Representations of SL_2(F_q)

Representations of SL2 (Fq ) Algebra and Applications Volume 13 Managing Editor: Alain Verschoren University of Antw...

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Representations of SL2 (Fq )

Algebra and Applications Volume 13

Managing Editor: Alain Verschoren University of Antwerp, Belgium Series Editors: Alice Fialowski Eötvös Loránd University, Hungary Eric Friedlander Northwestern University, USA John Greenlees Sheffield University, UK Gerhard Hiss Aachen University, Germany Ieke Moerdijk Utrecht University, The Netherlands Idun Reiten Norwegian University of Science and Technology, Norway Christoph Schweigert Hamburg University, Germany Mina Teicher Bar-llan University, Israel Algebra and Applications aims to publish well written and carefully refereed monographs with up-to-date information about progress in all fields of algebra, its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, as well as applications in related domains, such as number theory, homotopy and (co)homology theory, physics and discrete mathematics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, C ∗ -algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications. In addition, Algebra and Applications will also publish monographs dedicated to computational aspects of these topics as well as algebraic and geometric methods in computer science.

Cédric Bonnafé

Representations of SL2(Fq )

Cédric Bonnafé CNRS (UMR 5149) Université Montpellier 2 Institut de Mathématiques et de Modélisation de Montpellier Place Eugène Bataillon 34095 Montpellier Cedex France [email protected]

ISBN 978-0-85729-156-1 e-ISBN 978-0-85729-157-8 DOI 10.1007/978-0-85729-157-8 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2010938218 Mathematics Subject Classification (2010): 20, 20-02 © Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my parents.

Preface

If our sole purpose were to calculate the character table of the finite group G = SL2 (Fq ) (here, q is a power of a prime number p) by ad hoc methods, this book would only amount to a few pages. Indeed, this problem was solved independently by Jordan [Jor] and Schur [Sch] in 1907. The goal of this book is rather to use the group G to give an introduction to the ordinary and modular representation theory of finite reductive groups, and in particular to Harish-Chandra and Deligne-Lusztig theories. It is addressed in particular to students who would like to delve into Deligne-Lusztig theory with a concrete example at hand. The example of G = SL2 (Fq ) is sufficiently simple to allow a complete description, and yet sufficiently rich to illustrate some of the most delicate aspects of the theory. There are a number of excellent texts on Deligne-Lusztig theory (see for example Lusztig [Lu1], Carter [Carter], Digne-Michel [DiMi] for the theory of ordinary characters and Cabanes-Enguehard [CaEn] for modular representations). This book does not aim to offer a better approach, but rather to complement the general theory with an illustrated example. We have tried not to rely upon the above books and give full proofs, in the example of the group G , of certain general theorems of Deligne-Lusztig theory (for example the Mackay formula, character formulas, questions of cuspidality etc.). Although it is not always straightforward, we have tried to give proofs which reflect the spirit of the general theory, rather than giving ad hoc arguments. We hope that this shows how general arguments of Deligne-Lusztig theory may be made concrete in a particular case. At the end of the book we have included a chapter offering a very succinct overview (without proof) of Deligne-Lusztig theory in general, as well as making links to what has already been seen (see Chapter 12). Historically, the example of SL2 (Fq ) played a seminal role. In 1974, Drinfeld (at age nineteen!) constructed a Langlands correspondence for GL2 (K), where K is a global field of equal characteristic [Dri]. In the course of this work, he remarks that the cuspidal characters of G = SL2 (Fq ) may be found in the first -adic cohomology group (here, is a prime number different vii

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Preface

from p) of the curve Y with equation xy q − yx q = 1, on which G acts naturally by linear changes of coordinates (we call Y the Drinfeld curve). This example inspired Deligne and Lusztig (see their comments in [DeLu, page 117, lines 22-24]) who then, in their fundamental article [DeLu], established the basis of what has come to be known as Deligne-Lusztig theory. A large part of this book is concerned with unravelling Drinfeld’s example. Our principal is to rather shamelessly make use of the fundamental results of -adic cohomology (for which we provide an overview tailored to our needs in Appendix A) to construct representations of G in characteristic 0 or . In order to efficiently use this machinery, we conduct a precise study of the geometric properties of the action of G on the Drinfeld curve Y, with particular attention being paid to the construction of quotients by various finite groups. Having completed this study we do not limit ourselves to character theory. Indeed, a large part of this book is dedicated to the study of modular representation theory, most notably via the study of Broué’s abelian defect group conjecture [Bro]. This conjecture predicts the existence of an equivalence of derived categories when the defect group is abelian. For the representations of the group G in characteristic ∈ {2, p}, the defect group is cyclic, and such an equivalence can be obtained by entirely algebraic methods [Ric1], [Lin], [Rou2]. However, in order to stay true to the spirit of this book, we show that it is possible, when is odd and divides q + 1, to realise this equivalence of derived categories using the complex of -adic cohomology of the Drinfeld curve (this result is due to Rouquier [Rou1]). For completeness we devote a chapter to the study of representations in equal, or natural, characteristic. Here the Drinfeld curve ceases to be useful to us. We give an algebraic construction of the simple modules by restriction of rational representations of the group G = SL2 (Fq ), as may be done for an arbitrary finite reductive group. Moreover, in this case the Sylow psubgroup is abelian, and it was shown by Okuyama [Oku1], [Oku2] (for the principal block) and Yoshii [Yo] (for the nonprincipal block with full defect) that Broué’s conjecture holds. Unfortunately, the proof is too involved to be included in this book. P REREQUISITES – The reader should have a basic knowledge of the representation theory of finite groups (as contained, for example, in [Ser] or [Isa]). In the appendix we recall the basics of block theory. He or she should also have a basic knowledge of algebraic geometry over an algebraically closed field (knowledge of the first chapter of [Har], for example, is more than sufficient). An overview of -adic cohomology is given in Appendix A, while Appendix C contains some basic facts about reflection groups (this appendix will only be used when we discuss some curiosities connected to the groups SL2 (Fq ) for q ∈ {3, 5, 7}). We have also added a number of sections and subsections (marked with an asterisk) which contain illustrations, provided by G and Y, of related

Preface

ix

but more geometric subjects (for example the Hurwitz formula, automorphisms of curves, Abyankhar’s conjecture, invariants of reflections groups). These sections require a more sophisticated geometric background and are not necessary for an understanding of the main body of this book. Lastly, for results concerning derived categories, we will not need more than is contained in the economical and efficient summary in Appendix A1 of the book of Cabanes and Enguehard [CaEn]. The sections and subsections requiring some knowledge of derived categories are also marked with an asterisk. Besançon, France April 2009

Cédric Bonnafé

Acknowledgements

This book probably would have never seen the light of day had it not been for the suggestion of Raphaël Rouquier, who also offered constant encouragement whilst the project was underway. He also patiently answered numerous questions concerning derived categories, and carefully read large passages of this book. For all of this I would like to thank him warmly. I would also like to thank Marc Cabanes for numerous fruitful conversations on the subject of this book. This English version of the book is the translation, by Geordie Williamson, of a French manuscript I had written. I want to thank him warmly for the quality of his job, as well as for the hundreds of interesting remarks, comments, critics, suggestions. It has been a great pleasure to work with him during the translation process.

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Contents

Part I Preliminaries 1

Structure of SL2 (Fq ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Special Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Bruhat Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 The Non-Split Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Distinguished Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Conjugacy Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Centralisers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Parametrisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Sylow Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.1 Sylow p-Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.2 Other Sylow Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2

The Geometry of the Drinfeld Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Interesting Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Quotient by G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Quotient by U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Quotient by μq+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fixed Points under certain Frobenius Endomorphisms . . . . . . . 2.4 Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Curiosities* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Hurwitz Formula, Automorphisms* . . . . . . . . . . . . . . . . . 2.5.2 Abhyankar’s Conjecture (Raynaud’s Theorem)* . . . . . .

15 16 16 17 18 19 19 20 21 22 24

Part II Ordinary Characters 3

Harish-Chandra Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Harish-Chandra Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 xiii

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3.2.1 3.2.2 3.2.3 3.2.4 3.2.5

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mackey Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Restriction from GL2 (Fq ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 31 32 33 34

4

Deligne-Lusztig Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definition and First Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 The Character R (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Cuspidality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mackey Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Parametrisation of Irr G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Action of the Frobenius Endomorphism . . . . . . . . . . . . . . . . . . . . 4.4.1 Action on Hc1 (Y)e1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Action on Hc1 (Y)eθ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Action on Hc1 (Y)eθ ⊕ Hc1 (Y)eθ −1 . . . . . . . . . . . . . . . . . . . . .

37 37 37 38 39 39 40 45 46 46 47 48

5

The Character Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Characters of Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Calculation of Tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Calculation of Tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 The Characters R(α ) and R (θ ) . . . . . . . . . . . . . . . . . . . . . . 5.2 Restriction to U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 B-Invariant Characters of U . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Restriction of Characters of G . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Values of Υ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Character Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 51 52 53 53 54 54 56 57

Part III Modular Representations 6

More about Characters of G and of its Sylow Subgroups . . . . . . . . 6.1 Central Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Global McKay Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Characters of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Characters of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Characters of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Normalisers of Sylow 2-Subgroups . . . . . . . . . . . . . . . . . . 6.2.5 Verification of the Global McKay Conjecture . . . . . . . . . .

63 63 64 65 66 67 68 68

7

Unequal Characteristic: Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Blocks, Brauer Correspondents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Partition in -Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Brauer Correspondents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 71 72 74

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7.2 Modular Harish-Chandra Induction . . . . . . . . . . . . . . . . . . . . . . . . 75 7.3 Deligne-Lusztig Induction* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8

Unequal Characteristic: Equivalences of Categories . . . . . . . . . . . . . 8.1 Nilpotent Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Harish-Chandra Induction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Deligne-Lusztig Induction* . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Quasi-Isolated Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Harish-Chandra Induction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Deligne-Lusztig Induction* . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Principal Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 The Case when is Odd and Divides q − 1 . . . . . . . . . . . . 8.3.2 The Case when is Odd and Divides q + 1* . . . . . . . . . . . 8.3.3 The Case when = 2* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Alvis-Curtis Duality* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 86 86 87 88 88 88 89 89 90 93 94

9

Unequal Characteristic: Simple Modules, Decomposition Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.1.1 Induction and Decomposition Matrices . . . . . . . . . . . . . . 97 9.1.2 Dimensions of Modules and Restriction to U . . . . . . . . . 98 9.2 Nilpotent Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.3 Quasi-Isolated Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 9.4 The Principal Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.4.2 The Case when is Odd and Divides q − 1 . . . . . . . . . . . . 103 9.4.3 The Case when is Odd and Divides q + 1 . . . . . . . . . . . . 104 9.4.4 The Case when = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

10 Equal Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.1 Simple Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 10.1.1 Standard or Weyl Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 110 10.1.2 Simple Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 10.1.3 The Grothendieck Ring of G . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.2 Simple kG -Modules and Decomposition Matrices . . . . . . . . . . . 119 10.2.1 Simple kG -Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 10.2.2 Decomposition Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10.3 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.3.1 Blocks and Brauer Correspondents . . . . . . . . . . . . . . . . . . 122 10.3.2 Brauer Trees* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Part IV Complements 11 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 11.2 The Case when q = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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11.2.1 11.2.2 11.2.3 11.2.4 11.2.5

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Character Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 The Group SL2 (F3 ) as a Subgroup of SL2 (F ) . . . . . . . . . . 133 The Group SL2 (F3 ) as a Reflection Group of Rank 2 . . . 134 The Group PSL2 (F3 ) and the Isometries of the Tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 11.3 The Case when q = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11.3.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11.3.2 Character Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11.3.3 The Group SL2 (F5 ) as a Subgroup of SL2 (Fr ) . . . . . . . . . 137 11.3.4 The Group SL2 (F5 ) × Z/5Z as a Reflection Group of Rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 11.3.5 The Group PSL2 (F5 ), the Dodecahedron and the Icosahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 11.4 The Case when q = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 11.4.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 11.4.2 Character Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 11.4.3 The Isomorphism Between the Groups PSL2 (F7 ) and GL3 (F2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 11.4.4 The Group PSL2 (F7 ) × Z/2Z as a Reflection Group of Rank 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 12

Deligne-Lusztig Theory: an Overview* . . . . . . . . . . . . . . . . . . . . . . . . . 149 12.1 Deligne-Lusztig Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 12.2 Modular Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 12.2.1 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 12.2.2 Modular Deligne-Lusztig Induction . . . . . . . . . . . . . . . . . . 155 12.2.3 The Geometric Version of Broué’s Conjecture . . . . . . . . . 155

Appendix A -Adic Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 A.1 Properties of the Complex* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 A.2 Properties of the Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . 160 A.2.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 A.2.2 Cohomology with Coefficients in K . . . . . . . . . . . . . . . . . . 161 A.2.3 The Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 A.2.4 Action of a Frobenius Endomorphism . . . . . . . . . . . . . . . . 162 A.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 A.3.1 The Projective Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 A.3.2 The One-Dimensional Torus . . . . . . . . . . . . . . . . . . . . . . . . . 164 Appendix B Block Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 B.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 B.2 Brauer Correspondents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 B.2.1 Brauer’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 B.2.2 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Contents

xvii

B.2.3 Equivalences of Categories: Methods . . . . . . . . . . . . . . . . 170 B.3 Decomposition Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 B.4 Brauer Trees* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Appendix C Review of Reflection Groups . . . . . . . . . . . . . . . . . . . . . . . . . 177 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

List of Tables

1.1

Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

5.1 5.2 5.3 5.4

Values of Tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of Tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of the characters R(α ) and R (θ ) . . . . . . . . . . . . . . . . . . . . . . Character table of G = SL2 (Fq ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 53 54 58

6.1 6.2 6.3 6.4

Central characters of G = SL2 (Fq ) . . . . . . . . . . . . . . . . . . . . . . . . . . . Character table of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Character table of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Character table of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 65 66 67

9.1

Decomposition matrix for A1 when = 2 . . . . . . . . . . . . . . . . . . . . . 107

11.1 11.2 11.3 11.4

Character table of SL2 (F3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Character table of SL2 (F5 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Reflection groups having SL2 (F5 ) as derived group . . . . . . . . . . . 141 Character table of SL2 (F7 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

xix

General Notation

If E is a set, the cardinality of E (possibly infinite) will be denoted |E |. If ∼ is an equivalence relation on E , we will denote by [E /∼] a set of representatives for the equivalence classes under ∼. When this notation is used the reader will have no difficulty verifying that the statement does not depend on the choice of representative. In an expression of the form ∑x∈[E /∼] f (x), the reader will also be able to verify that the element f (x) depends only on the equivalence class of x. If Γ is a group, we will denote by Z(Γ) its centre and D(Γ) its derived subgroup. If E is a subset of Γ, we denote by NΓ (E ) (respectively CΓ (E )) its normaliser (respectively centraliser). If γ ∈ Γ, we will denote by ClΓ (γ ) its conjugacy class in Γ and we set γ E = γ E γ −1 . Of course we have |Γ| = |CΓ (γ )| · | ClΓ (γ )|. The order (possibly infinite) of γ will be denoted o(γ ). If γ ∈ Γ, we will denote by [γ , γ ] the commutator γγ γ −1 γ −1 . If A is a ring, we will denote by A+ its underlying additive group and by × A the (multiplicative) group of invertible elements in A. The centre of A will be denoted Z(A). We will denote by A−mod the category of left A-modules of finite type. Given another ring B, we denote by (A, B)−bimod the category of (A, B)-bimodules of finite type; that is, the category whose objects are simultaneously a left A-module of finite type and a right B-module of finite type satisfying (a · m) · b = a · (m · b) for all a ∈ A, b ∈ B and m ∈ M. Given a commutative ring R, an R-algebra A, a commutative R-algebra R and an Amodule M we will simplify the notation and denote by R M the extension of scalars R ⊗R M: it is an R A = R ⊗R A-module. If in addition R is a field, we will denote by K0 (A) the Grothendieck group of the category of A-modules which are finite dimensional over R. If M is a left A-module which is finite dimensional over R, we will denote by [M ] (or [M ]A if necessary) its class in K0 (A). We denote by Irr A a set of representatives for the isomorphism classes of simple (i.e. irreducible) A-modules which are finite dimensional over R; it follows that K0 (A) is the free Z-module with basis ([S ])S∈Irr A . We denote by Cb (A) the category of bounded complexes of A-modules, b K (A) the bounded homotopy category of A-modules and by Db (A) the dexxi

xxii

General Notation

rived category of A-modules (see [CaEn, §A1.2, A1.5 and A1.6]). After replacing A-modules by (A, B)-bimodules, we obtain the categories Cb (A, B), Kb (A, B) and Db (A, B). If C and C are two bounded complexes of Amodules (or of (A, B)-bimodules), we will write C C C (respectively C K C , respectively C D C ) if C and C are isomorphic in the category of complexes (respectively in the homotopy category, respectively in the derived category). We will denote by C [i] the complex shifted by i to the left [CaEn, §A1.2]. If M is a monoid and X is a subset of M , we denote by X mon the submonoid of M generated by X . If R is a commutative ring, the monoid algebra of M over R will be denoted by RM . If K is a field and M is a finite dimensional KM -module, we use [M ]M to denote the class of M in K0 (KM ) if there is no possible ambiguity as to the field K. If R is a ring and M is an R-module, we denote by GLR (M) the group of its R-linear automorphisms. If n is a non-zero natural number, we will denote by Matn (R) the R-algebra of n × n square matrices with coefficients in R and set GLn (R) = Matn (R)× . The identity matrix will be denoted In . Given M ∈ Matn (R), its transpose will be denoted t M. If R is commutative, the R-algebra Matn (R) as well as the group GLn (R) will be naturally identified with EndR (R n ) and GLR (R n ) respectively, using the canonical identification of R n with the R-module of n ×1 column vectors. We will denote by SLn (R) the subgroup of GLn (R) consisting of matrices of determinant 1. If K is a field, a matrix g ∈ GLn (K) will be said to be semisimple if it is diagonalisable over an algebraic closure of K. It will be said to be unipotent if g − In is nilpotent, that is, if (g − In )n = 0.

Part I

Preliminaries

The representation theory of SL2 (Fq ) will not commence until Chapter 3. Beforehand, in Chapters 1 and 2, we assemble the necessary preliminary facts needed for the study of representations. In Chapter 1 we are interested in the structure of the group SL2 (Fq ) (special subgroups, conjugacy classes, structure of centralisers, Sylow subgroups and their normalisers). Chapter 2 begins the study of the geometric properties of the Drinfeld curve: we essentially study certain interesting quotients as well as the action of the Frobenius endomorphism.

Chapter 1

Structure of SL2 (Fq )

Notation. Let p be an odd prime number, F an algebraic closure of the finite field Fp with p elements, q a power of p, Fq the subfield of F with q elements and G the group x z ∈ GL2 (Fq ) | xt − yz = 1}. G = SL2 (Fq ) = { y t If n is a non-zero natural number, we denote by μn the group of the n-th roots of unity in F× :

μn = {ξ ∈ F× | ξ n = 1}. We will denote by Tr2 : Fq 2 → Fq , ξ → ξ + ξ q and N2 : Fq×2 → Fq× , ξ → ξ 1+q the trace and norm respectively.

1.1. Special Subgroups Before commencing, recall that (1.1.1)

|G | = q(q − 1)(q + 1).

1.1.1. Bruhat Decomposition Let B (respectively T , respectively U) denote the subgroup of G consisting of upper triangular (respectively diagonal, respectively unipotent upper triangular ) matrices. Then B = T U. C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_1, © Springer-Verlag London Limited 2011

3

4

1 Structure of SL2 (Fq )

We have isomorphisms d:

and

Fq×

∼

−→ T ,

u:

Fq+

a → diag(a, a

∼

−→ U,

−1

a 0 )= 0 a−1

1x x→ . 01

Proposition 1.1.2. With the previous notation we have: (a) The groups T and U are abelian and the group B is solvable. (b) If a ∈ Fq× and x ∈ Fq , then d(a)u(x)d(a)−1 = u(a2 x). Now set s=

0 −1 . 1 0

Then (1.1.3)

s 2 = −I2

and s d(a) s −1 = d(a−1 ).

In particular, s normalizes T . The Bruhat decomposition of G is the partition of G into double cosets for the action of B: (1.1.4)

G = B ∪˙ BsB = B ∪˙ UsB.

Note also that B ∩ sB = T

and BsBsB = G . ab ∈ G \ B. Then c = 0 and therefore Proof (of 1.1.4 and 1.1.5). Let g = cd −1 s u(−a/c)g ∈ B. Hence g ∈ UsB. This proves 1.1.4. The first equality of 1.1.5 is immediate. To establish the second equality let X = BsBsB. Then X is stable by left and right multiplication by B and s 2 = −I2 ∈ X . It follows that X = B or X = G by 1.1.4. But su(1)s ∈ X \ B and so X = G .

(1.1.5)

In what follows we denote by N the group T , s . The exact sequence (1.1.6)

1 −→ T −→ N −→ Z/2Z −→ 1

is not split. In fact, all elements g ∈ N \ T satisfy g 2 = −I2 . We will see later that, if q > 3, then N is the normaliser of T (Corollary 1.3.2). It follows easily from the Bruhat decomposition that (1.1.7)

G=

˙ n∈N

UnU.

1.1 Special Subgroups

5

1.1.2. The Non-Split Torus The choice of a basis of the Fq -vector space Fq 2 induces an isomorphism ∼

d : GLFq (Fq 2 ) −→ GL2 (Fq ). The group Fq×2 acts on the Fq -vector space Fq 2 (which is of dimension 2) by multiplication. We may therefore view it as a subgroup of GLFq (Fq 2 ). It is then easy to verify that, if ξ ∈ Fq×2 , then (1.1.8)

Tr d (ξ ) = ξ q + ξ = Tr2 (ξ )

det d (ξ ) = ξ 1+q = N2 (ξ ).

and

(see Exercise 1.4 for an explicit construction of d and a proof). Therefore the image T of μq+1 under d is contained in G . The subgroup T (respectively T ) of G will be called the split torus (respectively the non-split torus, or anisotropic torus) of G . We have isomorphisms (1.1.9)

∼

d : μq−1 = Fq× → T

and

∼

d : μq+1 → T .

The Frobenius automorphism F : Fq 2 → Fq 2 , ξ → ξ q is Fq -linear, and is therefore an element of GLFq (Fq 2 ). Set ˜s = d (F ).We have ˜s 2 = I2

(1.1.10)

and ˜s d (ξ )˜s −1 = d (ξ q ).

On the other hand, det˜s = −1.

(1.1.11)

In fact, ˜s is of order 2 but does not commute with all the elements of GL2 (Fq ) by 1.1.10 and is therefore similar to the matrix diag(1, −1). Fix an element ξ0 of Fq×2 such that N2 (ξ0 ) = −1 (the norm N2 : Fq×2 → Fq× is surjective by Exercise 1.4(a)) and set s = d (ξ0 )˜s . Then s ∈ G and, if ξ ∈ μq+1 , (1.1.12)

s 2 = −I2

and

s d (ξ ) s −1 = d (ξ q ) = d (ξ )−1

In particular, s normalizes T . In what follows, the group T , s will be denoted N . The exact sequence (1.1.13)

1 −→ T −→ N −→ Z/2Z −→ 1

is not split. In fact, all the elements g ∈ N \ T satisfy g 2 = −I2 . We will see later that N is the normaliser of T (Corollary 1.3.2).

6

1 Structure of SL2 (Fq )

1.2. Distinguished Subgroups It is easy to verify that Z(G ) = {I2 , −I2 }.

(1.2.1)

To simplify notation, we set Z = Z(G ). We will show that, if q > 3, then G /Z is simple. Beforehand we will need the following two lemmas: Lemma 1.2.2. The group G is generated by U and sUs −1 . In particular, G = U, s . Proof. Set H = U, sUs −1 . By 1.1.4, it is enough to show that s ∈ H and T ⊆ H. Two simple calculations show that s = u(−1) · (su(1)s −1 ) · u(−1) ∈ H and, if a ∈ Fq× , then d(a) = u(a) · (su(−a−1 )s −1 ) · u(a)s ∈ H, which finishes the proof.

Lemma 1.2.3.

gBg −1 = Z .

g ∈G

Proof. Set H =

gBg −1 . It is clear that Z ⊆ H. On the other hand, note that

g ∈G

B ∩ s B = T . If we set u = u(1), then a calculation shows immediately that u T ∩ T = Z . The result then follows.

Theorem 1.2.4. We have: (a) If q > 3, then D(G ) = G and Z is the only non-trivial normal subgroup of G . In particular, G /Z is simple. (b) If q = 3, the non-trivial normal subgroups of G are Z and N . Moreover, G = N U and D(G ) = N . Proof. (a) Suppose that q > 3. We begin by showing that G = D(G ). As q > 3, there exists a ∈ Fq× such that a2 = 1. Now, if x ∈ Fq+ , it follows from Proposition 1.1.2(b) that [d(a), u(x)] = u((a2 − 1)x). This shows that U ⊆ D(G ) and therefore that D(G ) = G by Lemma 1.2.2. Let H be a non-trivial normal subgroup of G . If H is contained in B, then H is contained in the intersection of the conjugates of B, that is H ⊆ Z by Lemma 1.2.3. If H is not contained in B, then set G = HB. It is a subgroup of G which strictly contains B, and so G = G thanks to the Bruhat decomposition 1.1.4. It follows that G /H B/(B ∩ H) and, as D(G ) = G , we obtain that B/(B ∩ H) is equal to its derived group. As B is solvable, we must have

1.3 Conjugacy Classes

7

B ∩ H = B, and therefore H contains B and all of its conjugates. Therefore G = HB = H by Lemma 1.2.2. (b) is straightforward and will be shown in Chapter 11 (see Proposition 11.2.3), which treats particular cases related to small values of q.

1.3. Conjugacy Classes 1.3.1. Centralisers The following proposition describes the centralisers of certain elements of our group G . As we will see in the following section, the list of elements contained in the proposition is exhaustive up to conjugacy. Proposition 1.3.1. Let g ∈ G . Then: (a) (b) (c) (d)

If g If g If g If g

∈ {I2 , −I2 }, then CG (g ) = G . ∈ U \ {I2 }, then CG (g ) = {I2 , −I2 } × U = ZU. = d(a) with a ∈ μq−1 \ {1, −1}, then CG (g ) = T . = d (ξ ) with ξ ∈ μq+1 \ {1, −1}, then CG (g ) = T .

