 74 ψa , 91 RE/F , 194 Rep(G), 13 Representation — , admissible, 13 — , cuspidal, 47, 62, 71 — , discrete series, 117 — , equivalent, 13 — , essentially square-integrable, 117 — , γ-cuspidal, 70 — , induced, 18 — , irreducible, 13 — , principal series, 47, 61 — , special, 68 — , semisimple, 15, 185 — , smooth, 13 — , square-integrable, 75 — , Steinberg, 47, 68 Restriction-Induction Lemma, 67 Schur’s Lemma, 21 Schur’s orthogonality relation, 74 Semi-invariant measure, 31 Separation Property, 40 StG , 47, 68 Stability Theorem, 147, 162 Stratum, 92 — , essentially scalar, 96 — , fundamental, 92 — , ramiﬁed simple, 96

Index — , split, 96 — , unramiﬁed simple, 96 T, 254, 315 Tame lifting, 286 Tame Parametrization Theorem, 130 Totally ramiﬁed representation, 129, 214, 255

Translation action, 25 Twist, 65 UA , UAn , 88 WF , 182 Weil group, 182 Weil representation, 245 Whittaker model, 227

347

Some Common Symbols

aF : c-Ind: δ: ε(∗, s, ψ): Ind: ι: L(∗, s): (∗): λK/F (ψ): ωπ : πF : τ (∗, ψ):

Artin Reciprocity map compact induction functor module local constant (smooth) induction functor normalized induction functor L-function normalized level Langlands constant central character of π Langlands correspondence over F Gauss sum

Some Common Abbreviation

If E/F is a ﬁnite separable ﬁeld extension: χE ψE ρE IndE/F θ πE

= = = = =

χ ◦ NE/F , ψ ◦ TrE/F , ρ | WE , F IndW WE θ, LftE/F π,

χ a character of F × ; ψ ∈ F; ρ ∈ Gss n (F ); θ ∈ Gss m (E); π ∈ Awr 2 (F ).

The last only applies when E/F is tame and p = 2.

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