Proof. (a) is immediate, and (b) and (c) follow via elementary calculation. ∼ Let us now show (d). Using the isomorphism d : GLFq (Fq 2 ) −→ GL2 (Fq ), it is enough to show CGLFq (F 2 ) (ξ ) = Fq×2 . q

Firstly, it is clear that

Fq×2

is contained in the centraliser of ξ . On the other

hand, fix g ∈ GLFq (Fq 2 ) such that g ξ = ξ g . Set ξ0 = g (1) ∈ Fq×2 . We then have, for all a and b in Fq , g (a + bξ ) = g (a) + bg (ξ ) = ag (1) + b ξ g (1) = ξ0 (a + bξ ) as g and ξ commute. But, as ξ ∈ {1, −1}, we have ξ q = ξ −1 = ξ and therefore

Fq 2 = Fq ⊕ Fq ξ . It follows that g = ξ0 ∈ Fq×2 . Corollary 1.3.2. We have: (a) If q > 3, then CG (T ) = T and NG (T ) = N. If q = 3, then T = {I2 , −I2 } = Z and NG (T ) = CG (T ) = G . (b) CG (T ) = T and NG (T ) = N . Proof. (a) The case where q = 3 is immediate. Suppose therefore that q > 3. It is clear that T ⊆ CG (T ). Now choose g ∈ CG (T ). As q > 3, there exists an element a ∈ Fq× not equal to 1 or −1. As a consequence, g commutes with d(a) ∈ T and therefore g ∈ T by Proposition 1.3.1(c).

8

1 Structure of SL2 (Fq )

For the second equality, note first that N ⊆ NG (T ). On the other hand, let g ∈ NG (T ). Then there exists b ∈ Fq× such that g d(a) g −1 = d(b). This is only possible if a = b or a = b −1 . If a = b, then g commutes with d(a) and therefore belongs to T by Proposition 1.3.1(c). If a = b −1 , then sg commutes with d(a) and therefore belongs to T , again by Proposition 1.3.1(c). The result follows. (b) is shown in the same way after remarking that μq+1 must contain an element different from 1 and −1.

1.3.2. Parametrisation We denote by ≡ the relation on F× defined by x ≡ y if y ∈ {x, x −1 }. The equivalence classes of 1 and −1 contain a unique element, and all other classes contain two elements. We also fix an element z0 ∈ Fq which is not a square in Fq (which is possible as q is odd). Set 11 1 z0 and u− = . u+ = 01 0 1 Theorem 1.3.3. The group G consists of q + 4 conjugacy classes. A set of representatives is given by {I2 , −I2 } ∪ {u+ , u− , −u+ , −u− } ∪ {d(a) | a ∈ [(μq−1 \ {1, −1})/ ≡]} ∪ {d (ξ ) | ξ ∈ [(μq+1 \ {1, −1})/ ≡]}. Proof. Denote by E the set {I2 , −I2 } ∪ {u+ , u− , −u+ , −u− } ∪ {d(a) | a ∈ [(μq−1 \ {1, −1})/ ≡]} ∪ {d (ξ ) | ξ ∈ [(μq+1 \ {1, −1})/ ≡]}. Let g and g be two distinct elements of E . We will show that they are not conjugate in G . In fact, in most cases they are not conjugate in GL2 (Fq ) (which can be deduced by comparing the eigenvalues over F). The only delicate case is to show that u+ and u− (respectively −u+ and −u− ) are not conjugate in G (even though they are in GL2 (Fq )). We will only prove this for u+ and u− , the other case follows in the same −1 manner. Let us therefore suppose that there +h = exists h ∈ G such that hu 1 1 , h stabilises the line Fq . It u− . As Ker(u+ − I2 ) = Ker(u− − I2 ) = Fq 0 0 a b follows that h ∈ B. If we write h = with a ∈ Fq× and b ∈ Fq then a 0 a−1 straightforward calculation shows that z0 = a2 , which is impossible. It remains to show that every element of G is conjugate, in G , to an element of E . For this, one could prove it using linear algebra. We have chosen

1.4 Sylow Subgroups

9

to prove it by a counting argument: it is enough to show that

∑ | ClG (g )| = |G |.

g ∈E

Using Proposition 1.3.1, we obtain

∑ | ClG (g )| = ∑

g ∈E

g ∈E

|G | |CG (g )|

= 2+4× = |G |,

q−1 (q − 1)(q + 1) q − 3 + × q(q + 1) + × q(q − 1) 2 2 2

as expected.

The results of this section are summarised in Table 1.1. Table 1.1 Conjugacy classes

Representative

± I2

d(a)

d (ξ )

ε a 0ε

a ∈ μq−1 \ {±1} ξ ∈ μq+1 \ {±1} ε ∈ {±1}, a ∈ Fq× Number of classes

2

q −3 2

q −1 2

4

Order

o(±1)

o(a)

o(ξ )

p · o(ε )

Cardinality

1

q(q + 1)

q(q − 1)

q2 − 1 2

Centraliser

G

T

T

ZU

1.4. Sylow Subgroups In the course of the description of Sylow subgroups of G , we will see that some of the special subgroups introduced in Section 1.1 (B, U, T , N, T , N ...) will occur as normalisers or centralizers of Sylow subgroups. The situation is somewhat more complicated for Sylow 2-subgroups.

10

1 Structure of SL2 (Fq )

1.4.1. Sylow p-Subgroups The following proposition is immediate. Proposition 1.4.1. U is a Sylow p-subgroup of G . Moreover, CG (U) = ZU

and

NG (U) = B.

1.4.2. Other Sylow Subgroups Fix a prime number not equal to p and dividing the order of G . We denote by S (respectively S ) the -Sylow subgroup of T (respectively T ). Note that (1.4.2)

S and S are cyclic.

By 1.1.1, divides (q − 1)(q + 1). As gcd(q − 1, q + 1) = 2 (recall that q is odd), we have the following. Theorem 1.4.3. Let be a prime number different from p which divides the order of G . (a) If odd and divides q − 1, then S is a Sylow -subgroup of G . Moreover CG (S ) = T

and NG (S ) = N.

(b) If is odd and divides q + 1, then S is a Sylow -subgroup of G . Moreover CG (S ) = T

and NG (S ) = N .

(c) If q ≡ 1 mod 4 (respectively q ≡ 3 mod 4), then S2 , s (respectively S2 , s ) is a Sylow 2-subgroup of G . (d) Let S be a Sylow 2-subgroup of G . Then 1 if q ≡ ±1 mod 8, CG (S) = Z and |NG (S)/S| = 3 if q ≡ ±3 mod 8. Proof. (a) If is odd and divides q − 1, then divides neither q nor q + 1. This shows that S is a Sylow -subgroup of G . Let g ∈ S \ {I2 }. Proposition 1.3.1(c) tells us that CG (g ) = T , which allows us to conclude easily that CG (S ) = T . For the second equality, note that N ⊆ NG (S ). On the other hand, NG (S ) normalizes CG (S ) = T , and is therefore contained in N (as is odd and divides q − 1, we have q > 3 and we can therefore apply Corollary 1.3.2). (b) is shown in the same way as (a).

1.4 Sylow Subgroups

11

(c) It is enough to remark that, if q ≡ 1 mod 4 (respectively q ≡ 3 mod 4), then 4 does not divide q + 1 (respectively q − 1). This allows us to determine the 2-valuation of |G |. (d) First suppose that q ≡ 3 mod 4. Set S = S2 , s . By (c), S is a Sylow 2subgroup of G . Now there exists g ∈ S2 \ {I2 , −I2 } and therefore CG (g ) = T by Proposition 1.3.1(c). In particular, CG (S) ⊆ T . But, by 1.1.12, the only elements of T which commute with s are I2 and −I2 . The case where q ≡ 1 mod 4 is treated similarly. It remains to calculate the normaliser of S. • If q ≡ 1 mod 8, we may suppose, by (c), that S = S2 , s . Then S2 contains an element of order 8, which we denote by t. Now t and t −1 are the only elements of order 8 in S. Therefore, if g ∈ NG (S), then gtg −1 ∈ {t, t −1 }, which implies that g ∈ N. Remember that we hope to show that g ∈ S. After multiplying by s, we may suppose that g ∈ T . But then gsg −1 ∈ S and hence, as sgs −1 = g −1 , we have gsg −1 s −1 = g 2 ∈ S. Therefore g 2 ∈ S2 , which forces g ∈ S2 . It follows that NG (S) = S. • If q ≡ −1 mod 8, the result is shown in the same way. • If q ≡ ±3 mod 8, then |S| = 8 and, as −I2 is the only element of order 2 in G , we can deduce that S is the quaternionic group of order 8. It is well known that the group of outer automorphisms of S is isomorphic to S3 (via the action on the three conjugacy classes of elements of order 4), therefore NG (S)/SCG (S) = NG (S)/S is of order dividing 6. As it is, moreover, of odd order, it can only be equal to 1 or 3. In order to show that it is of order 3, we must construct an element of order 3 in the normaliser of S. We treat only the case where q ≡ 5 mod 8, whereas the case where q ≡ 3 mod 8 is relatively similar and left as an exercise (see Exercise 1.4). So assume that q ≡ 5 mod 8. We may suppose that S = S2 , s . Denote by i an element of Fq for which i 2 = −1 (which exists as q ≡ 1 mod 4). Set 1+i and α= 2 −α α g= . iα iα It is simple to verify that g belongs to G , normalizes S and satisfies g 3 = I2 . The proof of the theorem is finished.

Exercises 1.1. Let k be the subfield of Fq generated by (Tr2 (ξ ))ξ ∈μq+1 . Show that k = Fq .

(Hint: Set q = |k| and show that, if ξ ∈ μq+1 , then 1 + ξ 2 + ξ q + ξ q +1 = 0.) 1.2. Show that the map U × B → BsB, (u, b) → usb is bijective. 1.3. Show that NG (B) = B.

12

1 Structure of SL2 (Fq )

1.4. The purpose of this exercise is to choose a basis over Fq of Fq 2 which allows an easy and explicit construction of the matrices d (ξ ) for ξ ∈ Fq×2 and, in particular, to deduce a proof of 1.1.8. (a) Show that Tr2 and N2 are surjective. (b) For the rest of this exercise fix an element z ∈ Fq 2 such that z = 0 and Tr2 (z) = 0 (i.e. z q + z = 0). Show that z ∈ Fq and conclude that (1, z) is a ∼ Fq -basis of Fq 2 . We construct the isomorphism d : GLFq (Fq 2 ) −→ GL2 (Fq ) using the basis (1, z). (c) Show that then ˜s = diag(1, −1). (d) Let ξ ∈ Fq×2 . Show that ⎛

⎞ ξ q + ξ z(ξ − ξ q ) ⎜ 2 ⎟ 2 ⎜ ⎟ d (ξ ) = ⎜ ⎟. ⎝ξ −ξq ξq +ξ ⎠ 2z 2 (e) Show that Tr d (ξ ) = Tr2 (ξ ), det d (ξ ) = N2 (ξ ) and that the characteristic polynomial of d (ξ ) is (X − ξ )(X − ξ q ), where X is an indeterminate. (f) Show that d (ξ ) is conjugate, in GL2 (Fq 2 ), to diag(ξ , ξ q ). (g) Suppose that q ≡ 3 mod 8. Set S = S2 , s and denote by i an element of μq+1 such that i 2 = −1. (g1) Show that S2 = d (i) and calculate d (i). (g2) Find a matrix g ∈ G such that gs g −1 = d (i), g d (i)g −1 = s d (i) and show that g normalizes S and is of order 3 or 6. (g3) Conclude that |NG (S)/S| = 3 (which completes the proof of Theorem 1.4.3). 1.5. Denote by σ (respectively σ ) the automorphism of T (respectively T ) which sends an element to its inverse. Set M = σ T and M = σ T . Let R be a commutative ring in which 2 is invertible. Show that the group algebras RN and RM (respectively RN and RM ) are isomorphic. 1.6. Let H be a subgroup of T (respectively T ) which is not contained in Z . Show that CG (H) = T and NG (H) = N (respectively CG (H) = T and NG (H) = N ). 1.7. Let n be a non-zero natural number and g ∈ GLn (F). Show that g is semisimple (respectively unipotent) if and only if its order is prime to p (respectively a power of p). Remark: Note that g is of finite order as F = ∪r 1 Fp r . 1.8. Show that u+ and u− are conjugate in SL2 (Fq 2 ) (although they are not in G = SL2 (Fq )). Show that they are also conjugate in GL2 (Fq ). Show that two semi-simple elements of G are conjugate in G if and only if they are conjugate in GL2 (Fq ).

1.4 Sylow Subgroups

13

1.9. Show that the number of unipotent elements of G is q 2 . 1.10. Show that the group G has q conjugacy classes of semi-simple elements. 1.11. Let r be a prime number and let P and Q be two distinct Sylow r subgroup of G . Show that: (a) If r > 2, then P ∩ Q = {I2 }. (b) If r = 2, then P ∩ Q = {I2 , −I2 }. 1.12. Let P be a Sylow 2-subgroup of G . Show that P/D(P) Z/2Z × Z/2Z. 1.13. Let K be a commutative field of characteristic different from 2 and (e1 , e2 ) the canonical basis of K2 . Consider det : K2 × K2 → K, the alternating bilinear (i.e.symplectic) form given by the determinant in the canonical ba 0 1 sis. Let J = . Then J is the matrix of the symplectic form det in the −1 0 canonical basis. Finally, denote by Sp2 (K) the automorphism group of K2 which stabilise the symplectic form det. (a) Show that Sp2 (K) = SL2 (K). (b) Show that Sp2 (K) = {g ∈ GL2 (K) | t gJg = J}. (c) Deduce that the automorphism of SL2 (K) defined by g → t g −1 is inner (and is induced by conjugation by J). (d) Show that the automorphism of GL2 (K) given by g → t g −1 is not inner. 1.14*. In this exercise we calculate the group of outer automorphisms of

= GL2 (Fq ). Let φ : Fq → Fq , x → x p be the Frobenius auG = SL2 (Fq ) and G

induced tomorphism. We will also denote by φ the automorphism of G or G × by φ . Fix an element a0 ∈ Fq which is not a square and denote by σ the

. automorphism of G induced by conjugation by diag(a0 , 1) ∈ G (a) Show that σ and φ are non-inner automorphisms of G . Denote by Aut(G ) (respectively Inn(G ), respectively Out(G )) the group of automorphisms of G (respectively inner automorphisms of G , respectively outer automorphisms of G , i.e. Out(G ) = Aut(G )/ Inn(G )). If γ ∈ Aut(G ), we will denote by γ¯ its image in Out(G ). If g ∈ G , we set inn g : G → G , h → ghg −1 . (b) Show that σ¯ and φ¯ commute. ¯G. (c) Write q = p e . Show that σ¯ 2 = φ¯e = Id (d) Let γ be an automorphism of G . Show that there exists g ∈ G such that γ ◦ inn g stabilises B, U and T . (Hint: Use the fact that U is a Sylow psubgroup, that B is its normaliser and that all the Sylow p-subgroups of G are conjugate in G .)

14

1 Structure of SL2 (Fq )

(e) Deduce that Out(G ) = σ¯ × φ¯ Z/2Z × Z/eZ.

→G

, g → t g −1 . Recall that τ is not an inner automorphism Denote by τ : G

(see Exercise 1.13(d)). of G

) = ¯ τ × φ¯ Z/2Z × Z/eZ. (f) Show that Out(G

Chapter 2

The Geometry of the Drinfeld Curve

Let Y be the Drinfeld curve Y = {(x, y ) ∈ A2 (F) | xy q − yx q = 1}. It is straightforward to verify that: • G

acts linearly on A2 (F) (via g ·(x, y ) = (ax +by , cx +dy ) if g

ab = ∈ G) cd

and stabilises Y ; • μq+1 acts on A2 (F) by homotheties (via ξ · (x, y ) = (ξ x, ξ y ) if ξ ∈ μq+1 ) and stabilises Y ; • the Frobenius endomorphism F : A2 (F) → A2 (F), (x, y ) → (x q , y q ) stabilises Y. Moreover, if g ∈ G and ξ ∈ μq+1 , then, as endomorphisms of A2 (F) (or Y), we have g ◦ξ = ξ ◦g, g ◦F = F ◦g, F ◦ ξ = ξ −1 ◦ F . We can therefore form the monoid G × (μq+1 F mon ) which acts on A2 (F) and stabilises Y. The purpose of this chapter is to assemble the geometric properties of Y and the action of G × (μq+1 F mon ) which allows us to calculate its adic cohomology (as a module for the monoid G × (μq+1 F mon )). A large part of this chapter is dedicated to the construction of quotients of Y by the actions of the finite groups G , U and μq+1 .

C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_2, © Springer-Verlag London Limited 2011

15

16

2 The Geometry of the Drinfeld Curve

2.1. Elementary Properties The following proposition is (almost) immediate. Proposition 2.1.1. The curve Y is affine, smooth and irreducible. Proof. Y is affine because it is a closed subspace of the affine space A2 (F). It is irreducible because the polynomial XY q −YX q −1 in F[X , Y ] is irreducible (See Exercise 2.1). It is smooth because the differential of this polynomial is

given by the 1 × 2 matrix Y q −X q , which is zero only at (0, 0) ∈ Y. Proposition 2.1.2. The group G acts freely on Y. Proof. Let g ∈ G and (x, y ) ∈ Y be such that g · (x, y ) = (x, y ). It follows that 1 is an eigenvalue of g and, after conjugating g by an element of G , we may assume that there exists an a ∈ Fq such that 1a g= . 01 Then x + ay = x and, as y = 0 (since (x, y ) ∈ Y), we conclude that a = 0.

The next proposition is clear. Proposition 2.1.3. The group μq+1 acts freely on A2 (F) \ {(0, 0)} and therefore also on Y. Note however, that the group G × μq+1 does not act freely on Y: the pair (−I2 , −1) acts as the identity. (Even the quotient (G × μq+1 )/(−I2 , −1) does not act freely, see Exercise 2.3.)

2.2. Interesting Quotients We will now describe the quotients of Y by the finite groups G , U and μq+1 . In order to construct them we will use the following proposition, a proof of which can be found in [Bor, Proposition 6.6]. (Note that the proposition is far from optimal, but will be sufficient for our needs.) Proposition 2.2.1. Let V and W be two smooth and irreducible varieties, ϕ : V → W a morphism of varieties, and Γ a finite group acting on V. Suppose that the following three properties are satisfied: (1) ϕ is surjective; (2) ϕ (v ) = ϕ (v ) if and only if v and v are in the same Γ-orbit; (3) There exists v0 ∈ V such that the differential of ϕ at v0 is surjective. Then the morphism ϕ¯ : V/Γ −→ W induced by ϕ is an isomorphism of varieties.

2.2 Interesting Quotients

17

2.2.1. Quotient by G The map

γ:

Y −→ A1 (F) 2 2 (x, y ) −→ xy q − yx q

is a morphism of varieties. It is μq+1 F mon -equivariant (for the action of μq+1 on A1 (F) given by ξ · z = ξ 2 z and the action of F given by z → z q ). An elementary calculation shows that γ is constant on G -orbits. Even better, if we denote by γ¯ : Y/G → A1 (F) the morphism of varieties obtained by passing to the quotient, we have the following. Theorem 2.2.2. The morphism of varieties γ¯ : Y/G → A1 (F) is a μq+1 F mon equivariant isomorphism. Proof. The μq+1 F mon -equivariance is evident. In order to show that γ¯ is an isomorphism we must verify points (1), (2) and (3) of Proposition 2.2.1. Choose a ∈ F. To show (1) and (2), it is sufficient to show that |γ −1 (a)| = |G | (as G acts freely on Y by Proposition 2.1.2). After changing variables ∼ (z, t) = (x, y /x), we have a bijection γ −1 (a) −→ Ea , where Ea = {(z, t) ∈ F× × F× | t q − t =

1 z q+1

2

and t q − t =

a 2 z q +1

}.

2

As t q − t = (t q − t)q + (t q − t), we obtain Ea = {(z, t) ∈ F× × F× | t q − t =

1 1 1 a and q+1 + q 2 +q = q 2 +1 }. z q+1 z z z

Or equivalently Ea = {(z, t) ∈ F× × F× | z q

2 −1

− az q−1 + 1 = 0 and t q − t =

1 z q+1

}.

The polynomial z q −1 − az q−1 + 1 is coprime to its derivative, and therefore has q 2 − 1 distinct non-zero roots. For each of these roots, there are q non1 zero solutions t to the equation t q − t = q+1 . Therefore z 2

|γ −1 (a)| = |Ea | = (q 2 − 1)q = |G |, as expected. We now turn to (3). Let v = (x0 , y0 ) ∈ Y. The tangent space Tv (Y) to Y at v has equation y0q x − x0q y = 0 and the differential dv γ : Tv (Y) → F = Tγ (v ) (A1 (F)) is given by 2

2

dv γ (x, y ) = y0q x − x0q y .

18

2 The Geometry of the Drinfeld Curve

Therefore, if (x, y ) ∈ Ker dv γ , then 2

y0q x − x0q y = 0 and

2

y0q x − x0q y = 0. 2

2

The determinant of this system is −y0q x0q +x0q y0q = (x0 y0q −y0 x0q )q = 1, there

fore Ker dv γ = 0.

2.2.2. Quotient by U The morphism

υ:

Y −→ A1 (F) \ {0} (x, y ) −→ y

is well-defined and is a morphism of varieties. It is μq+1 F mon -equivariant (for the action of μq+1 on A1 (F) \ {0} given by ξ · z = ξ z and the action of F given by z → z q ). An elementary calculation show that υ is constant on U-orbits. Even better, if we denote by υ¯ : Y/U → A1 (F) \ {0} the morphism of varieties induced by passing to the quotient, we have the following. Theorem 2.2.3. The morphism of varieties υ¯ : Y/U → A1 (F) \ {0} is a μq+1 F mon -equivariant isomorphism. Proof. The μq+1 F mon -equivariance is evident. To show that υ¯ is an isomorphism, we verify points (1), (2) and (3) of Proposition 2.2.1. The surjectivity of υ is clear. We also have

υ (x, y ) = υ (x , y ) ⇐⇒ ∃ u ∈ U, (x , y ) = u · (x, y ). Indeed, if (x, y ) ∈ Y and (x , y ) ∈ Y are such that y = y , then x q y which shows that

−

x x q x = − , y y y

x − x x − x ∈ Fq . Now, if we set a = , then y y 1a x x · = . 01 y y

This shows (2). Point (3) is immediate.

2.3 Fixed Points under certain Frobenius Endomorphisms

19

2.2.3. Quotient by μq+1 The morphism

π:

Y −→ P1 (F) \ P1 (Fq ) (x, y ) −→ [x : y ]

is well-defined and is G × F mon -equivariant morphism of varieties (for the action of G induced by the natural action on P1 (F) and the action of F given by [x; y ] → [x q ; y q ]). An elementary calculation show that π is constant on μq+1 -orbits. Even better, if we denote by π¯ : Y/μq+1 → P1 (F) \ P1 (Fq ) the morphism of varieties induced by passage to the quotient, we have the following. Theorem 2.2.4. The morphism of varieties π¯ : Y/μq+1 → P1 (F) \ P1 (Fq ) is a G × F mon -equivariant isomorphism. Proof. The G × F mon -equivariance is evident. To show that π¯ is an isomorphism, we should verify points (1), (2) and (3) of Proposition 2.2.1, which is straightforward.

2.3. Fixed Points under certain Frobenius Endomorphisms In order to get the most out of the Lefschetz fixed-point theorem (see Theorem A.2.7(a) in Appendix A) we will need the following two results. Firstly, note that, if ξ ∈ μq+1 , we have Yξ F = ∅.

(2.3.1)

Indeed, (Y/μq+1 )F = ∅ by Theorem 2.2.4. On the other hand, we have the following. Theorem 2.3.2. Let ξ ∈ μq+1 . Then 0 ξF2 |Y | = q3 − q

−1, if ξ = if ξ = −1.

Proof. Let (x, y ) ∈ Yξ F . We then have 2

2

x = ξ xq ,

y = ξyq

2

and

xy q − yx q = 1.

As a consequence, 2

2

1 = (xy q − yx q )q = x q y q − y q x q = ξ (x q y − xy q ) = −ξ . This shows that, if ξ = −1, then Yξ F = ∅. 2

20

2 The Geometry of the Drinfeld Curve

Therefore suppose that ξ = −1. We are looking for the number of solutions to the system ⎧ q2 ⎪ (1) ⎨x = −x q q xy − yx = 1 (2) ⎪ 2 ⎩ y = −y q (3) However, if the pair (x, y ) satisfies (1) and (2), then it also satisfies (3). In1 + yx q and therefore deed, if (x, y ) satisfies (1) and (2), then x = 0, y q = x 2

yq =

1 + yx q q x

2

=

1 + y qxq 1 − xy q −yx q = = = −y . q q x x xq

It follows that it is sufficient to find the number of solutions to the system given by equations (1) and (2). Now, x being non-zero, there are q 2 − 1 possibilities for x to be a solution of (1). As soon as we have fixed x, there are q solutions to equation (2) (viewed as an equation in y ). Indeed, as an equation in y , xy q − yx q − 1 has derivative −x q = 0, and so this polynomial does not admit multiple roots. This gives therefore (q 2 − 1)q solutions to equations (1) and (2), and the theorem follows.

R EMARK – As G acts freely on Y, the set Y−F consists of a single G -orbit. 2

2.4. Compactification We will denote by [x; y ; z] homogeneous coordinates on the projective space P2 (F). We view A2 (F) as the open subset of P2 (F) defined by A2 (F) {[x; y ; z] ∈ P2 (F) | z = 0}. We identify P2 (F) \ A2 (F) with P1 (F) (using the canonical isomorphism [x; y ] → [x; y ; 0]).Theaction of G × (μq+1 F ) on A2 (F) extends uniquely ab , ξ ∈ μq+1 and [x; y ; z] ∈ P2 (F), then to P2 (F): if g = cd g · [x; y ; z] = [ax + by ; cx + dy ; z],

ξ · [x; y ; z] = [ξ x; ξ y ; z] and

F [x; y ; z] = [x q ; y q ; z q ].

Now let Y be the projective curve defined by Y = {[x; y ; z] ∈ P2 (F) | xy q − yx q = z q+1 }.

2.5 Curiosities*

21

The morphism Y −→ Y (x, y ) −→ [x; y ; 1] is an open immersion and allows us to identify Y with Y ∩ A2 (F). Proposition 2.4.1. The closed subvariety Y of P2 (F) is the closure of Y in P2 (F). It is smooth and stable under the action of of G × (μq+1 F ). Moreover, Y \ Y P1 (Fq ), with this isomorphism given by [x; y ] ∈ P1 (Fq ) → [x; y ; 0]. Proof. The only point needing a little work is the smoothness. The points of Y are smooth by Proposition 2.1.1. As G acts transitively on Y \ Y = P1 (Fq ) G /B (as a G -set), it is enough to show that [1; 0; 0] is a smooth point of Y. For this, let us consider the open subvariety defined by x = 0. In this open set (again isomorphic to A2 (F), this time via the morphism (y , z) → [1; y ; z]) Y is defined by the equation y − y q − z q+1 = 0 and the differential at (0, 0) of this polynomial is the 1 × 2 matrix (1 which is non-zero.

0),

We finish with a study of the quotient of Y by μq+1 . Consider the morphism π0 : Y −→ P1 (F) [x; y ; z] −→ [x; y ]. It is well-defined, G -equivariant, and surjective. Moreover, it is constant on μq+1 -orbits and therefore induces, after passing to the quotient, a morphism of varieties π¯0 : Y/μq+1 → P1 (F). Theorem 2.4.2. The morphism of varieties π¯0 : Y/μq+1 → P1 (F) is a G × F mon equivariant isomorphism. Proof. We omit the proof, as it follows the same arguments as those used in the proof of Theorem 2.2.4.

2.5. Curiosities* Independent of representation theory, the Drinfeld curve has interesting geometric properties which we discuss briefly here: it has a “large” automorphism group and gives a solution to a particular case of the Abhyankar’s Conjecture [Abh] about unramified coverings of the affine line in positive characteristic.

22

2 The Geometry of the Drinfeld Curve

2.5.1. Hurwitz Formula, Automorphisms* The group μq+1 acts trivially on Y \ Y = P1 (Fq ). Also, as μq+1 is of order prime to p, the morphism π0 is tamely ramified: it is only ramified at the points a ∈ P1 (Fq ) and ramification index at a is ea = q + 1. If we denote by g(Y) the genus of Y, then (2.5.1)

g(Y) =

q(q − 1) 2

as Y is a smooth plane curve of degree q + 1. Note also that π0 is a morphism of degree deg π0 = q + 1. We can therefore verify the Hurwitz formula [Har, Chapter IV, Corollary 2.4] 2g(Y) − 2 = (deg π0 )(2 · g(P1 (F)) − 2) +

∑

(ea − 1),

a∈P1 (Fq )

as g(P1 (F)) = 0. We will now extend the group G × μq+1 to a bigger group G still acting on Y (or Y). Set G = {(g , ξ ) ∈ GL2 (Fq ) × Fq×2 | det(g ) = ξ 1+q }. It is then straightforward to verify that, (2.5.2)

if (g , ξ ) ∈ G and (x, y ) ∈ Y, then g · (ξ x, ξ y ) ∈ Y.

This defines for us an action of G on Y which extends naturally to an action on Y. Set if q ≡ 3 mod 4, (−I2 , −1) √ √ D= ( −1 I2 , − −1) if q ≡ 1 mod 4. Then D is a central subgroup of G contained in the kernel of the action on Y (and on Y). Even better, we have the following. Lemma 2.5.3. The group G /D acts faithfully on Y (and Y). Proof. Let (g , ξ ) be an element of G which acts trivially on Y. Then (g , ξ ) acts trivially on Y (as Y is dense in Y) and, after passing to the quotient by {1} × μq+1 (which is a central subgroup of G ), we conclude that g acts trivially on on P1 (F) (by Theorem 2.4.2). Therefore g is a homothety: g = λ I2 , with λ ∈ Fq× . Now, if (x, y ) ∈ Y, we have (g , ξ ) · (x, y ) = (x, y ), that is λ ξ = 1. Therefore ξ = λ −1 . On the other hand, det(g ) = ξ q+1 , which implies that λ 2 = ξ q+1 or, in other words, λ q+3 = 1. As λ q−1 = 1, we collude that λ 4 = 1, which finishes the proof.

Let Δ = D ∩ (G × μq+1 ) = (−I2 , −1).

2.5 Curiosities*

23

Corollary 2.5.4. The group (G × μq+1 )/Δ acts faithfully on Y. Denote by p1 : G → GL2 (Fq ) and p2 : G → Fq×2 the canonical projections, and i1 : μq+1 → G , ξ → (I2 , ξ ) and i2 : G → G , g → (g , 1). The group G × μq+1 is contained in G and we set d : G → Fq× , (g , ξ ) → det(g ). We have a commutative diagram 1

1

Fq×2

μq+1

1

G × μq+1 i1

p2

i2

i2

d

G

N2 Fq×

p1

i1

1

det GL2 (Fq )

G

1

1

in which all straight lines of the form 1 → X → G → Y → 1 are exact sequences (which follows essentially from the surjectivity of N2 ). In particular, (2.5.5)

|G | = q(q 2 − 1)2 .

It follows from Lemma 2.5.3 that ⎧ q(q 2 − 1)2 ⎪ ⎪ ⎪ ⎨ 2 | Aut Y| ⎪ ⎪ 2 2 ⎪ ⎩ q(q − 1) 4

if q ≡ 3

mod 4,

if q ≡ 1

mod 4.

In particular, as soon as q 7, we have, by 2.5.1, | Aut Y| > 84(g(Y) − 1) = 42(q − 2)(q + 1). This illustrates the fact that the “Hurwitz bound” [Har, Chapter IV, Exercise 2.5] is not valid in positive characteristic.

24

2 The Geometry of the Drinfeld Curve

2.5.2. Abhyankar’s Conjecture (Raynaud’s Theorem)* It is not too difficult to show that if a finite group Γ is the Galois group of an unramified covering of the affine line A1 (F), then Γ is generated by its Sylow p-subgroups. The other implication was conjectured by Abhyankar and shown by Raynaud in a very difficult work [Ray]. Raynaud’s theorem (Abhyankar’s conjecture). A finite group Γ is the Galois group of an unramified Galois covering of the affine line A1 (F) if and only if it is generated by its Sylow p-subgroups. E XAMPLE – The morphism A1 (F) → A1 (F), x → x q − x is an unramified Galois covering of A1 (F) with Galois group Fq+ . By Proposition 1.4.1 and Lemma 1.2.2, the group G = SL2 (Fq ) is generated by its Sylow p-subgroups. By virtue of Raynaud’s theorem, G should be the Galois group of an unramified covering of A1 (F). In fact, in this particular case, the construction of such a covering is easy: the isomorphism Y/G A1 (F) and the fact that G acts freely on Y (see Proposition 2.1.2) tells us that (2.5.6) Y is an unramified Galois covering of A1 (F) with Galois group SL2 (Fq ).

Exercises 2.1. Show that the polynomial XY q − YX q − 1 in F[X , Y ] is irreducible (Hint: By performing the change of variables (Z , T ) = (X /Y , 1/Y ) reduce the problem to showing that T q+1 − Z q − Z in F[Z , T ] is irreducible. View this as a polynomial in T with coefficients F[Z ] and use Eisenstein’s criterion). 2.2*. Let F[X , Y ] a the polynomial ring in two variables, which we identify with the algebra of polynomial functions on A2 (F). If g ∈ G , P ∈ F[X , Y ] and v ∈ A2 (F), we set (g · P)(v ) = P(g −1 · v ). (a) Show that this does indeed give an action of G via F-algebra automorphisms. (b) Show that XY q−1 − X q and Y are algebraically independent and that F[X , Y ]U = F[XY q−1 − X q , Y ]. 2 2 (c) Show that XY q − YX q divides XY q − YX q . 2 2 XY q − YX q (d) Show that D1 = XY q − YX q and D2 = are algebraically XY q − YX q independent. (e) Show that F[X , Y ]G = F[D1 , D2 ] (Dickson invariants). (f) Use this to give another proof of Theorem 2.2.2.

2.5 Curiosities*

25

2.3. Denote by Δ the subgroup of G × μq+1 generated by (−I2 , −1). The purpose of this exercise is to show that (G × μq+1 )/Δ does not act freely on Y. To this end, choose ξ ∈ μq+1 \ {1, −1} and let v = (x, y ) ∈ A2 (F) be an eigenvector of d (ξ ) with eigenvalue ξ . (a) Show that xy q − yx q = 0 (Hint: xy q − yx q = x ∏a∈Fq (y + ax)). (b) Let κ ∈ F× be such that κ −1−q = xy q − yx q . Show that κ v ∈ Y. (c) Show that (d (ξ ), ξ −1 ) stabilises κ v ∈ Y. 2.4. † Let Z = {(x, y ) ∈ A2 (F) | x q+1 + y q+1 + 1 = 0}. We keep the notation F for the restriction to Z of the Frobenius endomorphism F of A2 (F). The purpose of this exercise is to construct an isomorphism of Y and Z which commutes with F 4 . 2

(a) Show that ZF = ∅. Deduce that there does not exist an isomorphism of ∼ varieties τ : Y −→ Z such that τ ◦ F 2 = F 2 ◦ τ . 1 Let z ∈ Fq 2 \ Fq and d ∈ F be such that d q+1 = − q . z −z (b) Show that d ∈ Fq 4 . q q d z dz . Show that g ∈ GL2 (Fq 4 ) and that g (Z) = Y. (c) Let g = dq d 2.5. Denote by τ : Y/U → Y/G the canonical projection. Set τ = γ¯ ◦ τ ◦ υ¯−1 : A1 (F) \ {0} −→ A1 (F), so that the diagram

τ

Y/U

Y/G

γ¯

υ¯ τ

A1 (F) \ {0}

A1 (F)

commutes. Show that τ (y ) = y −q (y q + y ). 2

†

The author is indebted to G. Lusztig to whom this exercise is due.

Part II

Ordinary Characters

The purpose of the next three chapters is to calculate the character table of G = SL2 (Fq ). To begin with, algebraic methods (in particular HarishChandra induction) give roughly half of the irreducible characters (see Chapter 3). The “other half” (the cuspidal characters) can be obtained in different ways (for example via ad hoc construction, as was done by Jordan [Jor] and Schur [Sch] in 1907). In this book we take the approach of constructing the cuspidal characters using the -adic cohomology of the Drinfeld curve (see Chapter 4). For this we will use results concerning -adic cohomology contained in Appendix A as well as some geometric properties obtained in Chapter 2. Once the parametrisation has been obtained, the calculation of the character table is completed in Chapter 5: it raises geometric questions (trace on -adic cohomology) as well as more arithmetic questions (Gauss sums). N OTATION – We fix a prime number different from p and K , an algebraic extension of the -adic field containing the |Γ|-th roots of unity, for all finite groups Γ encountered in this book. This ensures that the algebra K Γ is split (by Brauer’s theorem [Isa, Theorem 8.4]). If G is a finite group, we will denote by Irr Γ its set of irreducible characters over K (which we identify with Irr K Γ) and the Grothendieck group K0 (K Γ) will be identified with the character group Z Irr Γ. If M is a K Γ-module, we therefore identify its isomorphism class [M ]Γ with its character. We will denote by regΓ the character of the regular representation K Γ of Γ and by , Γ the scalar product on K0 (K Γ) for which the set Irr Γ is an orthonormal basis. We denote by 1Γ (or 1) the trivial character of Γ. The group of linear characters of Γ (with values in K × ) will be denoted by ∧ Γ (so that Γ∧ ⊆ Irr Γ). Given α ∈ Γ∧ , we denote by Kα the K Γ-module with underlying vector space K and on which Γ acts via the linear character α . If χ ∈ K0 (K Γ), we will denote by χ ∗ the virtual character defined by ∗ χ (γ ) = χ (γ −1 ) for all γ ∈ Γ: χ ∗ is the character dual to χ (and χ is auto-dual if χ = χ ∗ ). If M is a left K Γ-module of finite type, its K -dual M ∗ can either be viewed as a right K Γ-module or as a left K Γ-module (via the inverse map); in the second case, we have [M ∗ ] = [M ]∗ . If χ ∈ Irr Γ, we will denote by eχ (or eχΓ if necessary) the associated central primitive idempotent of K Γ: eχ =

χ (1) χ (γ −1 ) γ . |Γ| γ∑ ∈Γ

Chapter 3

Harish-Chandra Induction

In this chapter we study Harish-Chandra induction, which associates to a T -module the G -module obtained by first extending the T -module to a Bmodule (letting U act trivially) and then inducing to G . This construction allows us to obtain roughly half of the irreducible characters of G .

3.1. Bimodules Let Γ and Γ be two finite groups and let M be a (K Γ, K Γ )-bimodule of finite type. The dual bimodule M ∗ = Hom(M, K ) is naturally a (K Γ , K Γ)bimodule. We define two functors FM : K Γ −mod −→ K Γ−mod V −→ M ⊗K Γ V

and

∗F

M:

K Γ−mod −→ K Γ −mod V −→ M ∗ ⊗K Γ V .

Because K Γ and K Γ are semi-simple algebras, the bimodule M is projective both as a left K Γ-module and as a right K Γ -module. It follows that the functors FM and ∗FM are left and right adjoint: (3.1.1)

HomK Γ (V , FM V ) HomK Γ ( ∗FM V , V )

and (3.1.2)

HomK Γ (FM V , V ) HomK Γ (V , ∗FM V )

We denote by FM : K0 (K Γ ) → K0 (K Γ) and ∗FM : K0 (K Γ) → K0 (K Γ ) the Zlinear maps induced by FM and ∗FM respectively. They are characterised C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_3, © Springer-Verlag London Limited 2011

29

30

3 Harish-Chandra Induction

by (3.1.3)

FM [V ]Γ = [M ⊗K Γ V ]Γ

and

∗

FM [V ]Γ = [M ∗ ⊗K Γ V ]Γ .

Recall that, if V1 and V2 are two K Γ-modules of finite type, then [V1 ], [V2 ] Γ = dimK HomK Γ (V1 , V2 ).

(3.1.4)

Recall also that the Grothendieck group K0 (K Γ) is identified with the group of (virtual) characters of Γ. Under this identification, it follows from 3.1.1 and 3.1.4 that χ , FM (χ ) Γ = ∗FM (χ ), χ Γ

(3.1.5)

for all χ ∈ K0 (K Γ) and χ ∈ K0 (K Γ ). If (γ , γ ) ∈ Γ×Γ , we denote by TrM (γ , γ ) the trace of (γ , γ ) on M. We have (see, for example, [DiMi, Proposition 4.5]) (3.1.6)

FM χ (γ ) =

1 TrM (γ , γ ) χ (γ −1 ) |Γ | γ ∑ ∈Γ

and ∗

FM χ (γ ) =

(3.1.7)

1 TrM (γ , γ ) χ (γ −1 ). |Γ| γ∑ ∈Γ

3.2. Harish-Chandra Induction 3.2.1. Definition In our group G , Harish-Chandra induction is defined using the bimodule K [G /U]: by convention, K [G /U] is the K -vector space with basis G /U on which G (respectively T ) acts by left (respectively right) translations on the basis vectors. This is well-defined as T normalizes U. The dual of K [G /U] may be naturally identified with K [U\G ], with the action of the group G (respectively T ) given by right (respectively left) translations. In this way we obtain two functors KG −mod RK : KT −mod −→ V −→ K [G /U] ⊗KT V

and

∗R K

: KG −mod −→ KT −mod W −→ K [U\G ] ⊗KG W ,

3.2 Harish-Chandra Induction

31

called Harish-Chandra induction and restriction respectively. We denote by R : K0 (KT ) → K0 (KG ) and ∗R : K0 (KG ) → K0 (KT ), the induced Z-linear maps. An irreducible character g of G will be called cuspidal if there does not exist a character α of T μq−1 for which γ , R(α ) G = 0. In other words, taking into account 3.1.5, (3.2.1)

γ is cuspidal if and only if ∗R(γ ) = 0.

3.2.2. Other Constructions In order to study these functors, it will be useful to interpret them using classical induction and restriction. We will need the following notation: if V (respectively α ) is a KT -module (respectively a character of KT ), we will denote by VB (respectively αB ) its “restriction” to B via the natural projection B → T . Note that [VB ]B = ([V ]T )B . Proposition 3.2.2. Let V (respectively W ) be a KT -module (respectively a KG module). Then and ∗RK W W U . RK V IndG B VB Proof. We have IndG B VB = KG ⊗KB VB and RK (V ) = K [G /U] ⊗KT V . Set eU =

1 ∑ u. |U| u∈U

Denote by τU : KG → K [G /U] the canonical morphism and σU : K [G /U] → KG , gU → geU . We have

σU ◦ τU (g ) = geU Set

and

and τU ◦ σU (x) = x.

ϕ : KG ⊗KB VB −→ K [G /U] ⊗KT V a ⊗KB v −→ τU (a) ⊗KT v ψ : K [G /U] ⊗KT V −→ KG ⊗KB VB a ⊗KT v −→ σU (a) ⊗KB v .

It is easy to verify that ϕ and ψ are well-defined and morphisms of KG modules. It is also clear that ϕ ◦ ψ = IdK [G /U]⊗KT V . In order to show that ψ ◦ ϕ = IdKG ⊗KB VB , it is enough to note that a ⊗KB v = aeU ⊗KB v for all a ∈ KG and v ∈ VB . This shows the first isomorphism. Note that ∗RK W = K [U\G ] ⊗KG W . Set

32

3 Harish-Chandra Induction

ϕ : K [U\G ] ⊗KG W −→ W U a ⊗KG w −→ σU∗ (a)w

and

ψ : W U −→ K [U\G ] ⊗KG W w −→ U ⊗KG w .

Here, σU∗ (Ug ) = eU g for all g ∈ G . It is easy to verify that ϕ and ψ are welldefined and are mutually inverse isomorphisms of KT -modules. Corollary 3.2.3. Let α be a character of T . Then R(α ) = IndG B αB .

3.2.3. Mackey Formula It follows easily from the Bruhat decomposition G = B ∪˙ BsB (see 1.1.4 and 1.1.5), from Corollary 3.2.3 and from the Mackey formula for classical induction and restriction that, if α , β ∈ K0 (KT ), then R(α ), R(β ) G = α , β T + α , s β T . Here, s β (t) = β (s −1 ts). As s −1 ts = t −1 for all t ∈ T , we have, if α , β ∈ K0 (KT ), (3.2.4)

R(α ), R(β ) G = α , β T + α , β ∗ T .

Note that, if β is a linear character, then β ∗ = β −1 . Let us now fix α ∈ T ∧ = Irr T . By 3.2.4, we have: • If α 2 = 1, then R(α ) = R(α −1 ) ∈ Irr G . • If α = α0 (where α0 is the unique linear character of order 2 of T μq−1 , which exists because q is odd), then R(α0 ) = R+ (α0 ) + R− (α0 ), where R± (α0 ) ∈ Irr G and R+ (α0 ) = R− (α0 ). • 1G is a factor of R(1) and we denote by StG the other irreducible factor of R(1) (the Steinberg character). We have R(1) = 1G + StG ,

with deg 1G = 1 and deg StG = q.

• If α ∈ {β , β −1 }, then R(α ), R(β ) G = 0.

3.2 Harish-Chandra Induction

33

3.2.4. Restriction from GL2 (Fq ) In order to calculate the degrees of the characters R+ (α0 ) and R− (α0 ) we will = GL2 (Fq ). Denote by B show that they are conjugate under the action of G ) the subgroup of G consisting of upper triangular (respec(respectively T tively diagonal) matrices. Then =T U B

(3.2.5)

=B ∪˙ Bs B. and G

, we denote by α via the surjec˜B its “restriction” to B ˜ is a character of T If α →T . We have tion B ˜B = (ResT ResB Bα T )B . We set

α ˜. ˜ ) = IndG α R( B B

= G · B, we deduce from the Mackey formula for classical induction As G and restriction that

˜ ) = R(ResT α ResG G R(α T ˜ ).

(3.2.6)

On the other hand, it follows from 3.2.5 that β˜ ) = α α ˜ , β˜ T + α ˜ , s β˜ T . ˜ ), R( R(

(3.2.7) In particular, (3.2.8)

∧ is such that s α α ˜ , then R( ˜ ) is irreducible. ˜ = α ˜ ∈T if α

The following lemma is elementary. of α0 , then s α ˜0 . ˜0 = α ˜0 it an extension to T Lemma 3.2.9. If α The next corollary then follows immediately from 3.2.6 and 3.2.8. of α0 , then R( α ˜0 ) is an irreducible ˜0 is an extension to T Corollary 3.2.10. If α character of G . By 3.2.6 and Clifford theory [Isa, Theorem 6.2], we obtain the following. Corollary 3.2.11. The irreducible characters R+ (α0 ) and R− (α0 ) of G are conju. gate under the action of G In particular, (3.2.12)

deg R+ (α0 ) = deg R− (α0 ) =

q +1 . 2

Another proof of 3.2.12 will be given in Exercise 3.3.

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3 Harish-Chandra Induction

3.2.5. Summary We have therefore obtained: • • • •

one linear character 1G ; one character of degree q, the Steinberg character StG ; (q − 3)/2 characters of degree q + 1 (the characters R(α ), α 2 = 1); two characters R+ (α0 ) and R− (α0 ) of degree (q + 1)/2 (and conjugate un ). der G

This yields (q + 5)/2 irreducible characters. As the number of conjugacy classes (and therefore the number of irreducible characters) of G is q + 4, it remains to construct (q + 3)/2 irreducible characters: these are the cuspidal characters of G . Note also that (3.2.13)

1G + StG + R+ (α0 ) + R− (α0 ) + ∑ R(α ). R(regT ) = IndG U 1U = α 2 =1 R(1) R(α0 )

Exercises denote the centre of G . 3.1. Let Z = {diag(a, a) | a ∈ F× }. (a) Show that Z q /G · Z | = 2. (b) Show that |G 3.2. Use the fact that R(1) is the character of the permutation representation K [P1 (Fq )] in order to calculate its value on all conjugacy classes. Then verify that, if g ∈ G , |CG (g )|p if g is semi-simple, StG (g ) = 0 otherwise. Here, |Γ|p denotes the largest power of p which divides |Γ|. − 1 . Show that St is an irreducible character of Denote by StG = R(1) G G , that ResG St = StG and that, if g ∈ G , then G G

G

StG (g ) =

|CG (g )|p 0

if g is semi-simple, otherwise.

3.3*. The goal of this exercise is to give a new proof of 3.2.12. If g ∈ G , we set F (gU) =

∑ gusU ∈ K [G /U].

u∈U

3.2 Harish-Chandra Induction

35

(a) Show that this gives a well-defined K -linear endomorphism F of K [G /U]. (b) Show that, if g ∈ G , x ∈ K [G /U] and t ∈ T , then F (g · x · t) = g · F (x) · s t. If α ∈ T ∧ , we let Vα = RK (Kα ) (= K [G /U] ⊗KT Kα ). (c) Show that F induces an isomorphism of KG -modules Vα V s α . Now fix α ∈ T ∧ such that α 2 = 1. By (c), F induces an automorphism of the KG -module Vα which we will denote by Fα . (d) Show that Fα2 = α (−1)q IdVα + ∑ α (a) Fα and that Tr(Fα ) = 0. a∈Fq×

(e) Use (d) to deduce that (F1 − q IdV1 )(F1 + IdV1 ) = 0 and that, if we set I = Ker(F1 − q IdV1 ) and S = Ker(F1 + IdV1 ), then V1 = I ⊕ S is a decomposition of V1 as a sum of irreducible KG -modules such that [I ] = 1G and [S] = StG .

(f) Deduce from (d) that, if we set Vα+0 = Ker(Fα0 − α0 (−1)q IdVα0 ) and

Vα−0 = Ker(Fα0 + α0 (−1)q IdVα0 ), then Vα0 = Vα+0 ⊕ Vα−0 is a decomposiq+1 . tion of Vα0 as a sum of irreducible KG -modules and that dimK Vα±0 = 2 R EMARK – We have

α0 (−1) =

1 if q ≡ 1 mod 4, −1 if q ≡ 3 mod 4.

by the property that Consequently, the number α0 (−1) is characterised

1 ± α0 (−1)q is an algeα0 (−1)q ≡ 1 mod 4. Note that this implies that 2 braic integer. 3.4. Let V be a finite dimensional left KT -module. We view the dual V ∗ as a left KT -module. Show that we have an isomorphism of KG -modules RK V ∗ (RK V )∗ . Hence deduce that R(α ∗ ) = R(α )∗ for every character α of T . In particular, R(α0 )∗ = R(α0 ). Show that R+ (α0 ) if q ≡ 1 mod 4, R+ (α0 )∗ = R− (α0 ) if q ≡ 3 mod 4. Hint: Use Exercise 3.3 (and the remark that follows it, which implies that

α0 (−1)q is a real number if and only if q ≡ 1 mod 4).

Chapter 4

Deligne-Lusztig Induction

We will use the action of G × μq+1 on Y to construct a morphism between the Grothendieck groups K0 (K μq+1 ) and K0 (KG ). To this end, from now on we will view the monoid μq+1 F mon as acting on the right on the Drinfeld curve Y. It follows that the cohomology groups Hci (Y) inherit the structure of (KG , K [μq+1 F mon ])-bimodules. We will systematically use the results of Appendix A (which are referenced as A.x.y).

4.1. Definition and First Properties 4.1.1. Definition If θ is a character of μq+1 , we set R (θ ) = −

∑ (−1)i [Hci (Y) ⊗K μq+1 Vθ ]G ,

i 0

where Vθ is a K μq+1 -module admitting the character θ . If θ is linear, we take Vθ = Kθ . This defines a Z-linear map R : K0 (K μq+1 ) −→ K0 (KG ) which sends a character of μq+1 to a virtual character of G . The linear map R will be called Deligne-Lusztig induction. As the curve Y is affine and irreducible of dimension 1 it follows from Theorem A.2.1(b) that Hci (Y) = 0 if i ∈ {1, 2}. As a consequence, R (θ ) = [Hc1 (Y) ⊗K μq+1 Vθ ]G − [Hc2 (Y) ⊗K μq+1 Vθ ]G .

C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_4, © Springer-Verlag London Limited 2011

37

38

4 Deligne-Lusztig Induction

On the other hand, the irreducibility of Y (see Proposition 2.1.1) and Theorem A.2.1(c) tells us that [Hc2 (Y)]G ×μq+1 = 1G ×μq+1 .

(4.1.1)

The following result is then immediate. Corollary 4.1.2. If θ is a non-trivial linear character of μq+1 , then R (θ ) = [Hc1 (Y) ⊗K μq+1 Vθ ]G . In this case, R (θ ) is a character of G . Proposition 4.1.3. Let θ ∈ K0 (K μq+1 ). Then R (θ ) = R (θ ∗ ) = R (θ )∗ . Proof. Denote by ϕ : μq+1 → μq+1 the homomorphism ξ → ξ −1 . In order to show the first equality, it suffices to show that R (θ ) = R ( ϕ θ ), where ϕ θ (ξ ) = θ (ϕ (ξ )). Denote by ϕ H i (Y) the (KG , K μ q+1 )-bimodule on which c the action of μq+1 is twisted by ϕ . It is enough to show that the bimodules Hci (Y) and ϕ Hci (Y) are isomorphic. However, the endomorphism F of Y induces an K -linear automorphism of Hci (Y) (see Theorem A.2.7(c)). It is easy to see (taking account of the commutation relations between F and the elements of G and μq+1 ) that F induces an isomorphism of (KG , K μq+1 )bimodules Hci (Y) ϕ Hci (Y). The first equality is therefore proven. We now turn to the second equality. Let g ∈ G . Then, by 3.1.6, we have R (θ )(g ) =

1 Tr∗Y (g , ξ −1 ) θ (ξ ). q + 1 ξ ∈∑ μ q+1

Therefore R (θ )∗ (g ) =

1 Tr∗Y (g −1 , ξ −1 ) θ ∗ (ξ −1 ). q + 1 ξ ∈∑ μ q+1

Now, by A.2.5, we have Tr∗Y (g , ξ ) ∈ Z, therefore Tr∗Y (g , ξ ) = Tr∗Y (g −1 , ξ −1 ). It follows that R (θ )∗ (g ) =

1 Tr∗Y (g , ξ ) θ ∗ (ξ −1 ) = R (θ ∗ )(g ), q + 1 ξ ∈∑ μ q+1

as expected.

4.1.2. The Character R (1) We have, by definition, R (1) = [Hc1 (Y)μq+1 ]G − [Hc2 (Y)μq+1 ]G .

4.1 Definition and First Properties

39

Therefore, by 4.1.1 and A.2.3, R (1) = [Hc1 (Y/μq+1 )]G − 1G . Now Y/μq+1 P1 (F) \ P1 (Fq ) (see Theorem 2.2.4). Therefore, by Theorem A.2.6(a), Hc∗ (Y/μq+1 )G = Hc∗ (P1 (F))G − Hc∗ (P1 (Fq ))G . Now, Hc∗ (P1 (Fq ))G = [K [G /B]]G = 1G +StG and Hc∗ (P1 (F)) = 2·1G (see A.3.1 and A.3.2). Hence R (1) = StG −1G

(4.1.4) and therefore

(4.1.5)

⎧ ⎪ ⎨1G [Hci (Y/μq+1 )]G = StG ⎪ ⎩ 0

if i = 2, if i = 1, otherwise.

Although we have succeeded in calculating the character R (1), we still have not obtained any new characters of G .

4.1.3. Dimensions Let ξ be a non-trivial element of μq+1 . Then, by Theorem A.2.6(d), Tr∗Y (ξ ) = Tr∗Yξ (1). However Yξ = ∅, therefore

Tr∗Y (ξ ) = 0.

We deduce from this that, as a character of μq+1 , Hc∗ (Y) is a multiple of the character of the regular representation. So, for all θ ∈ (μq+1 )∧ , (4.1.6)

deg R (θ ) = deg R (1) = q − 1.

4.1.4. Cuspidality The goal of this subsection is to show that, as soon as θ is a non-trivial linear character of μq+1 , the irreducible components of R (θ ) are cuspidal. We first start by a result which shows that Harish-Chandra induction is orthogonal to Deligne-Lusztig induction.

40

4 Deligne-Lusztig Induction

Theorem 4.1.7. If α and θ are characters of T μq−1 and μq+1 respectively, then R(α ), R (θ )G = 0. Proof. We may suppose that θ is a linear character. We have R (1) = −1G + StG , R(1) = 1G + StG and therefore R(1), R (1)G = 0. On the other hand, if α is a character of T not containing 1 as an irreducible factor then R(α ), R (1)G = 0. We may therefore suppose that θ = 1. In this case, R (θ ) = [Hc1 (Y) ⊗KT Kθ ]G is a character (not only a virtual character) of G , therefore it suffices to show the result when α = regT . But, 1 U R (θ ), R(regT )G = R (θ ), IndG U 1U G = dimK (Hc (Y) ⊗K μq+1 Kθ ).

By A.2.3, we have Hc1 (Y)U = Hc1 (Y/U) = Hc1 (A1 (F) \ {0}), with the last equality following from Theorem 2.2.3. But, by A.3.4 and A.3.5, we have [Hc1 (A1 (F) \ {0})]μq+1 = 1μq+1 , and therefore dimK (Hc1 (Y)U ⊗K μq+1 Kθ ) = 0 as θ is a non-trivial linear character.

Theorem 4.1.7 shows us that, if θ is a non-trivial linear character of μq+1 , then the irreducible components of R (θ ) are cuspidal. Moreover, equality 4.1.6 shows that R (θ ) = 0. In order to obtain new irreducible characters of G all that remains is to decompose R (θ ).

4.2. Mackey Formula The goal of this section is to prove the following theorem (compare 3.2.4). Mackey formula. Let θ and η be two elements of K0 (K μq+1 ). We have (4.2.1)

R (θ ), R (η )G = θ , η μq+1 + θ , η ∗ μq+1 .

R EMARK – If η is a linear character of μq+1 , then η ∗ = η −1 . The rest of this section is dedicated to the proof of this Mackey formula. We begin with a crucial geometric result determining the class of the module Hc∗ ((Y × Y)/G ) in the Grothendieck group of μq+1 × μq+1 -modules. We (1) (2) will need the following notation. We denote by μq+1 (respectively μq+1 ) the μq+1 × μq+1 -set with underlying set μq+1 and such that, if ζ ∈ μq+1 and

4.2 Mackey Formula

41

(ξ , ξ ) ∈ μq+1 × μq+1 , then (ξ , ξ ) · ζ = ξ ξ ζ (respectively (ξ , ξ ) · ζ = ξ −1 ξ ζ ). This defines two permutation K (μq+1 × μq+1 )-modules. Then (4.2.2) Hc∗ ((Y × Y)/G )μq+1 ×μq+1 = [K [μq+1 ]]μq+1 ×μq+1 + [K [μq+1 ]]μq+1 ×μq+1 . (1)

(2)

Proof (of 4.2.2). Set Z = Y × Y = {(x, y , z, t) ∈ A4 (F) | xy q − yx q = 1 and zt q − tz q = 1}. Define

Z0 = {(x, y , z, t) ∈ Z | xt − yz = 0} Z=0 = {(x, y , z, t) ∈ Z | xt − yz = 0}.

and

Then Z0 and Z=0 are (G × μq+1 × μq+1 )-stable subvarieties of Z. By Theorem A.2.6(a), we therefore have (α ) [Hc∗ (Z/G )]μq+1 ×μq+1 = [Hc∗ (Z0 /G )]μq+1 ×μq+1 + [Hc∗ (Z=0 /G )]μq+1 ×μq+1 . It is enough to show that the two terms on the right of (α ) correspond to the two terms on the right of 4.2.2. We begin with the first term. It is very easy to see that the morphism

μq+1 × Y −→ Z0 (ξ , x, y ) −→ (x, y , ξ x, ξ y ) is an isomorphism of varieties. As Y/G A1 (F) (see Theorem 2.2.2), we conclude that Z0 /G μq+1 × A1 , with the action of (ξ , ξ ) ∈ μq+1 × μq+1 on (ζ , x) ∈ μq+1 × A1 being as follows: (ξ , ξ ) · (ζ , x) = (ξ −1 ξ ζ , x). Therefore, by Theorems A.2.1(c) and (f) and A.2.6(c), we have Hc∗ (Z0 /G ) = [K [μq+1 ]]μq+1 ×μq+1 . (2)

(β )

We now study the quotient Z=0 /G . To this end, consider the variety V = {(u, a, b) ∈ (A1 (F) \ {0}) × A2 (F) | u q+1 − ab = 1} together with the morphism

ν:

Z=0 −→ V (x, y , z, t) −→ (xt − yz, xt q − yz q , x q t − y q z).

42

4 Deligne-Lusztig Induction

A painstaking but easy calculation shows that ν does indeed take values in V and that ν (g · (x, y , z, t)) = ν (x, y , z, t) if (x, y , z, t) ∈ Z=0 and g ∈ G . To show that ν induces an isomorphism ν¯ : Z=0 /G V, all that remains it to show the following four properties (see Proposition 2.2.1): (a) ν is surjective. (b) ν (x, y , z, t) = ν (x , y , z , t ) if and only if there exists g ∈ G such that (x , y ) = g · (x, y ) and (z , t ) = g · (z, t). (c) V and Z=0 are smooth varieties. (d) If m ∈ Z=0 , then the differential dm ν is surjective. Let us first prove (a) and (b). Taking into account Proposition 2.1.2, it is enough to show that, if (u, a, b) ∈ V, then |ν −1 (u, a, b)| = |G |. Now, ν −1 (u, a, b) is the set of quadruples (x, y , z, t) satisfying ⎧ ⎪ (1) xy q − yx q = 1 ⎪ ⎪ ⎪ q q ⎪ ⎪ (2) ⎨zt − tz = 1 xt − yz = u (3) ⎪ ⎪ q q ⎪ (4) xt − yz = a ⎪ ⎪ ⎪ ⎩x q t − y q z = b. (5) The two equations (3) and (5), viewed as equations in t and z, form a system of linear equations which, by (1), has determinant −1. As a consequence, (x, y , z, t) ∈ ν −1 (u, a, b) if and only if (x, y , z, t) satisfy the system ⎧ ⎪ (1) xy q − yx q = 1 ⎪ ⎪ ⎪ q − tz q = 1 ⎪ ⎪ (2) zt ⎨ q (3 ) z = ux − bx ⎪ ⎪ ⎪ (4) xt q − yz q = a ⎪ ⎪ ⎪ ⎩t = uy q − by . (5 ) If we substitute z and t into (4), we obtain the equivalent system of equations (because u = 0): ⎧ ⎪ (1) xy q − yx q = 1 ⎪ ⎪ ⎪ q q ⎪ ⎪zt − tz = 1 (2) ⎪ ⎨ (3 ) z = ux q − bx q ⎪ a+b 2 2 ⎪ ⎪ (4 ) xy q − yx q = ⎪ ⎪ ⎪ uq ⎪ ⎩t = uy q − by . (5 ) On the other hand, it is easy to verify that (1), (3’), (4’) and (5’) imply (2). The system is therefore equivalent to

4.2 Mackey Formula

43

⎧ q xy − yx q = 1 ⎪ ⎪ ⎪ ⎪ ⎨z = ux q − bx 2 2 ⎪ xy q − yx q = ⎪ ⎪ ⎪ ⎩ t = uy q − by .

(1) (3 ) a + bq uq

(4 ) (5 )

But, by Theorem 2.2.2, the number of couples (x, y ) satisfying (1) to (4’) is equal to |G |. Because a couple (z, t) is determined by (u, a, b) and (x, y ), we indeed have |ν −1 (u, a, b)| = |G |, as expected. We now turn to (c) and (d). Let m = (x0 , y0 , z0 , t0 ) ∈ Z=0 . The tangent space T to Z=0 at m may be identified with T = {(x, y , z, t) ∈ F4 | y0q x − x0q y = 0 and t0q z − z0q t = 0}. Set (u0 , a0 , b0 ) = ν (m). The tangent space T of the variety V at ν (m) may be identified with T = {(u, a, b) ∈ F3 | u0q u − b0 a − a0 b = 0}. Using these identifications it is easy to verify that the morphism dm ν : T → T takes the following form: dm ν (x, y , z, t) = (t0 x + x0 t − y0 z − z0 y , t0q x − z0q y , x0q t − y0q z). It is enough to show that dm ν is injective. Now, if dm ν (x, y , z, t) = (0, 0, 0), then in particular ⎧ q q ⎪ ⎪y0 x − x0 y = 0 ⎪ ⎨t q z − z q t = 0 0 0 ⎪t0q x − z0q y = 0 ⎪ ⎪ ⎩ q x0 t − y0q z = 0. We indeed obtain x = y = z = t = 0 because, as m ∈ Z=0 , we have x0q t0q − y0q z0q = (x0 t0 − y0 z0 )q = 0. As (a), (b), (c) and (d) hold, the morphism ν : Z=0 → V induces an iso∼ morphism ν¯ : Z=0 /G −→ V and, under this isomorphism, the action of (ξ , ξ ) ∈ μq+1 × μq+1 is as follows: (ξ , ξ ) · (u, a, b) = (ξ ξ u, ξ ξ −1 a, ξ −1 ξ b). On the other hand, the torus F× acts on V by the formula:

λ · (u, a, b) = (u, λ a, λ −1 b), and this action commutes with that of μq+1 × μq+1 . As a consequence, by Theorem A.2.6(e), we have

44

4 Deligne-Lusztig Induction ×

Hc∗ (Z=0 /G )μq+1 ×μq+1 = Hc∗ (V)μq+1 ×μq+1 = Hc∗ (VF )μq+1 ×μq+1 . ×

Now VF = μq+1 × {0} × {0}. Therefore (γ )

Hc∗ (Z=0 /G )μq+1 ×μq+1 = [K [μq+1 ]]μq+1 ×μq+1 . (1)

Equation 4.2.2 follows immediately from (α ), (β ) and (γ ).

In order to prove the Mackey formula, it will be sufficient to deduce some algebraic consequences from the previous work, which was of a geometric nature. Proof (of the Mackey formula 4.2.1). We may and will suppose that θ and η are linear characters of μq+1 . By Proposition 4.1.3, we have R (θ ), R (η )G = 1G , R (θ ) · R (η )G . As a consequence, R (θ ), R (η )G = dimK

G Hc∗ (Y) ⊗K μq+1 Kθ ⊗K Hc∗ (Y) ⊗K μq+1 Kθ .

By Theorem A.2.6(b) and (c), we have R (θ ), R (η )G = dimK Hc∗ ((Y × Y)/G ) ⊗K (μq+1 ×μq+1 ) Kθ η , where θ η : μq+1 × μq+1 → K × , (ξ , ξ ) → θ (ξ )η (ξ ). It therefore follows from 4.2.2 that R (θ ), R (η )G = dimK K [μq+1 ] ⊗K (μq+1 ×μq+1 ) Kθ η (1)

(2)

+ dimK K [μq+1 ] ⊗K (μq+1 ×μq+1 ) Kθ η . Set μ (1) = {(ξ , ξ −1 ) | ξ ∈ μq+1 } and μ (2) = {(ξ , ξ ) | ξ ∈ μq+1 }. Then μ

(i)

K [μq+1 ] = Indμq+1 (i)

×μq+1

1μ (i) .

Therefore, by Frobenius reciprocity, μ

R (θ ), R (η )G = 1μ (1) , Resμq+1 (1) and the result follows.

×μq+1

μ

θ η μ (1) +1μ (2) , Resμq+1 (2)

×μq+1

θ η μ (2) ,

4.3 Parametrisation of Irr G

45

4.3. Parametrisation of Irr G The Mackey formula 4.2.1 together with 4.1.4 show that, if we denote by θ0 the unique character of μq+1 of order 2: • • • •

R (1) = −1G + StG . ∧ is such that θ 2 = 1. R (θ ) = R (θ −1 ) ∈ Irr G if θ ∈ μq+1 R (θ0 ) = R+ (θ0 ) + R− (θ0 ), where R± (θ0 ) ∈ Irr G and R+ (θ0 ) = R− (θ0 ). If θ 2 = 1, η 2 = 1 and θ ∈ {η , η −1 }, then R (θ ) = R (η ).

We obtain a formula analogous to 3.2.13: (4.3.1)

− R (reg μq+1 ) = −1G + StG + R+ (θ0 ) + R− (θ0 ) + ∑ R (θ ).

θ 2 =1 R (1) R (θ0 )

We have therefore obtained (q + 3)/2 cuspidal characters. In Chapter 3, we obtained (q + 5)/2 non-cuspidal irreducible characters. As | Irr G | = q + 4, we conclude that all the cuspidal characters of G have been obtained in the cohomology of Y. Therefore we have Irr G = {1G , StG } ∪˙ {R+ (α0 ), R− (α0 ), R+ (θ0 ), R− (θ0 )} (4.3.2)

∧ , α 2 = 1} ∪ ∧ , θ 2 = 1}. ˙ {R (θ ) | θ ∈ μq+1 ∪˙ {R(α ) | α ∈ μq−1

If we denote by d± the degree of R (θ0 )± , we have: |G | = 12 + q 2 +

q + 1 2 q − 1 q −3 2 (q + 1)2 + 2 (q − 1)2 + d+ + + d−2 . 2 2 2

Hence d+ + d− = q − 1

and

2 d+ + d−2 =

(q − 1)2 , 2

which implies that (4.3.3)

d+ = d− =

q−1 . 2

In the Exercises 4.3 and 4.4 we will give two other (more conceptual) proofs of this last equality: these exercises imply that R+ (θ0 ) and R− (θ0 ) are conju = GL2 (Fq ). gate under G

46

4 Deligne-Lusztig Induction

4.4. Action of the Frobenius Endomorphism We will now study the eigenvalues of the action of F on Hci (Y). By Theorem A.2.7(b), we have F = q on Hc2 (Y).

(4.4.1)

Taking this into account, in what follows we will only be interested in the eigenvalues of the action of F on Hc1 (Y). If θ is a linear character of μq+1 , the KG -modules Hci (Y) ⊗K μq+1 Kθ and i Hc (Y)eθ are canonically isomorphic. Recall that eθ =

1 θ (ξ −1 ) ξ ∈ K μq+1 . q + 1 ξ ∈∑ μ q+1

The commutation relations between F and μq+1 (and the fact that F is an automorphism of Hci (Y) by Theorem A.2.7(c)) show that (4.4.2)

F (Hci (Y)eθ ) = Hci (Y)eθ −1 .

In particular, F stabilises Hc1 (Y)e1 and Hc1 (Y)eθ0 .

4.4.1. Action on Hc1 (Y)e1 The KG -module Hc1 (Y)e1 Hc1 (Y/μq+1 ) is irreducible and therefore, by Schur’s lemma and the fact that the action of F commutes with that of G , F acts on Hc1 (Y)e1 by multiplication by a scalar ρ1 . To calculate ρ1 , it is enough to calculate the action of the Frobenius endomorphism F on Hc1 (Y/μq+1 ) = Hc1 (P1 (F) \ P1 (Fq )) (see Theorem 2.2.4). Now, the Lefschetz fixed-point theorem (see Theorem A.2.7(a)) shows that F 0 = | P1 (F) \ P1 (Fq ) | = q − ρ1 dimK Hc1 (Y/μq+1 ) = q(1 − ρ1 ). Hence (4.4.3)

ρ1 = 1.

4.4 Action of the Frobenius Endomorphism

47

4.4.2. Action on Hc1 (Y)eθ0 To simplify notation set Vθ0 = Hc1 (Y)eθ0 . Denote by Vθ±0 the irreducible subrepresentation of Vθ0 with character R± (θ0 ). By Schur’s lemma, F acts on Vθ±0 by multiplication by a scalar ρ± . We would like to calculate ρ± . Firstly, we have YF = ∅ and therefore, by the Lefschetz fixed-point formula, we obtain 0 = q − q ρ1 − But

(q − 1)(ρ+ + ρ− ) − Tr(F , ⊕ Hc1 (Y)eθ ). 2 θ 2 =1 Tr(F , ⊕ Hc1 (Y)eθ ) = 0 θ 2 =1

by 4.4.2. Hence

ρ− = −ρ+ .

(4.4.4)

To explicitly calculate ρ+ and ρ− , we will study the action of F 2 . As F 2 stabilises Hc1 (Y)eθ , it follows from Schur’s lemma that F 2 acts on Hc1 (Y)eθ by multiplication by a scalar λθ (in fact, if θ = θ0 , then this follows in fact 2 = ρ 2 ). from 4.4.4 as ρ+ − Theorem 4.4.5. Let θ ∈ (μq+1 )∧ . Then 1 λθ = −θ (−1)q

if θ = 1, if θ = 1.

Proof. The equality λ1 = 1 follows from 4.4.3. The Lefschetz fixed-point theorem shows that |Yξ F | = q 2 − q λ1 − 2

∑ (q − 1)θ (ξ )λθ

θ =1

for all ξ ∈ μq+1 . As a consequence, as λ1 = 1, we have |Yξ F | = q 2 − 1 − (q − 1) 2

It then follows from Theorem 2.3.2 that (Eξ )

∑∧

θ ∈μq+1

|Yξ F | = θ (ξ )λθ = (q + 1) − q −1 2

∑

∧ θ ∈μq+1

θ (ξ )λθ .

1 − q2 q +1

if ξ = −1, if ξ = −1.

The family of eigenvalues (λθ )θ ∈(μq+1 )∧ is therefore a solution of the system of linear equations (Eξ )ξ ∈μq+1 . The determinant of this system is invertible

48

4 Deligne-Lusztig Induction

(being the determinant of the character table of a cyclic group) and therefore it admits a unique solution. It remains only to verify that the solutions given in the corollary are valid, which is routine.

Corollary 4.4.6. We have ρ± = ± −θ0 (−1)q.

4.4.3. Action on Hc1 (Y)eθ ⊕ Hc1 (Y)eθ −1 ∧ Let θ ∈ μq+1 be such that θ 2 = 1. Then F stabilises Hc1 (Y)eθ ⊕ Hc1 (Y)eθ −1 2 and by −θ (−1)q. Therefore F has two eigenvalues, F acts as multiplication −θ (−1)q and − −θ (−1)q, each one having multiplicity q − 1 (because, as F (Hc1 (Y)eθ ) = Hc1 (Y)eθ −1 , the trace of F on the direct sum is zero).

Exercises 4.1* (Lusztig). In this exercise we show that R+ (θ0 ) si q ≡ 1 mod 4, ∗ (∗) R+ (θ0 ) = R− (θ0 ) si q ≡ 3 mod 4. We will use Poincaré duality, which is only possible using the (smooth) compactification Y of Y. Denote by , : Hc1 (Y)×Hc1 (Y) → Hc2 (Y) the perfect pairing A.2.4. (a) Show that, if θ = 1, then the KG -modules Hc1 (Y)eθ and Hc1 (Y)eθ are isomorphic (Hint: Use the open-closed exact sequence of Theorem A.2.1(d)). To simplify notation set Vθ0 = Hc1 (Y)eθ0 Hc1 (Y)eθ0 and denote by Vθ±0 the irreducible submodule of Vθ0 with character R± (θ0 ). (b) Show that , induces a perfect KG -equivariant pairing between Hc1 (Y)eθ and Hc1 (Y)eθ −1 . Deduce that R (θ )∗ = R (θ ). We will denote by , 0 the perfect pairing on Vθ0 obtained by restriction from , . 1 if q ≡ 3 mod 4, (c) Show that θ0 (−1) = −1 if q ≡ 1 mod 4. (d) Suppose that q ≡ 1 mod 4. Show that Vθ+0 is orthogonal to Vθ−0 . (Hint: Use the F -equivariance of , 0 and the knowledge of the eigenvalues of F ). Deduced that , 0 induces an isomorphism of KG -modules Vθ+0 (Vθ+0 )∗ (and therefore that R+ (θ0 )∗ = R+ (θ0 )).

4.4 Action of the Frobenius Endomorphism

49

(e) Suppose now that q ≡ 3 mod 4. Show that the restriction of , θ0 to Vθ+0 is zero (Hint: Use the F -equivariance of , 0 and the knowledge of the eigenvalues of F ). Deduce that , 0 induces an isomorphism of KG modules Vθ+0 (Vθ−0 )∗ (and therefore that R+ (θ0 )∗ = R− (θ0 )). R EMARK – Part (c) shows that −θ0 (−1) = α0 (−1) (see the remark following Exercise 3.3). 4.2. Show that G has four irreducible characters of odd degree. /G · Z(G ). Recall that A is of order 2 (see Exercise 3.1). 4.3. Denote by A = G The group A acts on the conjugacy classes and on the irreducible characters of G . In this exercise, we will give another proof of 4.3.3. (a) Show that A stabilises the q semi-simple conjugacy classes of G and permutes the other four without fixed points. (b) Show that A stabilises the q characters 1G , StG , R(α ) (α ∈ (μq−1 )∧ , α 2 = 1) and R (θ ) (θ ∈ (μq+1 )∧ , θ 2 = 1). (Hint: Use the group G .) (c) Deduce that A permutes the four characters R± (α0 ) and R± (θ0 ) without fixed points. (Hint: Apply some theorem of Brauer [Isa, Theorem 6.32].) and that d+ = (d) Deduce that R+ (θ0 ) and R− (θ0 ) are conjugate under G d− = (q − 1)/2. and deduce 4.4*. In this exercise, we define a Deligne-Lusztig induction for G that the characters R+ (θ0 ) and R− (θ0 ) are conjugate under G (which gives a new proof of 4.3.3). Set = {(x, y ) ∈ A2 (F) | (xy q − yx q )q−1 = 1}. Y × F×2 acts naturally on Y. We let F×2 act on the right. (a) Show that G q q =G ×G Y = Y ×μ F×2 . Here, A ×Γ B denotes the quotient (b) Show that Y q+1

q

of A × B by the diagonal action of Γ.

: K0 (K F×2 ) → K0 (K G ), [M ] × → − ∑i 0 (−1)i [H i (Y) ⊗ × M ] . Set R c F KF G q q2

F×2

q2

◦ Resμq = ResG ◦R . (c) Deduce from (b) that R G q+1 (d∗ ) Show that (θ ), R (η ) = θ , η × + θ , F η × . R F F G q2

q2

(Hint: Mimic the proof of the Mackey formula 4.2.1 for the group G .) (e) Let θ˜0 be an extension of θ0 to Fq×2 . Show that F θ˜0 = θ˜0 .

(θ˜0 ) is irreducible, that R (θ0 ) = ResG R (θ˜0 ), and that (f) Deduce that R G . R+ (θ0 ) and R− (θ0 ) are conjugate under G (g) Show that d+ = d− = (q − 1)/2.

50

4 Deligne-Lusztig Induction

4.5. Let m be a non-zero natural number. Show that |Y and

Fm

m

| = |YF | + q + 1

⎧ m ⎪ ⎨q + 1 Fm |Y | = q m + 1 − q(q − 1)q m/2 ⎪ ⎩ m q + 1 − (q − 1)q m/2

if m is odd, if m ≡ 0 mod 4, if m ≡ 2 mod 4.

m

Verify that |YF | is divisible by |G |. ˜ = {(u, a, b) ∈ A3 (F) | u q+1 − ab = 1} and denote by ν˜ : Z → V ˜ the 4.6. Let V morphism defined by

ν˜ (x, y , z, t) = (xt − yz, xt q − yz q , x q t − y q z). We also use the notation introduced in the proof of the Mackey formula (Z, V, ν ,. . . ). ˜ and that ν˜ is a well-defined (a) Show that V is an open dense subset of V extension of the morphism ν : Z=0 → V. ˜ and Z are smooth. (b) Show that V (c) Show that ν˜ is constant on G -orbits. (d) Nevertheless, show that ν does not induce an isomorphism between ˜ Z/G and V.

Chapter 5

The Character Table

Having parametrised the irreducible characters of the group G in the last chapter, it is natural to turn to the question of determining their values on the elements of G . For this, an important step is the calculation of the characters of the bimodules K [G /U] and Hci (Y). For K [G /U], a little elementary linear algebra is sufficient. For Hci (Y) it is necessary to invoke certain results from Appendix A. These calculations allow us to calculate the majority of the irreducible characters of G , but it does not allow us to determine the values of R± (α0 ) and R± (θ0 ). In the latter case, we study their restriction to U and utilise certain elementary arithmetic results (on Gauss sums).

5.1. Characters of Bimodules Denote by

and

Tr : G × μq−1 −→ K (g , a) −→ Tr((g , d(a)), K [G /U]) K Tr : G × μq+1 −→ (g , ξ ) −→ − Tr∗Y (g , ξ ).

The goal of this section is to calculate Tr and Tr .

5.1.1. Calculation of Tr The bimodule K [G /U] is a permutation bimodule. Therefore, if (g , t) ∈ G × T , then Tr((g , t), K [G /U]) = |{xU ∈ G /U | x −1 gx ∈ t −1 U}|. C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_5, © Springer-Verlag London Limited 2011

51

52

5 The Character Table

From this we easily deduce the values of Tr(g , b), which are given in Table 5.1. Table 5.1 Values of Tr

ε I2 ε ∈ {±1} b

d(a)

d (ξ )

ε x 0ε

a ∈ μq−1 \ {±1} ξ ∈ μq+1 \ {±1} ε ∈ {±1}, x ∈ Fq×

(q 2 − 1)δb=ε (q − 1)δb∈{a,a−1 }

0

(q − 1)δb=ε

In this and following tables, if P is a statement, then δP has value 1 if P is true and is 0 otherwise.

5.1.2. Calculation of Tr Let (g , ξ ) ∈ G × μq+1 . We write g = tu = ut, where t is of order prime to p and u is of order a power of p (Jordan decomposition). By Theorem A.2.6(d), we have Tr (g , ξ ) = − Tr∗Y(t,ξ ) (u). Hence we must calculate the fixed points of (t, ξ ) on Y: this is easy and the result is given in the following lemma. Lemma 5.1.1. We have ⎧ ⎪ ⎨Y (t,ξ ) Y = μq+1 vξ ⎪ ⎩ ∅

if ξ 2 = 1 and t = ξ I2 , if ξ 2 = 1 and t is conjugate to d (ξ ), otherwise.

Here, vξ ∈ Y satisfies t · vξ = ξ −1 vξ . Corollary 5.1.2. If ξ is not an eigenvalue of g , then Tr (g , ξ ) = 0. Recall that u+ and u− are representatives (in U) of the conjugacy classes of non-trivial unipotent elements of G . Lemma 5.1.3. Tr (u+ , 1) = Tr (u− , 1) = −(q + 1). Proof. Set λ = Tr (u+ , 1). As u+ and u− are conjugate under GL2 (Fq ), the elements (u+ , 1) and (u− , 1) of G (see §2.5.1 for the definition of G ) are conjugate under G . Therefore λ = Tr (u− , 1). Now, taking into account 5.1.2, we have

5.2 Restriction to U

53

⎧ 2 ⎪ ⎨q − 1 if g = 1, R (reg μq+1 )(g ) = λ if g is conjugate to u+ or u− , ⎪ ⎩ 0 otherwise. Moreover, by Exercise 3.2, q +1 R(1)(g ) = 1

if g = 1, if g is conjugate to u+ or u− .

But, by Theorem 4.1.7, we have R(1), R (reg μq+1 ) G = 0. In other words, (q + 1)(q 2 − 1) + (q 2 − 1)λ = 0. The result now follows easily.

Thus we have obtained all necessary information to easily determine the values of Tr (g , ξ ), which are given in Table 5.2. Table 5.2 Values of Tr

ξ

ε x 0ε

ε I2

d(a)

d (ξ )

ε ∈ {±1}

a ∈ μq−1 \ {±1}

ξ ∈ μq+1 \ {±1}

ε ∈ {±1}, x ∈ Fq×

(q 2 − 1)δξ =ε

0

−(q + 1)δξ ∈{ξ ,ξ −1 }

−(q + 1)δξ =ε

5.1.3. The Characters R(α ) and R (θ ) Fix two linear characters α and θ of μq−1 and μq+1 respectively. Then the values of R(α ) and R (θ ) are given in Table 5.3 (as follows, using 3.1.6 and Tables 5.1 and 5.2).

5.2. Restriction to U The information contained in Table 5.3 gives much of the character table of G . The only characters which remain to be determined are R± (α0 ) and R± (θ0 ). For this, we will determine their restriction to U.

54

5 The Character Table Table 5.3 Values of the characters R(α ) and R (θ )

ε I2

d (ξ )

d(a)

ε x 0ε

ε ∈ {±1} a ∈ μq−1 \ {±1} ξ ∈ μq+1 \ {±1} ε ∈ {±1}, x ∈ Fq× R(α )

(q + 1)α (ε ) α (a) + α (a−1 )

R (θ )

(q − 1)θ (ε )

0

0

α (ε )

−θ (ξ ) − θ (ξ )−1

−θ (ε )

5.2.1. B-Invariant Characters of U We fix a non-trivial linear character χ+ of Fq+ . The morphism (Fq+ )∧ Fq+ −→ z −→ (z → χ+ (zz ))

(5.2.1)

is an isomorphism of groups. (It is enough to check injectivity, which follows easily from the non-triviality of χ+ .) Denote by C the squares in Fq× and let z0 ∈ Fq× \ C . We set Υ+ :

U −→ u(z) −→

∑

K χ+ (cz)

and

Υ− :

c∈C

U −→ u(z) −→

∑

K χ+ (cz0 z).

c∈C

Taking into account Proposition 1.1.2(b), Υ+ and Υ− are B-invariant characters of U. Even better, one can easily verify the following lemma (recall that is the subgroup of G = GL2 (Fq ) formed by upper triangular matrices). B Lemma 5.2.2. With notation as above, we have: (a) (1U , Υ+ , Υ− ) is a Z-basis of K0 (KU)B . (b) (1U , Υ+ + Υ− ) is a Z-basis of K0 (KU)B . Note that (5.2.3)

Υ+ + Υ− = regU −1U .

5.2.2. Restriction of Characters of G Firstly, we have the following.

5.2 Restriction to U

55

Proposition 5.2.4. Let α and θ be linear characters of μq−1 and μq+1 respectively. Then ResG U R(α ) = regU +1U = 2 · 1U + Υ+ + Υ− and

ResG U R (θ ) = regU −1U = Υ+ + Υ− .

Proof. This follows from Table 5.3.

Corollary 5.2.5. ResG U StG = regU . Denote by ψ+ = χ+ ◦ u−1 : U → Fq× . We have

ψ+

1z = χ+ (z). 01

The previous proposition shows easily the next corollary. G Corollary 5.2.6. R(α ), IndG U ψ+ G = R (θ ), IndU ψ+ G = 1.

Up until now, the irreducible components of R± (α0 ) and R± (θ0 ) of R(α0 ) and R (θ0 ) have not yet been singled out. Corollary 5.2.6 gives us a possibility to do so. Notation. From now on, we will denote by R+ (α0 ) (respectively R+ (θ0 )) the unique common irreducible component of R(α0 ) (respectively R (θ0 )) and IndG U ψ+ .

Proposition 5.2.7. With the above choice, we obtain ResG U R± (α0 ) = 1U + Υ±

and

ResG U R± (θ0 ) = Υ± .

Proof. By construction, ResG U R+ (θ0 ) contains ψ+ and therefore “contains” Υ+ . As deg R+ (θ0 ) = deg Υ+ = (q − 1)/2, we conclude the second equality. We now turn to the first equality. Set Ψ± = ResG U R± (α0 ) and write Ψ± = , Ψ+ λ± 1U + μ± Υ+ + ν± Υ± . As R+ (α0 ) and R− (α0 ) are conjugate under G Therefore λ+ = λ− . Moreover, λ+ + λ− = 2 and Ψ− are conjugate under B. by Proposition 5.2.4, therefore λ± = 1. On the other hand, by construction, μ+ 1. As deg Ψ+ = (q + 1)/2 = deg(1U + Υ+ ), we conclude that Ψ+ = 1U + Υ+ . Similarly, Ψ− = 1U + Υ− .

56

5 The Character Table

5.2.3. Values of Υ± In order to use the information gathered in the previous subsection, we will calculate the values of the characters Υ± at the unipotent elements u± . For this we will need to use Gauss sums but, luckily, we will only need to consider the simplest case. Let us temporarily define

γ=

∑

α0 (z)χ+ (z).

z∈Fq×

Then

γ 2 = α0 (−1)q.

(5.2.8) Proof (of 5.2.8). We have

∑ × α0 (zz )χ+ (z + z ).

γ2 =

z,z ∈Fq

Let z = z −1 z . We obtain

∑ × α0 (z )χ+ (z(1 + z )).

γ2 =

z,z ∈Fq

Hence

γ2 =

∑×

α0 (z )

z ∈Fq

∑× χ+ (z(1 + z ))

z∈Fq

= (q − 1)α0 (−1) +

∑ ×

α0 (z )

z ∈Fq \{−1}

∑× χ+ (z(1 + z ))

.

z∈Fq

But, if z = −1, then

∑× χ+ (z(1 + z )) = −1 + ∑+ χ+ (z(1 + z )) = −1.

z∈Fq

z∈Fq

Finally,

γ 2 = (q − 1)α0 (−1) −

∑ ×

α0 (z ) = q α0 (−1) −

z ∈Fq \{−1}

as expected.

∑ × α0 (z ) = qα0 (−1),

z ∈Fq

This allows us to choose a square root of α0 (−1)q in K . We set α0 (−1)q = ∑ α0 (z)χ+ (z). z∈Fq×

5.3 Character Table

57

Then we have the following lemma. −1 ± α0 (−1)q 1 ± α0 (−1)q and Υ− (u± ) = − . Lemma 5.2.9. Υ+ (u± ) = 2 2 Proof. It is enough to note that Υ+ (u+ ) + Υ− (u+ ) = −1 (see 5.2.3) and that Υ+ (u+ ) − Υ− (u+ ) = γ (by definition of γ ).

5.3. Character Table The previous section allows us to calculate the values of the characters R± (α0 ) and R± (θ0 ) at non-trivial unipotent elements. To complete the character table, we need only the following results. ∧ ∧ . Then: and θ ∈ μq+1 Proposition 5.3.1. Let g ∈ G , α ∈ μq−1

(a) R(α )(−g ) = α (−1)R(α )(g ) and R (θ )(−g ) = θ (−1)R (θ )(g ). (b) R± (α0 )(−g ) = α0 (−1)R± (α0 )(g ) and R± (θ0 )(−g ) = θ0 (−1)R± (θ0 )(g ). (c) If g is semi-simple, then R± (α0 )(g ) = 12 R(α0 )(g ) and R± (θ0 )(g ) = 12 R (θ0 )(g ). Proof. (a) is clear and (b) follows immediately from (a). The last assertion follows from the fact that R+ (α0 ) and R− (α0 ) (respectively R+ (θ0 ) and R− (θ0 )) by Exercise 4.3 or 4.4.

are conjugate under G Having now collected all necessary information, the character table of G is given in Table 5.4. To simplify notation in this table we have set q0 = α0 (−1)q. In other words, q0 is the unique element of {q, −q} such that q0 ≡ 1 mod 4. In particular, as has already been pointed out in the remark following Exercise 3.3, the numbers √ ±1 ± q0 2 are algebraic integers.

Exercises 5.1. Show that, if α ∈ (μq−1 )∧ and θ ∈ (μq+1 )∧ , then StG ·R(α ) = IndG Tα Conclude that StG =

and

et

StG ·R (θ ) = IndG T θ.

1 G IndT 1T − IndG T 1T 2

StG · StG =

1 G IndT 1T + IndG T 1T . 2

58

5 The Character Table Table 5.4 Character table of G = SL2 (Fq )

ε I2

g

ε ∈ {±1}

d (ξ )

d(a)

ε uτ

a ∈ Fq× \ {±1} ξ ∈ μq+1 \ {±1} ε ∈ {±1}, τ ∈ {±}

| ClG (g )|

1

q2 + q

q2 − q

q2 − 1 2

o(g )

o(ε )

o(a)

o(ξ )

p · o(ε )

CG (g )

G

T

T

{±I2 } × U

1G

1

1

1

1

StG

q

1

−1

0

0

α (ε ) −θ (ε )

R(α ),

α 2 = 1

(q + 1)α (ε ) α (a) + α (a)−1

R (θ ),

θ 2 = 1

(q − 1)θ (ε )

0

−θ (ξ ) − θ (ξ )−1

Rσ (α0 ), σ ∈ {±}

(q + 1)α0 (ε ) 2

α0 (a)

0

Rσ (θ0 ), σ ∈ {±}

(q − 1)θ0 (ε ) 2

0

−θ0 (ξ )

5.2. Set

√ 1 + σ τ q0 2 √ −1 + σ τ q0 θ0 (ε ) 2

α0 (ε )

+ Γ+ = IndG Uψ .

Show that Γ+ is a multiplicity-free character of G and that Γ+ , Γ+ G = q + 1. R EMARK – The character Γ+ defined in this exercise is called a Gelfand-Graev character of G . 5.3. It was shown in Exercise 3.4 (respectively Exercise 4.1) that R+ (α0 ) (respectively R+ (θ0 )) is self-dual if and only if q ≡ 1 mod 4. Rediscover this result by inspecting the character table. = GL2 (Fq ). 5.4*. Calculate the character table of G

Part III

Modular Representations

In the next five chapters we study the modular representations of G . In Appendix B we recall the necessary facts about modular representations of “abstract” finite groups. For general reductive finite groups, the nature of this study is radically different, depending on whether one is in unequal characteristic (where one studies -modular representations, where is a prime number different from p) or in equal characteristic (where one studies p-modular representations). Here p denotes the characteristic of the “field of definition” of the group. In unequal characteristic, the geometric methods of Deligne-Lusztig theory remain as powerful as ever; this cohomology theory supplies representations not just over K , but over O and k as well (see the definition of O and k in the notation below). It has also been observed that the family of natural numbers d such that divides Φd (q) and Φd (q) divides the order of the group (Φd denotes the d-th cyclotomic polynomial) play a fundamental role. In the case of our group G (whose cardinality is qΦ1 (q)Φ2 (q)) it will be convenient to distinguish three cases: = 2, is odd and divides q − 1, is odd and divides q + 1. The first case is the most subtle, because the Sylow 2-subgroup is non-abelian. The second case is the simplest and all O-blocks of the group with non-trivial defect group are Morita equivalent to O-blocks of N, or even to T for some of them (see Corollary 8.3.2). In the last case, all non-principal O-blocks of G are Morita equivalent to blocks of N , or even of T for some of them (see Propositions 8.1.4 and 8.2.2), while the principal O-block is Rickard equivalent to the principal O-block of N . This can be seen by algebraic methods, valid for all abstract finite groups (see [Ric1], [Lin] and [Rou2]) but, in order to remain true to the spirit of this book and to illustrate Deligne-Lusztig theory, we construct these equivalences with the help of the cohomology RΓc (Y, O) of the Drinfeld curve (with coefficients in O). This construction was conjectured by Broué, Malle and Michel [BrMaMi, “Zyklotomische Heckealgebren”, §1A] for an arbitrary reductive group, and shown in the case of the group G = SL2 (Fq ) by Rouquier [Rou1] (see Corollary 8.3.5). To be precise, Rouquier treats the case of the group GL2 (Fq ) but his method can be easily adapted to the group SL2 (Fq ). Finally, in Chapter 10, we give a brief account of the representations in equal characteristic. The majority of this chapter is devoted to the construction of the simple modules. In accordance with the general theory of finite reductive groups, all simple modules are obtained by restriction from rational representations of the algebraic group G = SL2 (F). N OTATION – In the previous chapters, the irreducible characters of G have been parametrised, using a prime number different to p and an -adic field K . Once this parametrisation has been obtained, the irreducible characters may be viewed as KG -modules, where K is any sufficiently large field of characteristic zero. We may also take K to be a finite extension of Qp . In other words, if we do not make it explicit, we keep our field K , a sufficiently large extension of Q , but we allow the possibility that be equal to p.

We denote by O the ring of integers of K over Z and l denotes its maximal ideal. Set k = O/l: it is a finite field of characteristic . If M is an O-module, we denote by M its reduction modulo l, that is M = k ⊗O M. If m ∈ M, the image of m in M will be denoted m.

Chapter 6

More about Characters of G and of its Sylow Subgroups

The purpose of this chapter is to assemble some preliminary results (exclusively concerning irreducible characters) which will be useful in the study of modular representations. We describe central characters, congruences and the character tables of normalisers of Sylow subgroups. We make use of this information to verify the global McKay conjecture for G in all characteristics.

6.1. Central Characters In order to determine the blocks of G , we will use Proposition B.1.5 applied to Table 6.1 below, which gives the central characters of G . Table 6.1 shows that we will need some results about congruences modulo l of cyclotomic numbers. To this end, denote by μ (O) (respectively μ ∞ (O)) the group of roots of unity of O which are of order prime to (respectively of order a power of ) and denote by μ (O) the group of roots of unity of O. Note that μ (O) = μ (O) × μ ∞ (O). It is well known that (6.1.1)

the kernel of the canonical morphism μ (O) → k × is μ ∞ (O).

We deduce the following result. Lemma 6.1.2. Let a and b be two roots of unity in O such that a + a−1 ≡ b + b −1 mod l. Then ab−1 ∈ μ ∞ (O) or ab ∈ μ ∞ (O). −1

Proof. The hypothesis implies that a + a−1 = b + b . Now, a a−1 = b b −1 and therefore a ∈ {b, b }. The result then follows from 6.1.1.

C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_6, © Springer-Verlag London Limited 2011

−1

= 1,

63

64

6 More about Characters of G and of its Sylow Subgroups Table 6.1 Central characters of G = SL2 (Fq )

g

ε I2

d (ξ )

ε uτ

ξ ∈ μq+1 \ {±1}

ε ∈ {±1}, τ ∈ {±}

d(a)

ε 2 = 1 a ∈ μq−1 \ {±1} | ClG (g )|

1

q2 + q

q2 − q

q2 − 1 2

o(g )

o(ε )

o(a)

o(ξ )

p · o(ε )

CG (g )

G

T

T

{±I2 } × U

1G

1

q(q + 1)

q(q − 1)

q2 − 1 2

StG

1

q+1

−(q − 1)

0

R(α ), α 2 = 1

α (ε ) q(α (a) + α (a)−1 )

R (θ ), θ 2 = 1

θ (ε )

0

Rσ (α0 ), σ = ± α0 (ε )

2q α0 (a)

Rσ (θ0 ), σ = ± θ0 (ε )

0

q−1 α (ε ) 2 q + 1 −q(θ (ξ ) + θ (ξ )−1 ) − θ (ε ) 2 √ 1 + σ τ q0 0 (q − 1)α0 (ε ) 2 √ −1 + σ τ q0 −2q θ0 (ξ ) (q + 1)θ0 (ε ) 2 0

Corollary 6.1.3. Let α and β be two linear characters of a finite abelian group Γ such that α (γ ) + α (γ )−1 ≡ β (γ ) + β (γ )−1 mod l for all γ ∈ Γ. Then α −1 β or αβ is of order a power of .

6.2. Global McKay Conjecture Here we verify the global McKay conjecture (stated in Appendix B) in order to lay the groundwork for the study of more structural questions, like equivalences of categories. To establish equivalences between certain blocks of G and blocks of normalisers of Sylow subgroups, we will use Theorems B.2.5 and B.2.6. It is therefore necessary to understand the characters of such normalisers. Recall that N (respectively N , respectively B) is the normaliser of a Sylow -subgroup when is odd and divides q − 1 (respectively is odd and divides q + 1, respectively = p).

6.2 Global McKay Conjecture

65

6.2.1. Characters of N We will always identify the groups T and μq−1 using the isomorphism d. If α ∈ T ∧ , we set χα = IndN T α. As s α = α −1 , a linear character α of T is N-invariant if and only if α 2 = 1. Moreover, χα = χα −1 . Denote by 1N the trivial character of N, ε the linear character of order 2 which is trivial on T , χα±0 the unique extension of α0 to N such that χα±0 (s) = ± α0 (−1) (recall that s 2 = d(−1)). Then Clifford theory [Isa, Theorem 6.16] shows that Irr N = {1N , ε , χα+0 , χα−0 } ∪˙ {χα | α ∈ [T ∧ / ≡], α 2 = 1}. The character table of N is then easy to calculate and is given in Table 6.2 (we set s+ = s and s− = sd(z0 ), where z0 is a non-square in Fq× ). Table 6.2 Character table of N

ε I2

sτ

d(a)

ε ∈ {±1}

τ ∈ {±}

a ∈ μq−1 \ {±1}

| ClN (g )|

1

q−1 2

2

o(g )

o(ε )

4

o(a)

CN (g )

N

sτ

T

1N

1

1

1

ε

1

−1

1

χασ0 , σ ∈ {±}

α0 (ε )

χα , α 2 = 1

2α (ε )

g

στ

α0 (−1) 0

In particular, (6.2.1)

| Irr N| =

q +5 . 2

α0 (a) α (a) + α (a)−1

66

6 More about Characters of G and of its Sylow Subgroups

6.2.2. Characters of N The character table of N is obtained in essentially the same way as for that of N. If θ is a linear character of T , we set

χθ = IndN T θ.

As s θ = θ −1 , we conclude that χθ = χθ −1 and that θ is N -invariant if and only if θ 2 = 1. Denote by 1N the trivial character of N , ε the unique linear character of order 2 which is trivial on T and χθ±0 the unique extension of θ0 with value ± θ0 (−1) on s . Clifford theory [Isa, Theorem 6.16] shows that , χθ−0 } ∪˙ {χθ | θ ∈ [(T )∧ / ≡], θ 2 = 1}. Irr N = {1N , ε , χθ+ 0 The character table of N is then easily calculated and is given in Table 6.3 = s and s = sd (ξ ), where ξ is a non-square in μ (we set s+ 0 0 q+1 ). − Table 6.3 Character table of N

ε I2

sτ

d (ξ )

ε ∈ {±1}

τ ∈ {±}

ξ ∈ μq+1 \ {±1}

| ClN (g )|

1

q+1 2

2

o(g )

o(ε )

4

o(ξ )

CN (g )

N

sτ

T

1N

1

1

1

ε

1

−1

1

χθσ0 , σ ∈ {±}

θ0 (ε )

σ τ θ0 (−1)

θ0 (ξ )

χθ , θ 2 = 1

2θ (ε )

0

θ (ξ ) + θ (ξ )−1

g

In particular, (6.2.2)

| Irr N | =

q +7 . 2

6.2 Global McKay Conjecture

67

6.2.3. Characters of B Denote by 1+ (respectively 1− ) the trivial (respectively non-trivial) linear character of Z . Recall that χ+ is a non-trivial linear character of Fq+ and that ψ+ is the corresponding non-trivial linear character of U under the isomor∼ phism u : Fq+ → U. We denote by ψ− an irreducible component of the character Υ− defined in 5.2. If σ , τ ∈ {+, −}, we set

χσB,τ = IndB ZU 1σ ψτ . One may easily verify that B B B B ˜ | α ∈ T ∧ } ∪˙ {χ+,+ , χ+,− , χ−,+ , χ−,− } Irr B = {α

and that the character table of B is given by Table 6.4 (recall that q0 = α0 (−1)q). Table 6.4 Character table of B

ε I2

d(a)

ε uτ

ε ∈ {±1}

a ∈ Fq× \ {±1}

ε , τ ∈ {+, −}

| ClB (g )|

1

q

q−1 2

o(g )

o(σ )

o(a)

o(σ ) · p

CB (g )

B

T

ZU

α (ε )

α (a)

α (ε )

(q − 1)1σ (ε ) 2

0

g

˜, α

α ∈ T∧

χσB,τ , σ , τ ∈ {±}

In particular, (6.2.3)

| Irr B| = q + 3.

1σ (ε )

√ −1 + ττ q0 2

68

6 More about Characters of G and of its Sylow Subgroups

6.2.4. Normalisers of Sylow 2-Subgroups As the Sylow 2-subgroups of G are non-abelian, we will be content to state the following result, leaving its proof and the calculation of the character table as an exercise (see Exercise 6.1). (6.2.4)

If S is a Sylow 2-subgroup of G , then | Irr2 NG (S)| = 4.

6.2.5. Verification of the Global McKay Conjecture We can now verify the following result. Proposition 6.2.5. If S is a Sylow -subgroup of G , then | Irr G | = | Irr NG (S)|. Proof. We may assume that divides the order of G . Set E = {R(α ) | α ∈ T ∧ , α 2 = 1} and Note that |E | = We have

Irr (G ) =

q −3 2

∧ E = {R (θ ) | θ ∈ μq+1 , θ 2 = 1}.

and

⎧ ⎪ {1G , StG , R+ (α0 ), R− (α0 )} ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨{1G , StG , R+ (θ0 ), R− (θ0 )}

|E | =

{1G , StG , R+ (α0 ), R− (α0 )} ∪˙ E ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ⎪{1G , StG , R+ (θ0 ), R− (θ0 )} ∪ E ⎪ ⎩ (Irr G ) \ {StG }

It follows that

| Irr G | =

⎧ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪q +5 ⎪ ⎪ ⎪ ⎨ 2 ⎪ ⎪ q +7 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩q + 3

q −1 . 2

if = 2 and q ≡ 1

mod 4,

if = 2 and q ≡ 3

mod 4,

if is odd and divides q − 1, if is odd and divides q + 1, if = p.

if = 2, if is odd and divides q − 1, if is odd and divides q + 1, if = p.

6.2 Global McKay Conjecture

69

Moreover, when S is abelian, it follows from a theorem of Ito [Isa, Theorem 6.14] that | Irr NG (S)| = | Irr NG (S)| because does not divide |NG (S)/S|. We have therefore verified the global McKay conjecture thanks to 6.2.1, 6.2.2, 6.2.3 and 6.2.4.

Exercises 6.1. Let S be a Sylow 2-subgroup of G . Calculate the character table of NG (S) and verify 6.2.4. 6.2. Show that, if divides |G | and is different from p, then StG is in the principal -block of G . R EMARK – On the other hand, the Steinberg character StG is not in the principal p-block of G (in fact, it is alone in its p-block by B.2.4). = GL2 (Fq ). 6.3. Verify McKay’s conjecture for G 6.4*. Determine, as a function of , the radical filtration of the kG -module IndG B k.

Chapter 7

Unequal Characteristic: Generalities

Hypotheses. In this and the following two chapters, we assume that is a prime number different from p. The purpose of this chapter is to assemble results valid in all unequal characteristics. We determine, for example, the decomposition into O-blocks as well as the Brauer correspondents. We also introduce modular (that is over O, or even over Z ) and structural versions of Harish-Chandra and DeligneLusztig induction. These are functors between categories, rather than being simply maps between Grothendieck groups. The preliminary work on these two functors will be useful in the next chapter, where we study the equivalences of categories which they induce (which turn out to be either Morita or derived equivalences).

7.1. Blocks, Brauer Correspondents 7.1.1. Partition in -Blocks Using Table 6.1 (central characters) together with Corollary 6.1.3, we can determine the partition of Irr G into -blocks. We will need the following notation: we denote by T and T the maximal subgroups of T and T of order prime to , so that T = S × T

and

T = S × T .

These isomorphisms allow us to identify T ∧ and T ∧ with S∧ × T∧ and S∧ × T∧ respectively. On the other hand, we identify T and μq+1 via the isomorphism d . If α (respectively θ ) is a linear character of T (respecC. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_7, © Springer-Verlag London Limited 2011

71

72

7 Unequal Characteristic: Generalities

tively T ), denote by Bα (respectively Bθ ) the set formed by the irreducible components of all R(αλ ) (respectively R (θ λ )), where λ ∈ S∧ (respectively λ ∈ S∧ ). The following theorem describes the decomposition of Irr G into -blocks (recall that = p). Theorem 7.1.1. If = p, then: (a) If α ∈ T∧ is such that α 2 = 1, then Bα is an -block of G with defect group S . 2 (b) If θ ∈ T∧ is such that θ = 1, then Bθ is an -block of G with defect group S . (c) If is odd and divides q − 1, then: (c1) {R+ (θ0 )} and {R− (θ0 )} are -blocks of G with defect group 1 = S (note that Bθ 0 = {R+ (θ0 ), R− (θ0 )}). (c2) B1 and Bα0 are -blocks of G with defect group S (B1 being the principal -block). (d) If is odd and divides q + 1, then: (d1) {R+ (α0 )} and {R− (α0 )} are -blocks of G with defect group 1 = S (note that Bα0 = {R+ (α0 ), R− (α0 )}). (d2) B1 and Bθ 0 are -blocks of G with defect group S (B1 being the principal -block). (e) If = 2, then B1 ∪ B1 is the principal -block of G , with defect group a Sylow 2-subgroup of G (note that B1 ∩ B1 = {1G , StG }). Proof. This follows from a careful inspection of Table 6.1, Corollary 6.1.3 and the fact that q is invertible in O.

7.1.2. Brauer Correspondents Let α (respectively θ ) be a linear character of T (respectively T ) such that α 2 = 1 (respectively θ 2 = 1) and denote by Aα (respectively Aθ ) the O-block of G such that Irr KAα = Bα (respectively Irr KAθ = Bθ ). The principal Oblock of G will be denoted A1 or A1 . If moreover is odd, we denote by Aα0 (respectively Aθ0 ) the sum of the O-blocks of G satisfying Irr KAα0 = Bα0 (respectively Irr KAθ0 = Bθ 0 ). R EMARK – If is odd and divides q − 1, then Aα0 is an O-block with defect group S while Aθ0 is the sum of two O-blocks with defect group 1 = S . If is odd and divides q + 1, then Aα0 is the sum of two O-blocks with defect group 1 = S while Aθ0 is an O-block with defect group S . If A is the sum of multiple O-blocks of G with the same defect group S, we still use Brauer correspondent to refer to the sum of the associated Brauer correspondents (i.e. a sum of O-blocks of NG (S)).

7.1 Blocks, Brauer Correspondents

73

If α (respectively θ ) is a linear character of T (respectively T ) such that = 1 (respectively θ 2 = 1), we denote by bα (respectively bθ ) the primitive central idempotent of OT (respectively OT ) equal to

α2

bα =

1 ∑ α (t) t −1 |T | t∈T

(respectively

bθ =

1 θ (t ) t −1 |T | t ∑ ∈T

).

It is now straightforward to calculate the Brauer correspondents (the reader is referred to Exercise 1.6 for the calculation of the normalisers of subgroups of T and T ). These are the object of the following theorem, where the correspondents are given only for those O-blocks with non-central defect group. Theorem 7.1.2. Let α and θ be linear characters of T and T respectively. Let S be a Sylow 2-subgroup of G . (a) If is odd and divides q − 1, then: (a1) If α 2 = 1, then ONbα is the Brauer correspondent of Aα . (a2) If α 2 = 1, then ON(bα + bα −1 ) is the Brauer correspondent of Aα . (b) If is odd and divides q + 1, then: (b1) If θ 2 = 1, then ON bθ is the Brauer correspondent of Aθ . (b2) If θ 2 = 1, then ON (bθ + bθ −1 ) is the Brauer correspondent of Aθ . (c) If = 2, then the principal O-block of NG (S) is the Brauer correspondent of A1 = A1 . Moreover: (c1) If q ≡ 1 mod 4 and α = 1, then ON(bα + bα −1 ) is the Brauer correspondent of Aα . (c2) If q ≡ 3 mod 4 and θ = 1, then ON (bθ + bθ −1 ) is the Brauer correspondent of Aθ . R EMARK – In statements (c1) and (c2) above, the condition α = 1 (respectively θ = 1) is equivalent to α 2 = 1 (respectively θ 2 = 1): indeed, in (c), we have = 2, therefore α and θ are of odd order. Proof. We will prove (a), the other cases are treated in a similar manner and are left to the reader. Therefore suppose that is odd and divides q − 1. Denote by iα the primitive central idempotent in OG such that Bα = OGiα . We will show that bα if α 2 = 1, BrS (i α ) = bα + b α −1 if α 2 = 1.

74

7 Unequal Characteristic: Generalities

As CG (S ) = T , it is enough to calculate the coefficients of iα on elements of the group T . If t ∈ T , the coefficient of t in iα is equal to ⎧ 1 ⎪ 1 × 1 + q × 1 + (q + 1) × ∑ λ (t) + λ (t −1 ) ⎪ ⎪ |G | ⎪ ⎪ λ ∈[S∧ /≡] ⎪ ⎪ ⎪ λ =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 q +1 2× × α0 (t) + (q + 1) × ∑ α0 (t)(λ (t) + λ (t −1 )) 2 ⎪ |G | λ ∈[S∧ /≡] ⎪ ⎪ ⎪ λ =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ q +1 ⎪ −1 ⎪ ⎪ ⎩ |G | ∑∧ λ (t) + λ (t) λ ∈S

if α = 1,

if α = α0 ,

if α 2 = 1.

This coefficient is therefore equal to ⎧ (q + 1)|S | ⎪ ⎪ α (t)δt∈T ⎪ ⎪ |G | ⎨ ⎪ ⎪ (q + 1)|S | ⎪ ⎪ ⎩ (α (t) + α (t −1 ))δt∈T |G |

if α 2 = 1, if α 2 = 1,

which shows the desired result, because (q + 1)|S |/|G | = 1/q|T | ≡ 1/|T | mod l (since q ≡ 1 mod l).

7.1.3. Terminology If α (respectively θ ) is a linear character of T (respectively T ) such that α 2 = 1 (respectively θ 2 = 1), we will say that the block Aα (respectively Aθ ) is nilpotent. (The reader may verify that they are indeed nilpotent in the sense of Broué and Puig [BrPu, definition 1.1].) If is odd, the (sum of) blocks Aα0 and Aθ0 will be called quasi-isolated. (One may verify that this agrees with the definition of a quasi-isolated semisimple element given in [Bon, §1.B] and the Jordan decomposition of blocks of reductive groups in unequal characteristic [BrMi, Theorem 2.2].) R EMARK – The notion of nilpotent block is algebraic (as it is defined for any block of any finite group) whereas the notion of quasi-isolated block is of more geometric nature (as it is defined only for finite reductive groups and requires Deligne-Lusztig theory).

7.2 Modular Harish-Chandra Induction

75

7.2. Modular Harish-Chandra Induction The functors of Harish-Chandra induction and restriction are defined by R : Z T −mod −→ Z G −mod M −→ Z [G /U] ⊗Z T M

and

∗R :

Z G −mod −→ Z T −mod M −→ Z [U\G ] ⊗Z G M.

If Λ is a commutative Z -algebra, we denote by RΛ and ∗RΛ the extension of scalars to Λ of the functors R and ∗R respectively. For example, if Λ = K , we obtain the functors RK and ∗RK defined in Section 3.2. Proposition 7.2.1. The (Z G , Z T )-bimodule Z [G /U] is projective both as a left Z G -module and as a right Z T -module. Proof. Let eU = (1/q) ∑u∈U u. Then eU is idempotent and belongs to Z U ⊆ Z G as q is invertible in Z . Now, Z [G /U] Z GeU as a (Z G , Z T )-bimodule. The proposition follows, as Z G is free both as a Z G and Z T -module. Corollary 7.2.2. The functors R and ∗R are adjoint. We now turn to a study of the endomorphism algebra of the Z G -module Z [G /U]. To this end we revisit a part of Exercise 3.3, this time working over the ring Z rather than the field K . Firstly, note that the right action of T on G /U induces a morphism of Z -algebras Z T −→ EndZ G Z [G /U] t ∈ T −→ (x → x · t −1 ). We will view the algebra EndZ G Z [G /U] as a Z T -module via this morphism. On the other hand, denote by F the unique Z -linear endomorphism of Z [G /U] such that F (gU) = ∑ gusU u∈U

0 −1 for all g ∈ G . Recall that s = ∈ N ⊆ NG (T ). It is easy to verify that, if 1 0 g ∈ G , t ∈ T and x ∈ Z [G /U], then (7.2.3)

F (g · x · t) = g · F (x) · s t.

In particular, F ∈ EndZ G Z [G /U]. Thanks to the endomorphisms induced by T and F we may describe the endomorphism algebra EndZ G Z [G /U] as a kind of “quadratic extension” of the group algebra Z T .

76

7 Unequal Characteristic: Generalities

Theorem 7.2.4. Set E = ∑t∈T t ∈ Z T . Then: (a) We have the following equality between elements of EndZ G Z [G /U]: F 2 = qd(−1) + F · E . Moreover, F ◦ t = s t ◦ F for all t ∈ Z T . (b) If τ , τ ∈ Z T satisfies τ + τ F = 0, then τ = τ = 0. (c) EndZ G Z [G /U] = Z T ⊕ Z T · F . (d) If Λ is a commutative Z -algebra, then EndΛG Λ[G /U] = Λ⊗Z EndZ G Z [G /U]. Proof. (a) Let g ∈ G . Then F 2 (gU) =

∑

gusvsU.

u,v ∈U

Now, if a, b ∈ Fq , then ⎧ ⎪

⎨g d(−1)U 1a 1b g s sU = 1 −a 01 01 ⎪ sd(b−1 )U ⎩g 0 1

if b = 0, if b = 0.

It follows that, F 2 (gU) = qg d(−1)U +

∑ gus ∑ t

u∈U

U.

t∈T

Hence the result. We now turn to (b), (c) and (d). By using the isomorphism of (ΛG , ΛT )bimodules Λ[G /U] ΛGeU . The map EndΛG ΛGeU −→ (eU ΛGeU )opp f −→ f (eU ) is an isomorphism of Λ-algebras. Now the set (eU neU )n∈N gives a Λ-basis of eU ΛGeU by 1.1.7. Via this isomorphism, F corresponds to qeU seU . Now, if t ∈ T and n ∈ N, we have (eU neU )(eU teU ) = eU nteU , which shows (b), (c) and (d). In Chapter 8, we will use Theorem 7.2.4 to show that, if is odd and divides q − 1, then EndZ G Z [G /U] Z N (see Theorem 8.3.1): this arithmetic condition is related to Broué’s conjecture.

7.3 Deligne-Lusztig Induction*

77

7.3. Deligne-Lusztig Induction* As G and μq+1 act freely on the Drinfeld curve Y (see Propositions 2.1.2 and 2.1.3), it follows from Theorem A.1.5 that (7.3.1)

the complex RΓc (Y, Z ) is left and right perfect,

that is, quasi-isomorphic to a bounded complex of (Z G , Z μq+1 )-bimodules projective as left and right modules. In order to maintain consistent notation we will identify μq+1 and T via the isomorphism d and view Y as a variety equipped with an action of G × (T F mon ), with F acting on T as conjugation by s ∈ N . In this way RΓc (Y, Z ) will be viewed as a complex of (Z G , Z T )-bimodules, inducing a functor between the bounded derived categories Db (Z G ) R : Db (Z T ) −→ C −→ RΓc (Y, Z ) ⊗Z T C . If Λ is a commutative Z -algebra we denote by RΛ the extension of scalars to Λ of the functor R . On the other hand, as Y is a smooth curve, it follows from A.1.4 that (7.3.2)

Hci (Y, Z ) is a free Z -module.

Recall also that, by Theorem A.2.1, we have (7.3.3)

Hc2 (Y, Z ) = Z

(equipped with the trivial action of G )

and, by A.2.3, 7.3.2 and Theorem 2.2.2, (7.3.4)

Hc1 (Y, Z )G = 0.

From these properties we will deduce the following result. Proposition 7.3.5 (Rouquier). There exist two projective Z G -modules P and Q together with a morphism d : P → Q satisfying the following properties. (a) In the derived category Db (Z G ), we have an isomorphism d

RΓc (Y, Z ) D (0 −→ P −→ Q −→ 0), the term P being in degree 1. (b) The Z G -module Q is the projective cover of Z (regarded as a trivial Z G module). (c) The Z G -module P does not admit a subquotient isomorphic to Z (or, equivalently, (KP)G = 0). Proof. By 7.3.1 and 7.3.2, there exists projective Z G -modules P and Q together with a morphism of Z G -modules d : P → Q such that RΓc (Y, Z ) is

78

7 Unequal Characteristic: Generalities

quasi-isomorphic to the complex d C = (0 −→ P −→ Q −→ 0). Here, the terms P and Q are situated in degrees 1 and 2. By 7.3.3, we have Q/ Im d Z . As Q is projective, we can write Q = P1 ⊕ Q , where P1 is the projective cover of Z and Q is contained in Im d. The projectivity of Q implies that the canonical surjective morphism −1 d (Q ) → Q splits: we denote by Q the Z G -submodule of P which is the image of this section. It is isomorphic to Q via the restriction of d. On the other hand, the Z -module P/Q is torsion-free: if x ∈ P satisfies x ∈ Q , then d(x) ∈ Q and therefore d(x) ∈ Q as Q/Q is torsion-free. This shows that x ∈ d −1 (Q ) and in conclusion we remark that d −1 (Q )/Q d −1 (Q ) ∩ Ker d is torsion-free. As P/Q is torsion-free, the exact sequence of Z G -modules 0 → Q → P → P/Q → 0 splits over Z , and therefore splits as a sequence of Z G modules as Q is projective, therefore relatively (G , 1)-projective and therefore relatively (G , 1)-injective [CuRe, Theorem 19.2]. Denote by P a complement of Q in P and denote by d : P → P1 the composition of the morphism P → Q → P1 , the last arrow being the projection Q = P1 ⊕ Q → P1 . The complex d

0 −→ P −→ Q −→ 0 is therefore homotopic to a complex d

0 −→ P −→ P1 −→ 0. This shows that we may (and will) assume that Q = P1 . All that remains is to verify the third assertion. We have an exact sequence of Z G -modules d

0 −→ Hc1 (Y, Z ) −→ P −→ Q −→ Hc2 (Y, Z ) −→ 0, which, after tensoring with the flat Z -algebra K , yields a short exact sequence d

K KQ −→ Hc2 (Y) −→ 0, 0 −→ Hc1 (Y) −→ KP −→

where dK : KP → KQ is the canonical extension of d. Because the algebra KG is semi-simple, the functor “invariants under G ” is exact, which gives us a final exact sequence 0 −→ Hc1 (Y)G = 0 −→ (KP)G −→ (KQ)G −→ Hc2 (Y)G = K −→ 0.

7.3 Deligne-Lusztig Induction*

79

Now, (KQ)G = K as Q is the projective cover of Z , which shows property (c). Encouraged by Proposition 7.3.5, we will study the endomorphism algebra of RΓc (Y, Z ), viewed as an object in the derived category Db (Z G ). We will commence with the following important result, which will also prove useful in the construction of equivalences of derived categories in the following chapter. Corollary 7.3.6 (Rouquier). If i = 0, then HomDb (Z G ) (RΓc (Y, Z ), RΓc (Y, Z )[i]) = 0. Moreover, the Z -module EndDb (Z G ) RΓc (Y, Z ) is torsion-free. Proof. We keep the notation (P, Q, d) used in the statement of Proposition 7.3.5 and denote by C the complex d C = (0 −→ P −→ Q −→ 0). As P and Q are projective Z G -modules, we have HomDb (Z G ) (RΓc (Y, Z ), RΓc (Y, Z )[i]) = HomKb (Z G ) (C , C [i]). We begin with the first assertion. In virtue of the previous equality, it suffices to show that HomKb (Z G ) (C , C [i]) = 0 for all i = 0. As the result is evident for |i| 2, we assume that |i| = 1. • Suppose that i = 1. Then HomCb (Z G ) (C , C [1]) HomZ G (P, Q). Therefore, let ϕ ∈ HomZ G (P, Q) and view ϕ as a morphism of complexes C

0

d

P

Q

0

ϕ

C [1]

0

P

d

Q

0.

Suppose first that ϕ is not contained in the image of d. Then ϕ induces a surjection KP → Hc2 (Y) K . It follows that (KP)G = 0, which is impossible by Proposition 7.3.5(c). Therefore Im ϕ ⊆ Im d. The projectivity of P then allows one to construct a morphism ϕ : P → P such that ϕ = d ◦ ϕ . This shows that the morphism of complexes ϕ is homotopic to zero. • Now suppose that i = −1. To give a morphism of complexes ϕ : C → C [−1] is the same as giving a morphism of Z G -modules ϕ : Q → P such that Im ϕ ⊆ Ker d and Im d ⊆ Ker ϕ :

80

7 Unequal Characteristic: Generalities

C

0

P

d

Q

0

ϕ

C [−1]

0

d

P

Q

0.

Therefore ϕ factorises to give a morphism Q/ Im d → P and hence to a morphism Z → P. This last morphism is necessarily zero by virtue of Proposition 7.3.5(c). Therefore ϕ = 0. We now turn to the last assertion, concerning the absence of torsion in the endomorphism algebra RΓc (Y, Z ). So let ϕ ∈ EndCb (Z G ) (C ) be such that ϕ is homotopic to zero. We have to show that ϕ itself is homotopic to zero. Let α : P → P and β : Q → Q be the morphisms of Z G -modules such that ϕ is equal to the following morphism C

0

d

P α

C

0

P

Q

0

β d

Q

0.

By the hypotheses, there exists a morphism of Z G -modules ρ : Q → P such that α = ρ ◦ d and β = d ◦ ρ . As P and Q are torsion-free it is enough to show that Im ρ ⊆ P. Denote by ρ¯ : Q → P/P the composition of ρ : Q → P with the canonical projection P → P/P. Then ρ¯ is zero on Im d, and therefore ρ¯ factorises to give a morphism ρ˜ : Hc2 (Y, Z ) → P/P. If ρ˜ is non-zero, then P/P contains an F G -submodule isomorphic to F . But P/P = F P is a projective F G -module. Therefore if it contains the trivial module in its socle, then it contains the projective cover of the trivial module as a direct summand. This implies that P admits the Z G -module Q as a direct summand, and therefore P G = 0, contradicting Proposition 7.3.5(c). Therefore ρ˜ = 0 and ρ¯ = 0, which concludes the proof of the corollary. Denote by Hc• (Y, Z ) the graded Z G -module ⊕i 0 Hci (Y, Z ) (in fact, we have Hc• (Y, Z ) = Hc1 (Y, Z ) ⊕ Hc2 (Y, Z )). The previous proposition has the following consequence. Corollary 7.3.7 (Rouquier). The natural morphism of O-algebras • EndDb (Z G ) RΓc (Y, Z ) −→ Endgr Z G Hc (Y, Z )

is injective (here, Endgr ? (−) denotes the algebra of graded endomorphisms). Proof. By Corollary 7.3.6, the Z -algebra EndDb (Z G ) RΓc (Y, Z ) injects naturally into EndDb (Q G ) RΓc (Y, Q ). But as Q G is semi-simple, this last algebra • is isomorphic to Endgr Q G Hc (Y, Q ). The commutativity of the diagram

7.3 Deligne-Lusztig Induction*

81 • Endgr Z G Hc (Y, Z )

EndDb (Z G ) RΓc (Y, Z )

EndDb (Q G ) RΓc (Y, Q )

∼

• Endgr Q G Hc (Y, Q )

implies the corollary. Having established these preliminary results, we now turn to the question of determining the endomorphism algebra EndDb (Z G ) RΓc (Y, Z ), using similar techniques to those used in the determination of EndZ G Z [G /U]. The right action of T on Y commutes with the left action of G , and we therefore have a canonical morphism of Z -algebras Z T → EndDb (Z G ) RΓc (Y, Z ). Using this morphism we view EndDb (Z G ) RΓc (Y, Z ) as a Z T -module. On the other hand, the Frobenius endomorphism F of the Drinfeld curve induces an endomorphism, denoted F , of the complex RΓc (Y, Z ). As the action of G commutes with that of F , this yields an element of EndDb (Z G ) RΓc (Y, Z ). It follows from the relations between the actions of F , G and T on Y that F ◦ (g , t ) = (g , t −1 ) ◦ F

(7.3.8)

for all (g , t ) ∈ G × T . The following theorem describes the structure of the algebra EndDb (Z G ) RΓc (Y, Z ) in an analogous fashion to the description of EndZ G Z [G /U] in Theorem 7.2.4. Theorem 7.3.9 (Rouquier). Set E = ∑t ∈T t ∈ OT . Then: (a) We have the following equality between elements of EndDb (Z G ) RΓc (Y, Z ): F 2 = −qd (−1) + F ◦ E .

Moreover, F ◦ t = s t ◦ F for all t ∈ Z T . (b) If τ , τ ∈ Z T satisfy τ + τ F = 0, then τ = τ = 0. (c) We have EndDb (Z G ) RΓc (Y, Z ) = Z T ⊕ Z T · F . (d) If Λ is a commutative Z -algebra, then EndDb (ΛG ) RΓc (Y, Λ) Λ ⊗Z EndDb (Z G ) RΓc (Y, Z ). Proof. (a) By virtue of Corollary 7.3.7, it is enough to verify this equality on the cohomology groups Hc1 (Y, Z ) and Hc2 (Y, Z ). But, by 7.3.2, it is enough to verify this equality on the cohomology groups Hc1 (Y, K ) = Hc1 (Y) and Hc2 (Y, K ) = Hc2 (Y). This is done using the calculation of the eigenvalues of F and F 2 completed in Section 4.4. We have:

82

7 Unequal Characteristic: Generalities

• On Hc2 (Y), μq+1 acts trivially (therefore E acts as multiplication by q + 1) and F acts as multiplication by q (see 4.4.1). • On Hc1 (Y)e1 , μq+1 acts trivially (therefore E acts as multiplication by q + 1) and F acts as multiplication by 1 (see 4.4.3). • On Hc1 (Y)eθ0 , an element ξ ∈ μq+1 acts as multiplication by θ0 (ξ ) (therefore E acts as multiplication by 0) and F 2 acts as multiplication by −q θ0 (−1) (see 4.4.5). • If θ is a linear character of μq+1 such that θ 2 = 1, then, on Hc1 (Y)(eθ + eθ −1 ), an element ξ ∈ μq+1 acts as multiplication by θ (ξ ) (therefore E acts as multiplication by 0) and F 2 acts as multiplication by −q θ (−1) (see 4.4.5). The result follows immediately from these observations. (b) If τ + τ F = 0, then this equality is still true on Hc1 (Y) and Hc2 (Y). If θ ∈ T ∧ , we denote by θˆ its linear extension to the group algebra KT . On Hc2 (Y), τ + τ F acts as multiplication by ˆ1(τ )+q ˆ1(τ ). On Hc1 (Y)e1 , τ + τ F acts as multiplication by ˆ1(τ )− ˆ1(τ ). As a consequence, ˆ1(τ ) = ˆ1(τ ) = 0. by θˆ0 (τ ) and θˆ0 (τ ) On the space Hc1 (Y)eθ0 , τ and τ act as multiplication

θ0 (−1)q and − −θ0 (−1)q. respectively, while F has two eigenvalues, − ˆ ˆ ˆ ˆ Hence, θ0 (τ ) + θ0 (τ ) −θ0 (−1)q = θ0 (τ ) − θ0 (τ ) −θ0 (−1)q = 0, which implies that θˆ0 (τ ) = θˆ0 (τ ) = 0. We now study the action of τ + τ F on Hc1 (Y)(eθ + eθ −1 ) if θ 2 = 1. Let (v1 , ... , vr ) be a basis of Hc1 (Y)eθ . Then (F (v1 ), ... , F (vr )) is a basis of Hc1 (Y)eθ −1 . In the basis (v1 , ... , vr , F (v1 ), ... , F (vr )) of Hc1 (Y)(eθ + eθ −1 ), the matrix of F has the form

0 −θ (−1)qIr F → Ir 0 while an element τ of KT has matrix

τ → diag(θˆ(τ ), ... , θˆ(τ ), θˆ∗ (τ ), ... , θˆ∗ (τ )). r times

r times

The equality τ + τ F = 0 therefore implies θˆ(τ ) = θˆ(τ ) = 0. ˆ τ ) = Θ( ˆ τ ) = 0 for all linear characters Θ of Hence we have shown that Θ( T . Hence, τ = τ = 0. ˜ = End b ˜ (c) Set A D (Z G ) RΓc (Y, Z ) and A = Z T ⊕ Z T · F ⊆ A. We want to ˜ Note first that, by Proposition 7.3.6, the O-modules A and show that A = A. ˜ ˜ As K is flat A are torsion-free. On the other hand, we claim that KA = K A. ˜ over Z , K A = EndDb (KG ) RΓc (Y, K ), so it follows from (b) that it is enough to show that dimK EndDb (KG ) RΓc (Y, K ) = 2 · |T |.

But

7.3 Deligne-Lusztig Induction*

83

dimK EndDb (KG ) RΓc (Y, K ) = [Hc1 (Y)]KG , [Hc1 (Y)]KG G +[Hc2 (Y)]KG , [Hc2 (Y)]KG G . As [Hc1 (Y)]KG , [Hc2 (Y)]KG G = 0, it is enough to show that R (reg μq+1 ), R (reg μq+1 )KG = 2 · |T |, which follows immediately from the Mackey formula 4.2.1. We have there˜ fore shown the claim, namely that KA = K A. ˜ On the other hand, if we set, for ϕ ∈ K A, T (ϕ ) =

1 Tr(ϕ , Hc2 (Y)) − Tr(ϕ , Hc1 (Y)) |G |

∈ K.

The key to the proof of (c) is the following lemma. ˜ then T (ϕ ) ∈ Z . Moreover, the restriction of T to a Lemma 7.3.10. If ϕ ∈ A, morphism T : A → Z is a symmetrising form. Proof (of Lemma 7.3.10). As the complex RΓc (Y, Z ) is perfect as an object of Db (Z G ) (see 7.3.1), it suffices to show that, if P is a projective Z G -module and if ϕ ∈ EndZ G (P), then Tr(ϕ , KP)/|G | ∈ O. To show this we may assume that P is indecomposable. Then there exists an idempotent e ∈ OG such that P Z Ge and we may assume that P = Z Ge. If we denote by κ : Z G → Z the symmetrising form such that κ (g ) = δg =1 for all g ∈ G , then it is easy to verify that Tr(ϕ , KP) = |G |κ (ϕ (e)). This shows the first claim. We turn to the second claim. Denote by B the O-basis of A obtained by the concatenation of (t )t ∈T and (t · F )t ∈T and set M = (T (bb ))b,b ∈B . It is enough to show that det M ∈ Z× . Now, for all t1 , t2 ∈ T , we have ⎧ ⎨0 if t1 t2 = 1, T (t1 t2 ) = 1 ⎩ if t1 t2 = 1, q

T (t1 t2 F ) = |Yt1 t2 F | = 0 and T

−1 2 (t1 F t2 F ) = |Yt1 t2 F | =

0 if t1 t2−1 = −I2 , 1 if t1 t2−1 = −I2 .

Indeed, the first equality follows from Table 5.2, while the second and third follow from the Lefschetz fixed-point theorem combined with, respectively, 2.3.1 and Theorem 2.3.2. Hence, det(M) = ±(1/q)|T | ∈ O × , as required. We now show why (c) follows from Lemma 7.3.10. Let (a1 , ... , ad ) be a Z -basis of A and let (a1∗ , ... , ad∗ ) be the dual Z -basis of A for the form T . Let

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7 Unequal Characteristic: Generalities

˜ Then, by (∗), there exists λ1 ,. . . , λd ∈ K such that a = λ1 a1 + · · · + λd ad . a ∈ A. ˜ as A ˜ is an algebra and so By duality we obtain λi = T (aai∗ ). Now aai∗ ∈ A ∗ ˜ λi = T (aai ) ∈ O by Lemma 7.3.10, therefore a ∈ A. Hence A = A. (d) follows immediately from the fact that RΓc (Y, Z ) is perfect.

In Chapter 8, we will use the previous theorem to show that, if is odd and divides q + 1, then EndZ G RΓc (Y, Z ) Z N (see Theorem 8.3.4).

Exercises 7.1. Complete the proof of 7.1.2. 7.2. We identify the Z G -modules Z [G /B] with the space of T -invariants (for the right action) in Z [G /U]. Show that F restricts to an endomorphism of Z [G /B]. We will denote by FB this restriction. Denote by I the identity endomorphism of Z [G /B]. Show that FB2 = qI + (q − 1)FB and that EndZ G Z [G /B] = Z · I ⊕ Z · FB . 7.3 (Howlett-Lehrer, Dipper-Du). Denote by U − the subgroup of G formed by lower triangular matrices. Let R be a ring in which p is invertible. Identify the (RG , RT )-bimodules R[G /U] and R[G /U − ]) with RGeU and RGeU − respectively. Show that the map RGeU − −→ RGeU a −→ aeU is an isomorphism of (RG , RT )-bimodules. (Hint: As both bimodules are free over R of the same rank, it is sufficient to show surjectivity, which follows from the formula qeU eU − eU = eU +

∑

u∈U − \{I2 }

eU ueU

and the fact that U − \ {I2 } ⊆ TUsU.) 7.4. Let R be a commutative ring in which p is invertible. If α : T → R × is a linear character, we denote by Rα the RT -module defined as follows: the underlying R-module is R itself and an element t ∈ T acts as multiplication by α (t). Use the previous exercise (or the automorphism F , extended to the ring R) to show that R[G /U] ⊗RT Rα R[G /U] ⊗RT Rα −1 . 7.5. Show that, if V is a Z T -module, then ∗R(R(V )) V ⊕ s V , where s V denotes the Z T -module with underlying Z -module V but on which an element t ∈ T acts as s t = t −1 .

Chapter 8

Unequal Characteristic: Equivalences of Categories

Hypothesis. In this chapter, as in the next and the previous chapters, we assume that is a prime number different from p. The purpose of this chapter is to verify Broué’s abelian defect conjecture (see Subsection B.2.2). In the case of non-principal blocks (which all have abelian defect groups), the equivalences of categories predicted by Broué’s conjecture are always Morita equivalences (see Sections 8.1 and 8.2). While it is possible to obtain this result using Brauer trees and Brauer’s theorem B.4.2, we give instead a concrete construction of these equivalences using HarishChandra and Deligne-Lusztig induction. In the case of principal blocks, treated in Section 8.3, the situation is more interesting. If is odd and divides q − 1, then the principal block is Morita equivalent to its Brauer correspondent and Harish-Chandra induction induces an equivalence. If is odd and divides q + 1, then the principal block is Rickard equivalent to its Brauer correspondent and Deligne-Lusztig induction induces the required equivalence. If = 2, the situation is more complicated: when q ≡ ±3 mod 8, then the principal block of G is Rickard equivalent to its Brauer correspondent; when q ≡ ±1 mod 8, the derived category of the principal block is equivalent to the derived category of an A∞ -algebra. These two final results are due to Gonard [Go]. The last section is dedicated to Alvis-Curtis duality, viewed as an endofunctor of the homotopy category Kb (Z G ) or of the derived category Db (Z G ). For an arbitrary finite reductive group it was shown by Cabanes and Rickard [CaRi] that this duality is an equivalence of the derived category. More recently, Okuyama [Oku3] improved this result by showing that it was in fact an equivalence of the homotopy category (see also the work of Cabanes [Ca] for a simplified treatment of Okuyama’s theorem). We give a very concrete proof of Okuyama’s result in the case of our small group G = SL2 (Fq ). C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_8, © Springer-Verlag London Limited 2011

85

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8 Unequal Characteristic: Equivalences of Categories

8.1. Nilpotent Blocks 8.1.1. Harish-Chandra Induction Let α be a linear character of T such that α 2 = 1. We will show the following. Proposition 8.1.1. Harish-Chandra induction RO induces a Morita equivalence between Aα and OTbα . Proof. Denote by fα the primitive central idempotent of OG such that Aα = OGfα . Set M = O[G /U]bα . Then M is an (OG , OTbα )-bimodule. Moreover, the irreducible factors of the KG -module KM are the irreducible characters of KAα . Moreover, fα M = M and therefore M is in fact an (Aα , OTbα )bimodule, which is projective as a left and right module by Proposition 7.2.1. We would like to show that the functor M ⊗O Tbα − is an equivalence of categories. By virtue of Broué’s theorem B.2.5, it is enough to show that the functor KM ⊗KTbα − induces a bijection between Irr KTbα and Irr KAα . As Irr(KTbα ) = {αλ | λ ∈ S∧ }, this amounts to showing that Irr KAα = {R(αλ ) | λ ∈ S∧ } which is precisely the definition of Aα . Corollary 8.1.2. The O-algebras Aα and ON(bα + bα −1 ) are Morita equivalent. Proof. By Proposition 8.1.1, it is enough to show that the O-algebras A = ON(bα + bα −1 ) and A = OTbα are Morita equivalent. Set P = ONbα . Then P is an (A, A )-bimodule which is projective as a left and right module. We would like to show that the functor P ⊗A − is an equivalence of categories. By virtue of Broué’s theorem B.2.5, it is enough to show that the functor KP ⊗KA − induces a bijection between Irr KA and Irr KA. However this follows from the (easily verified) fact that Irr KA = {χαλ | λ ∈ S∧ } and [KP ⊗KA Kαλ ]KN = χαλ for all λ ∈ S∧ . C OMMENTARY – When S is not contained in Z (which occurs when is odd and divides q − 1 or when = 2 and q ≡ 1 mod 4), then NG (S ) = N by Exercise 1.6 and ON(bα + bα −1 ) is the Brauer correspondent of Aα (by Theorem 7.1.2). As a consequence, Corollary 8.1.2 shows that Broué’s conjecture (see Appendix B) is verified in a stronger form. Indeed, the predicted equivalence of derived categories is induced by a Morita equivalence. Another consequence of Theorem 8.1.1 is that (8.1.3)

Aα Matq+1 (OTbα ) Matq+1 (OS ),

which agrees with structure theorems for nilpotent blocks (for the case of an abelian defect group, see for example [BrPu, §1]).

8.1 Nilpotent Blocks

87

8.1.2. Deligne-Lusztig Induction* Let θ be a linear character of T such that θ 2 = 1. We will show the following result. induces a Morita equivalence Proposition 8.1.4. Deligne-Lusztig induction RO between Aθ and OTbθ .

Proof. Denote by fθ the primitive central idempotent of OG such that A θ = OGfθ . Set M = Hc1 (Y, O)bθ . Then M is an (OG , OT bθ )-bimodule and the irreducible factors of the KG -module KM are the irreducible characters of KA θ . Furthermore, fθ M = M and therefore M is in fact an (A θ , OT bθ )bimodule. We would like to show that the functor M ⊗O T b − is an equivθ alence of categories. We first show that M is projective as a left and right module. By 7.3.1, the complex fθ RΓc (Y, O)bθ is perfect as a left and right complex. Its i-th cohomology group is fθ Hci (Y, O)bθ , which is non-zero only when i = 1. Therefore fθ RΓc (Y, O)bθ is quasi-isomorphic to a complex of bimodules consisting of one non-zero term fθ Hc1 (Y, O)bθ = M occurring in degree 1. Therefore M is projective as a left and right module. By virtue of Broué’s theorem B.2.5, it is enough to show that the functor KM ⊗KT b − gives a bijection between Irr KT bθ and Irr KA θ . As Irr(KT bθ ) = θ {θ λ | λ ∈ S ∧ }, this amounts to saying that Irr KA θ = {R (θ λ ) | λ ∈ S ∧ }, which is precisely the definition of A θ . Corollary 8.1.5. The O-algebras A θ and ON (bθ + bθ −1 ) are Morita equivalent. Proof. The proof follows the same lines as that of Corollary 8.1.2, the goal being to show that the (ON (bθ + bθ −1 ), OT bθ )-bimodule ON bθ induces a Morita equivalence. C OMMENTARY – When S is not contained in Z (which occurs when is odd and divides q + 1 or when = 2 and q ≡ 3 mod 4), then NG (S ) = N by Exercise 1.6 and ON (bθ + bθ −1 ) is the Brauer correspondent of A θ (by Theorem 7.1.2). We conclude that Corollary 8.1.5 shows Broué’s conjecture (see Appendix B) in a stronger form. Indeed, the predicted equivalence of derived categories is induced by a Morita equivalence. Another consequence of Theorem 8.1.4 is that (8.1.6)

A θ Matq−1 (OT bθ ) Matq−1 (OS ),

which agrees with structure theorems for nilpotent blocks (for the case of an abelian defect group, see for example [BrPu, §1]).

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8 Unequal Characteristic: Equivalences of Categories

8.2. Quasi-Isolated Blocks Hypothesis. In this and only this section we assume that is odd (and different from p). This hypothesis is necessary for the existence of quasi-isolated blocks.

8.2.1. Harish-Chandra Induction In this subsection we will show the following result. Proposition 8.2.1. Harish-Chandra induction RO induces a Morita equivalence between the O-algebras Aα0 and ONbα0 . √ Proof. Denote by q a root of q in O (recall that O is sufficiently large). Set M = O[G /U]bα0 . As in the proof of Proposition 8.1.1, M is an (Aα0 , OTbα0 )bimodule. Moreover, by 7.2.3, M is stable under the endomorphism F . De√ note by S the restriction of F / q to M. As the element E = ∑t∈T t acts as multiplication by 0 on M, we have S 2 = d(−1) by Theorem 7.2.4(a). Therefore, by 7.2.3, we can extend the right OT -module structure on M to an ON-module structure, letting s act as the automorphism S . We may then view M as an (Aα0 , ONbα0 )-bimodule. As N/T is of order 2 and is odd, it follows from Proposition 7.2.1 that M is projective as a left and right module. By virtue of Broué’s Theorem B.2.5, it is enough to show that EndAα0 M ONbα0 (via the canonical morphism). This fact is a consequence of Theorem 7.2.4(c). C OMMENTARY – If is odd and divides q − 1, then ONbα0 is the Brauer correspondent of Aα0 (see Theorem 7.1.2), and Proposition 8.2.1 shows that Broué’s conjecture (see Appendix C) is true in a stronger form: the predicted equivalence of derived categories is induces by a Morita equivalence.

8.2.2. Deligne-Lusztig Induction* In this subsection we will show the following result. Proposition 8.2.2. Deligne-Lusztig induction R induces a Morita equivalence between the Z -algebras A θ0 and ON bθ 0 .

8.3 The Principal Block

89

Proof. The proof is similar to that of Proposition 8.2.1. First, we define M = Hc1 (Y, O)bθ 0 . It is an (A θ0 , OT bθ 0 )-bimodule which is projective as a left and right module by the same argument as in the proof of Proposition 8.1.4. We may then give it the structure of an (A θ0 , ON bθ 0 )-bimodule, √ letting s act as F / −q (by 7.3.8 and Theorem 7.3.9(a)). This bimodule is still projective as a left and right module, because N /T is of order 2 and is odd. It then results from Theorem 7.3.9(c) that EndA (M ) ON bθ 0 . We may then conclude, thanks to Theorem B.2.5.

θ0

C OMMENTARY – If is odd and divides q + 1, then ON bθ 0 is the Brauer correspondent of A θ0 (see Theorem 7.1.2), and Proposition 8.2.2 shows Broué’s conjecture (see Appendix B) in a stronger form. Indeed, the predicted equivalence of derived categories is induced by a Morita equivalence.

8.3. The Principal Block As described in the introduction to this chapter, the case of the principal block is the most interesting. Here we give a proof of the results described in the introduction when is odd. When = 2, we will content ourselves with stating the results of Gonard [Go] without proof. In the richest case for which we give a complete treatment, that is when is odd and divides q + 1, the equivalence predicted by Broué’s conjecture will be constructed using the complex of cohomology RΓc (Y, O). For this we will make crucial use of the results of the previous chapter, most notably the description of the algebra of endomorphisms of this complex.

8.3.1. The Case when is Odd and Divides q − 1 We begin by showing a result of a “global” nature which refines Theorem 7.2.4. Theorem 8.3.1. If is odd and divides q − 1, then EndZ G Z [G /U] Z N. Proof. As q ≡ 1 mod and is odd, the polynomial X 2 − q is split over F . √ Therefore, by Hensel’s lemma, there exists an element + q of Z such that √ √ 2 + q ≡ 1 mod Z and ( + q) = q. Set √ + q −1 1 1 S = √ (F − E ) · (1 − E ), √ + q 2 ( + q + 1)(q + 1) and recall that E = ∑t∈T t ∈ Z T . As is odd and longs to EndZ G Z [G /U].

√ + q ≡ 1 mod Z , S be

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8 Unequal Characteristic: Equivalences of Categories

On the other hand a tedious but straightforward calculation shows that S 2 = d(−1) and S t = t −1 S for all t ∈ T . As a consequence, the right Z T to a right Z N-module strucmodule structure on Z [G /U] may be extended √ + q−1 ture by letting s act as S . Moreover, 1 − ( +√q+1)(q+1) E ∈ (OT )× because, if λ λ ∈ Z , then the inverse of the element 1 + λ E ∈ OT is 1 − 1+(q−1) λ E ∈ OT . Theorem 7.2.4(c) then shows that EndZ G Z [G /U] = Z T ⊕ Z T · S Z N, the isomorphism being induced by the right action of Z N on Z [G /U].

Corollary 8.3.2. If is odd and divides q − 1, then Harish-Chandra induction induces a Morita equivalence between the O-algebras ⊕α ∈[T ∧ /≡] Aα and ON. Proof. Set A=

⊕

α ∈[T ∧ /≡]

Aα .

It follows from 3.2.13 that the irreducible factors of K [G /U] are exactly the elements of Irr KA. The result then follows from Theorem 8.3.1 and Broué’s theorem B.2.5. C OMMENTARY – When is odd and divides q − 1 the Morita equivalence of the previous corollary induces an equivalence between Aα and its Brauer correspondent, after cutting by block idempotents. Thanks to Corollary 8.3.2, we rediscover when α = 1 the results of Corollary 8.1.2 and Proposition 8.2.1 (which were valid in all unequal characteristic) by a more direct route. The reader may verify that the Morita equivalences constructed in these corollaries agree with those constructed in Corollary 8.3.2. Corollary 8.3.3. If is odd and divides q − 1, then Harish-Chandra induction induces a Morita equivalence between the principal O-block of G and that of N. R EMARK – Even when is odd and divides q − 1, the O-blocks Aα0 and OTbα0 are not Morita equivalent. Similarly, the O-blocks A1 and OTb1 are not Morita equivalent. There is even no equivalence of derived categories, because | Irr KA? | = |S | + 1 = |S | = | Irr KTb? | when ? ∈ {1, α0 } (see Remark B.2.7).

8.3.2. The Case when is Odd and Divides q + 1* We begin by showing a result of a “global” nature which refines Theorem 7.3.9. Theorem 8.3.4. If is odd and divides q + 1, then EndDb (Z G ) RΓc (Y, Z ) Z N . Proof. The proof is analogous to that of Theorem 8.3.1. Indeed, set √ + 1 1 −q − 1 √ (F E E ), − ) · (1 − S = √ + + −q 2 ( −q + 1)(−q + 1)

8.3 The Principal Block

91

√ where we recall that E = ∑t ∈T t ∈ Z T and + −q denotes a square root √ of −q in Z such that + −q ≡ 1 mod Z (this root exists because −q ≡ 1 mod ). One then shows that S 2 = d (−1) and S t = t −1 S for all t ∈ T . Thanks to Theorem 7.3.9(c) this allows us to show that EndDb (Z G ) RΓc (Y, Z ) Z N , the isomorphism being obtained by letting s ∈ N act through the automor phism S . Corollary 8.3.5 (Rouquier). If is odd and divides q + 1, then Deligne-Lusztig induction induces a Rickard equivalence between ⊕θ ∈[T ∧ /≡] A θ and ON . Proof. The proof consists of two steps. In the first step we construct, using the building blocks at our disposal, a complex of (OG , ON )-bimodules. This is the most delicate step. The second step consists of showing that this complex verifies the conditions of Theorem B.2.6, which is almost a formality using the results at our disposal. First step: construction of the complex. By Appendix A (section A.1), the complex RΓc (Y, O) is a well-defined element in the category of complexes of bimodules Cb (OG , OT F mon ). Moreover, it is homotopic, in the category Kb (OG , OT ), to a bounded complex C of (OG , OT )-bimodules of finite type. After eliminating direct factors homotopic to zero, we may suppose that C is a complex of (OG , OT )-bimodules without a direct factor homotopic to zero. This property has the following consequence. Lemma 8.3.6. In the algebra EndCb (O G ,O T ) C the two-sided ideal of morphisms homotopic to zero is contained in the radical of EndCb (O G ,O T ) C . Proof (of Lemma 8.3.6). Denote by A = EndCb (O G ,O T ) C , I the two-sided ideal of morphisms which are homotopic to zero, and R the radical of A . If I is not contained in R, then (I + R)/R is a non-zero two-sided ideal in the finite-dimensional semi-simple algebra A /R. In particular, (I +R)/R contains a non-zero idempotent e¯ of A /R. The theorem on lifting idempotents [The, Theorem 3.1 (h)] implies that there exists an idempotent e ∈ I whose image in A /R is e¯. It follows that eC is a direct factor of C and e : eC → eC is the identity, and therefore eC is homotopic to zero. This contradicts our hypothesis. Hence I is contained in R as claimed. Denote by f : C → RΓc (Y, O) and f : RΓc (Y, O) → C two morphisms of complexes which are mutually inverse in the category Kb (OG , OT ). Denote by S˜ the element OT F mon which is equal to

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8 Unequal Characteristic: Equivalences of Categories

√ + 1 1 −q − 1 ˜ √ (F E E ), S = √ − ) · (1 − + + −q 2 ( −q + 1)(−q + 1) by analogy with the proof of Theorem 8.3.4. We then define S : C → C by S = f ◦ S˜ ◦ f , where S˜ is viewed as a morphism S˜ : RΓc (Y, O) → RΓc (Y, O). Then, if (g , t ) ∈ G × T , we have

(g , t ) ◦ S = S ◦ (g , s t ). Moreover, as shown during the proof of Theorem 8.3.4, the element S − d (−1) in EndCb (O G ,O T ) C is homotopic to zero. Set h = d (−1)S 2 −IdC . Then 2

S 2 = d (−1)(1 + h), where h ∈ EndCb (O G ,O T ) C is homotopic to zero. By Lemma 8.3.6, h belongs to the radical of EndCb (O G ,O T ) C and we may therefore extract a square root √ of 1 + h thanks to the standard formal series for 1 + X . This series converges because = 2. Therefore there exists h ∈ EndCb (O G ,O T ) C which is homotopic to zero and such that (1 + h )2 = 1 + h. We now define σ : C → C by σ = S ◦ (1 + h ). Then σ 2 = d (−1), which shows that we can equip C with the structure of a complex of (OG , ON ) bimodules, letting s act by σ . The equality σ ◦ t = s t ◦ σ may be easily verified (for t ∈ T ). Second step: verification of the criteria of Theorem B.2.6. Let us define A =

⊕

θ ∈[T ∧ /≡]

A θ

and denote by e the primitive central idempotent of OG such that A = OGe . Set C = e C . Then C is a complex of (A , ON )-bimodules which is quasi-isomorphic to C because e acts as the identity on cohomology groups by 7.3.2. By Corollary 7.3.6, we have HomDb (A ,O N ) (C , C [i]) = 0 for all i = 0. Moreover, every irreducible character of KA is an irreducible factor of H • (K C ) = Hc1 (Y) ⊕ Hc2 (Y) and the natural morphism ON → (EndDb (O G ) C )opp

8.3 The Principal Block

93

is an isomorphism by Theorem 8.3.4. The proof of the corollary is now complete by virtue of Theorem B.2.6. Corollary 8.3.7. If is odd and divides q + 1, then Deligne-Lusztig induction induces a Rickard equivalence between the principal O-block of G and that of N . R EMARK – Even when is odd and divides q + 1, the O-blocks A θ0 and OT bθ 0 are not Morita equivalent. Similarly, the O-blocks A 1 and OT b1 are not Morita equivalent. There is even no equivalence of derived categories, because | Irr KA ? | = |S | + 1 = |S | = | Irr KT b? | where ? ∈ {1, θ0 } (see Remark B.2.7).

8.3.3. The Case when = 2* In this subsection we will content ourselves to mention without proof two results of Gonard [Go]. Denote by S a Sylow 2-subgroup of G . Even though S is not abelian, the following result remains valid. Theorem 8.3.8 (Gonard). If = 2 and q ≡ ±3 mod 8 the principal O-blocks of G and NG (S) are Rickard equivalent. C OMMENTARY – If = 2 and q ≡ 3 mod 8, recall that |S| = 8 and that |NG (S)| = 24 by Theorem 1.4.3. In particular, S/Z Z/2Z × Z/2Z is abelian and is a Sylow 2-subgroup of G /Z . As predicted by Broué’s conjecture, it is possible to construct a Rickard equivalence between the principal blocks of G /Z and NG /Z (S/Z ) = NG (S)/Z . Moreover, this equivalence may be lifted to give the Rickard equivalence of Theorem 8.3.8. = GL2 (Fq ), from which it To be precise, Gonard works with the group G is easy to deduce Theorem 8.3.8. Set

A = End•Db (O G ) (O[G /U] ⊕ RΓc (Y, O)).

This algebra is an A∞ -algebra (that is to say, an algebra equipped with higher products mn : A ⊗n → A verifying certain compatibility conditions). In his thesis, Gonard [Go, §4.2] gives a complete description of this A∞ -algebra. For a review of A∞ -algebras, the reader is referred to [Go, §4.1]. Theorem 8.3.9 (Gonard). If = 2, then the derived category Db (OG ) is equivalent to the derived category of the A∞ -algebra A . This A∞ -algebra satisfies dimO A = 4(q + 1)

and

mn = 0 if n 4.

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8 Unequal Characteristic: Equivalences of Categories

8.4. Alvis-Curtis Duality* I would like to thank Marc Cabanes for suggesting this section to me, as well as for having provided a simple argument for the proof of the main result (Theorem 8.4.1). Hypothesis. In this section we fix a commutative ring R in which p is invertible. R EMARK – Q , Z , F , as well as K , O and k are rings in which p is invertible. This hypothesis implies that the idempotent eU can be viewed as an element of the group algebra RG . Recall that the (RG , RT )-bimodules R[G /U] and RGeU are isomorphic. Denote by

δ : RGeU ⊗RT eU RG −→ RG −→ ab. a ⊗RT b It is a morphism of (RG , RG )-bimodules. We set

δ ∗ : RG −→ a −→

RGeU ⊗RT eU RG ageU ⊗RT eU g −1

∑

g ∈[G /B]

the dual morphism. One may easily verify that δ ∗ is indeed a morphism of (RG , RG )-bimodules (indeed, it is enough to verify that δ ∗ (g ) = g δ ∗ (1) = δ ∗ (1)g for all g ∈ G ). Denote by D the complex of (RG , RG )-bimodules δ D = 0 −→ RGeU ⊗RT eU RG −→ RG −→ 0 and set

δ∗ D ∗ = 0 −→ RG −→ RGeU ⊗RT eU RG −→ 0 ,

the dual complex. Even though this will not influence the principal result of this section we suppose, in accordance with standard conventions, that the non-zero terms of D (respectively D ∗ ) occur in degree 0 and 1 (respectively −1 and 0). If M is a (RG , RG )-bimodule, we also use the notation M for the complex 0 → M → 0, where M is in degree 0. Theorem 8.4.1. In the homotopy category Kb (RG , RG ), we have D ⊗RG D ∗ K D ∗ ⊗RG D K RG .

8.4 Alvis-Curtis Duality*

95

Proof. To simplify notation set e = eU and M = RGe ⊗RT eRG . Furthermore, we will write ⊗ for the tensor product ⊗RT . We have g f D ⊗RG D ∗ = 0 −→ M −→ RG ⊕ (RGe ⊗ eRGe ⊗ eRG ) −→ M −→ 0 , where

f (a ⊗ b) = ab ⊕

∑

a ⊗ bge ⊗ eg −1

g ∈[G /B]

and

g (x ⊕ (a ⊗ b ⊗ c)) = δ ∗ (x) − ab ⊗ c.

The endomorphism F of the RG -module RGe defined by F (x) = qxese is an isomorphism. Indeed, the calculation completed over Z in Theorem 7.2.4 remains valid here and shows that F ◦ (F − E ) = qd(−1). We will need the following lemma. Lemma 8.4.2. The morphism M ⊕M −→ RGe ⊗ eRGe ⊗ eRG (x ⊗ x ) ⊕ (y ⊗ y ) −→ x ⊗ e ⊗ x + F −1 (y ) ⊗ ese ⊗ y is an isomorphism of (RG , RG )-bimodules. Proof (of Lemma 8.4.2). By Bruhat decomposition 1.1.4, we have eRGe = eRBe ⊕ eR[BsB]e (as (RT , RT )-bimodules). As a consequence, RGe ⊗ eRGe = RGe ⊗ eRBe ⊕ RGe ⊗ eR[BsB]e . It is enough to show that the morphisms RGe −→ RGe ⊗ eRBe x −→ x ⊗e

and

RGe −→ RGe ⊗ eR[BsB]e y −→ F −1 (y ) ⊗ ese

are isomorphisms of (RG , RT )-bimodules. This is a straightforward consequence of the fact that, in both cases, the inverse is given by the formula a ⊗ b → ab. Using the isomorphism of Lemma 8.4.2, the complex D ⊗RG D ∗ is isomorphic (in the category Cb (RG , RG )) to the complex g f 0 −→ M −→ RG ⊕ M ⊕ M −→ M −→ 0, where

f (m) = δ (m) ⊕ m ⊕ δ ∗ (δ (m)) − m

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8 Unequal Characteristic: Equivalences of Categories

g (a ⊕ m ⊕ m ) = δ ∗ (a) − m − m .

and

This uses the fact that δ ∗ δ (a ⊗ b) = a ⊗ b + qase ⊗ es −1 b. Now consider

ϕ : RG ⊕ M ⊕ M −→ RG ⊕ M ⊕ M a ⊕ m ⊕ m −→ a − δ (m) ⊕ m ⊕ (δ ∗ (a) − m − m ). Then ϕ is an isomorphism of (RG , RG )-bimodules, which shows that the complex D ⊗RG D ∗ is isomorphic (in the category Cb (RG , RG )) to the complex g f 0 −→ M −→ RG ⊕ M ⊕ M −→ M −→ 0, where f (m) = 0 ⊕ m ⊕ 0 and g (a ⊕ m ⊕ m ) = m . In other words, D ⊗RG D ∗ C RG ⊕ C ⊕ C [1], Id

M M → 0 in which the non-zero terms are where C is the complex 0 → M −→ concentrated in degrees 0 and 1. As C (and therefore C [1]) is homotopic to zero, we deduce that D ⊗RG D ∗ is homotopic to RG . The fact that D ∗ ⊗RG D is also homotopic to RG is shown in exactly the same fashion.

Corollary 8.4.3. The functor D ⊗RG − induces an auto-equivalence of the homotopy category Kb (RG , RG ) (as well as of the derived category Db (RG , RG )).

Exercises 8.1. If α ∈ T ∧ (respectively θ ∈ T ∧ ), show that OTbα OS (respectively OT bθ OS ). Now suppose that is odd. Show that the O-algebras ONb1 and ONbα0 are isomorphic, and similarly for the O-algebras ON b1 and ON bθ 0 . 8.2. Show that, if divides |G |/(q − 1), then the algebras EndZ G Z [G /U] and Z N are not isomorphic. Show that, if divides |G |/(q + 1), then the algebras EndDb (Z G ) RΓc (Y, Z ) and Z N are not isomorphic. 8.3*. If R is a field of characteristic different from p, the equivalence of homotopy categories D ⊗RG − induces an isomorphism between the Grothendieck ∼ groups that we will denote DR : K0 (RG ) −→ K0 (RG ). Show that DR [M ] = [RR ( ∗RR M)] − [M ]. Calculate DK .

Chapter 9

Unequal Characteristic: Simple Modules, Decomposition Matrices

Hypothesis. In this chapter, as in the two previous chapters, we suppose that is a prime number different from p.

In this chapter we will determine the simple kG -modules and the decomposition matrix Dec(OG ), as a function of the prime number . This study will be carried out block by block, or more precisely by type of block (nilpotent, quasi-isolated, principal). When the defect group is cyclic we will also give the Brauer tree. We refer the reader to Appendix B for the definitions of these concepts. To carry out this study, we will use the equivalences of categories constructed in the previous chapter. When these equivalences are Morita equivalences the simple modules correspond to one another and the decomposition matrices are preserved (as is the Brauer tree). When the equivalences in question are genuine Rickard equivalences (which only occurs for the principal block when is odd and divides q + 1) the only property that is conserved is the number of simple modules.

9.1. Preliminaries 9.1.1. Induction and Decomposition Matrices By extension of scalars from to K to k, Harish-Chandra induction R (respectively Deligne-Lusztig induction R ) induces two functors RK and Rk (respectively RK and Rk ). These functors induce linear maps, denoted R and Rk (respectively R and Rk ) between the Grothendieck groups of T (reC. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_9, © Springer-Verlag London Limited 2011

97

98

9 Unequal Characteristic: Simple Modules, Decomposition Matrices

spectively T ) and G :

and

R : K0 (KT ) −→ K0 (KG ), Rk : K0 (kT ) −→ K0 (kG ), R : K0 (KT ) −→ K0 (KG ) Rk : K0 (kT ) −→ K0 (kG ).

Because the functor R (respectively R ) is induced by an OG -module which is O-free (respectively a complex of OG -modules which are O-free), the diagrams K0 (KT ) (9.1.1)

R

decO T

K0 (KG ) decO G

K0 (kT )

Rk

K0 (kG )

and K0 (KT ) (9.1.2)

R

decO T

K0 (KG ) decO G

K0 (kT )

Rk

K0 (kG )

are commutative.

9.1.2. Dimensions of Modules and Restriction to U In 5.2.1 we introduced a non-trivial linear character χ+ of the additive group χ+ Fq+ . We denote by χk : Fq+ → k × the composition Fq+ −→ O × − k × . If x ∈ Fq we denote by ψx the linear character of U (with values in k × ) defined by

ψx (u(z)) = χk (xz) for all z ∈ Fq . We set ex =

1 ∑ ψx (u−1 )u ∈ kU. q u∈U

9.2 Nilpotent Blocks

99

If a ∈ Fq× , then d(a)ex d(a−1 ) = ea−2 x

(9.1.3)

(see Proposition 1.1.2(b)). Consequently, if M is a kG -module, then (9.1.4)

d(a)ex M = ea−2 x M.

On the other hand, as the order of U is invertible in k, we have e0 M = M U

(9.1.5)

and

M = ⊕ ex M. x∈Fq

The next proposition follows from these observations. Proposition 9.1.6. Let M be a kG -module and let x and y be two elements of Fq× such that xy is not a square. Then dimk M = dimk M U +

q −1 (dimk ex M + dimk ey M). 2

In particular, (9.1.7)

dimk M ≡ dimk M U

mod

q −1 . 2

9.2. Nilpotent Blocks Let α (respectively θ ) be a linear character of T (respectively T ) such that α 2 = 1 (respectively θ 2 = 1). By Proposition 8.1.1 (respectively 8.1.4), the bimodule O[G /U]bα (respectively Hc1 (Y, O)bθ ) induces a Morita equivalence between Aα and OTbα (respectively between Aθ and OT bθ ). By extension of scalars we deduce that tensoring with k[G /U]bα (respectively Hc1 (Y, k)bθ ) induces a Morita equivalence between kAα and kTbα (respectively kAθ and kT bθ ). Now, kTbα (respectively kT bθ ) has only one simple module kα (respectively kθ ), that is to say the k-vector space k on which T α (respectively T ) acts via the linear character T −→ O × k × (respectively θ

T −→ O × k × ). As a consequence, Irr kAα = {Rk kα } = {IndG ˜} B kα and

Irr kAθ = {Hc1 (Y, k) ⊗kT kθ }.

100

9 Unequal Characteristic: Simple Modules, Decomposition Matrices

Note that | Irr kAα | = | Irr kAθ | = 1, as should be the case for any nilpotent block. The decomposition matrices are given by ⎛ ⎞ ⎛ ⎞ 1 1 ⎜1⎟ ⎜1 ⎟ ⎜ ⎟ ⎜ ⎟ Dec(Aα ) = ⎜ . ⎟ and Dec(Aθ ) = ⎜ . ⎟ , ⎝ .. ⎠ ⎝ .. ⎠ 1 1 where the number of lines of Dec(Aα ) (respectively Dec(Aθ )) is |S | (respectively |S |) because Irr KAα = {R(αη ) | η ∈ S∧ } Irr KAθ = {R (θ η ) | η ∈ S∧ }.

and

Brauer trees*. The only non-exceptional character of Aα (respectively Aθ ) is R(α ) (respectively R (θ )). The Brauer trees are therefore given by TAα

i

y

TA

i

y.

and θ

9.3. Quasi-Isolated Blocks Hypothesis. In this and only this section we suppose that is odd (and different from p). We recall that this hypothesis is necessary for the existence of quasi-isolated blocks. First, by Proposition 8.2.1 (respectively 8.2.2), the (Aα0 , OTbα0 )-bimodule O[G /U]bα0 (respectively the (Aθ0 , OT bθ 0 )-bimodule Hc1 (Y, O)bθ 0 ) can be extended to an (Aα0 , ONbα0 )-bimodule (respectively to an (Aθ0 , ON bθ 0 )bimodule), and the latter induces a Morita equivalence. The action of the ele√ √ ment s of N (respectively s of N ) is given by F / q (respectively F / −q). Now, the k-algebra kNbα0 (respectively kN bθ 0 ) admits two simple modules kα±0 (respectively kθ±0 ) associated to the linear characters χα±0 (respectively χθ±0 ) of N (respectively N ) defined in Section 6.2. Via the Morita equivalence they correspond to two simple modules of kAα0 (respectively kAθ0 ) that we

9.4 The Principal Block

101

(θ )). By 9.1.1 (respectively 9.1.2), these denote by R± (α0 ) (respectively R± 0 are the reductions modulo l of O-free OG -modules admitting R± (α0 ) (respectively R± (θ0 )) as characters. As

Irr KAα0 = {R+ (α0 ), R− (α0 )} ∪˙ {R(α0 η ) | η ∈ [S∧ / ≡], η = 1} and

Irr KAθ0 = {R+ (θ0 ), R− (θ0 )} ∪˙ {R(θ0 η ) | η ∈ [S∧ / ≡], η = 1},

we deduce that

Irr kAα0 = {R+ (α0 ), R− (α0 )}, (θ0 ), R− (θ0 )} Irr kAθ0 = {R+

that

and that the decomposition matrices are given by ⎛ ⎞ ⎛ 10 1 ⎜0 1⎟ ⎜0 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ Dec(Aα0 ) = ⎜1 1⎟ and Dec(Aθ0 ) = ⎜1 ⎜ .. .. ⎟ ⎜ .. ⎝. .⎠ ⎝. 11

⎞ 0 1⎟ ⎟ 1⎟ ⎟. .. ⎟ .⎠

11

Here, the first two lines of Dec(Aα0 ) (respectively Dec(Aθ0 )) correspond to the characters R± (α0 ) (respectively R± (θ0 )). The number of lines of Dec(Aα0 ) (respectively Dec(Aθ0 )) is equal to (|S | + 3)/2 (respectively (|S | + 3)/2). Brauer trees*. The non-exceptional characters of Aα0 (respectively Aθ0 ) are R± (α0 ) (respectively R± (θ0 )). Therefore, the Brauer trees are given by TAα0

i

y

i

i

y

i.

and TA

θ0

9.4. The Principal Block 9.4.1. Preliminaries We identify Rk k with k[P1 (Fq )] and, via this identification, we denote by StkG the k-vector subspace of k[P1 (Fq )] equal to

102

9 Unequal Characteristic: Simple Modules, Decomposition Matrices

∑

StkG = {

∑

κl l |

l∈P1 (Fq )

κl = 0}.

l∈P1 (Fq )

We denote by v the element v=

∑ 1

l ∈ k[P1 (Fq )].

l∈P (Fq )

Then kv and StkG are kG -submodules of k[P1 (Fq )], with kv being isomorphic to the trivial module. We have (9.4.1)

decO G StG = [ StkG ]kG .

Moreover, as |P1 (Fq )| = q + 1, we deduce that (9.4.2)

kv ⊆ StkG if and only if divides q + 1.

We finish with the following result. Proposition 9.4.3. If = 2, then kv and StkG are the only two non-trivial kG submodules of k[P1 (Fq )]. Proof. We begin with some notation. If z ∈ Fq , we set lz = [z; 1] ∈ P1 (Fq ) and l∞ = [1; 0]. We then have s · lz = l−z −1

u(x) · lz = lz+x

and

for all x ∈ Fq . Note that, with the usual conventions, the two previous formulas make sense and are valid even when z ∈ {0, ∞}. If x ∈ Fq , we set vx = ∑ χk (−xz)lz . z∈Fq

Then

k[P1 (Fq )] = kv ⊕

⊕ kvx .

x∈Fq

The action of the subgroup U is entirely determined by the following formulas: (∗)

e0 · v = v

and

ex · vy = δx=y vy

for all x, y ∈ Fq . Let M be a non-trivial kG -submodule of k[P1 (Fq )]. If U acts trivially on M, then G acts trivially on M because G is generated by the conjugates of U (see Lemma 1.2.2). In this case, M = kv and the proposition holds. We may therefore suppose that U (i.e. G ) does not act trivially on M. As a consequence, there exists an element x ∈ Fq× such that

9.4 The Principal Block

103

ex M = 0.

(1)

In other words, vx ∈ M by (∗). Again by (∗), we have ey s · vx = ey l∞ +

1 q

∑× χk

z+

z∈Fq

xy vy . z

But, by Exercise 9.1 (which is easy) and the fact that = 2, there exists y ∈ Fq× such that xy is not a square in Fq and

∑× χk

z∈Fq

xy = 0. z+ z

As a consequence, (2)

ey M = 0.

On the other hand, the injective morphism of kG -modules M → IndG B k, which, by adjunction, furnishes a non-trivial morphism of kB-modules ResG B M → k. Hence, (3)

e0 M = 0.

It then follows from (1), (2), (3) and Proposition 9.1.6 that dimk M q. Now set M = M ∩ StkG . Then dimk M q − 1 2 and again M is a kG submodule of k[P1 (Fq )]. As its dimension is at least 2, G does not act trivially on M and the previous argument applies again to M . We then obtain that dimk M q and, as M ⊆ StkG , we conclude that M = StkG , that is to say StkG ⊆ M.

9.4.2. The Case when is Odd and Divides q − 1

Hypothesis. In this and only this subsection we assume that is odd and divides q − 1. This case may be studied in a similar manner to the quasi-isolated block Aα0 . As does not divide q + 1, we deduce from 9.4.2 that k[G /B] = StkG ⊕kv .

104

9 Unequal Characteristic: Simple Modules, Decomposition Matrices

Hence the two simple modules of kA1 which correspond to the two simple kN-modules k1 and kε via the Morita equivalence are kv and StkG : Irr kA1 = {k, StkG }. As

Irr KA1 = {1G , StG } ∪˙ {R(η ) | η ∈ [S∧ / ≡], η = 1},

we conclude that the decomposition matrix is ⎛ ⎞ 10 ⎜0 1⎟ ⎜ ⎟ ⎜ ⎟ Dec(A1 ) = ⎜1 1⎟ . ⎜ .. .. ⎟ ⎝. .⎠ 11 Here, the two first lines are indexed by 1G and StG and the number of lines is (|S | + 1)/2. Brauer trees*. The non-exceptional characters of A1 are 1G and StG , and the Brauer tree is given by TA1

i

y

i .

9.4.3. The Case when is Odd and Divides q + 1

Hypothesis. In this and only this subsection we suppose that is odd and divides q + 1. In this case it follows from 9.4.2 that StkG is not simple. Indeed, it contains kv k

as a submodule. Let us denote StG = StkG /kv . Proposition 9.4.4. The kG -module k[G /B] is projective and indecomposable. The k kG -module StG is simple. k

Proof. As is odd, the simplicity of StG follows from Proposition 9.4.3. The projectivity of k[G /B] follows because does not divide the cardinality of B. The fact that k[G /B] is indecomposable follows from Proposition 9.4.3 and the fact that kv ⊆ StkG .

9.4 The Principal Block

105

As a consequence, k

Irr kA1 = {k, StG }.

(9.4.5) As

Irr KA1 = {1G , StG } ∪˙ {R(η ) | η ∈ [S∧ / ≡], η = 1},

we conclude that the decomposition matrix has the form ⎛ ⎞ 10 ⎜1 1⎟ ⎜ ⎟ ⎜ ⎟ Dec(A1 ) = ⎜0 1⎟ . ⎜ .. .. ⎟ ⎝. .⎠ 01 Brauer trees*. The non-exceptional characters of A1 are 1G and StG and the Brauer tree is given by y

TA1

i

i

9.4.4. The Case when = 2

Hypothesis. In this and only this subsection we suppose that = 2. Fix an O-free OG -module V± such that KV± admits the character R± (θ0 ). Set k

St± = kV± . Then (9.4.6)

k

dimk St± =

q −1 2

and, by Proposition 5.2.7, (9.4.7)

k

dimk eU St± = 0.

In order to show the last equality one uses the fact that |U| is invertible in k. It then follows from Proposition 9.1.6 that

106

9 Unequal Characteristic: Simple Modules, Decomposition Matrices k

(9.4.8)

St± is a simple kG -module. k

k

k

k

Proposition 9.4.9. If = 2, then Irr kA1 = {k, St+ , St− }. Moreover, St+ St− . k

k

Proof. The fact that St+ St− follows immediately from considering the restriction to U and Proposition 5.2.7. On the other hand, recall that Irr KA1 is the set of irreducible characters of R(α ) (for α ∈ S2∧ ) and R (θ ) (for θ ∈ S2∧ ). But, for such characters, it follows from 9.1.1 and 9.1.2 that decO G R(α ) = decO G R(1) = [k ]kG + decO G StG and As

decO G R (θ ) = decO G R (1) = −[k ]kG + decO G StG . k

k

decO G R (θ0 ) = [St+ ]kG + [St− ]kG ,

the proposition then follows from 9.4.8 and the fact that the isomorphism class of every simple kA1 -module appears in at least one decO G χ , where χ ∈ Irr KA1 (because decO G [KA1 ]KG = [kA1 ]kG ). The calculations completed in the proof of the previous proposition allow us to compute the decomposition matrix of A1 (using also Proposition 5.2.7), which is given in Table 9.1 below. In this table, m = (|S2 | − 2)/2 and m = (|S2 |−2)/2 and α1 ,. . . , αm (respectively θ1 ,. . . , θm ) give a set of representatives for (S2∧ \ {1, α0 })/ ≡ (respectively (S2∧ \ {1, θ0 })/ ≡). Note that, if q ≡ 1 mod 4 (respectively q ≡ 3 mod 4), then m = 0 (respectively m = 0). R EMARK 9.4.10 – Note that there is no Brauer tree for A1 because the defect group of A1 is a Sylow 2-subgroup of G and is therefore not cyclic (see Theorem 1.4.3). = GL2 (Fq ) has only R EMARK 9.4.11 – When = 2, the principal k-block of G two simple modules, contrarily to the principal k-block of G = SL2 (Fq ).

Exercises 9.1 (Katz). † In this and only this exercise we suppose that = 2. If a ∈ Fq , set Kl(a) =

∑× χk

z∈Fq †

a z+ . z

I would like to thank N. Katz for suggesting this exercise to me.

9.4 The Principal Block

107

Table 9.1 Decomposition matrix for A1 when = 2. k

k

k

St+

St−

1G

1

0

0

StG

1

1

R+ (θ0 )

0

R− (θ0 )

k

k

k

St+

St−

1G

1

0

0

1

StG

1

1

1

1

0

R+ (θ0 )

0

1

0

0

0

1

R− (θ0 )

0

0

1

R+ (α0 )

1

1

0

R+ (α0 )

1

1

0

R+ (α0 )

1

0

1

R+ (α0 )

1

0

1

R(α1 )

2

1

1

R (θ1 )

0

1

1

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

R(αm )

2

1

1

R (θm )

0

1

1

Dec(A1 ) when q ≡ 1 mod 4

Dec(A1 ) when q ≡ 3 mod 4

The purpose of this exercise is to show that there exists an element a of Fq× which is not a square and such that Kl(a) = 0 (this fact is used in the proof of Proposition 9.4.3). The value Kl(a) is called a Kloosterman sum. The following exercise does not make use of the finer properties of such sums.

a (a) Show that, if a ∈ Fq and b ∈ Fq× , then Kl(ab2 ) = ∑ χk b(z + ) . z z∈Fq× a a (b) Let a, z and z be three elements of Fq× such that z + = z + . Show z z that z = z or zz = a. (c) From now on fix an element a0 of Fq which is not a square. Show that a0 are of cardinality 0 or 2. the fibres of the map ω : Fq× → Fq , z → z + z (d) If θ ∈ Fq we set f (θ ) = ∑ χk (−bθ ) Kl(a0 b 2 ). b∈Fq×

Using (a), show that

f (θ ) =

1 − q if |ω −1 (θ )| = 0, q + 1 if |ω −1 (θ )| = 2.

108

9 Unequal Characteristic: Simple Modules, Decomposition Matrices

(e) Using (c) and the fact that q − 1 and q + 1 cannot both be zero as = 2, show that there exists an element a of Fq× which is not a square and such that Kl(a) = 0. 9.2. In this and only this exercise we suppose that = 2. Fix a non-square a0 ∈ Fq× and consider C+ = {b 2 | b ∈ Fq× } and C− = a0 C+ . Set Stk± = kv ⊕

⊕ kvx .

x∈C±

Use the argument of the proof of Proposition 9.4.3 to show that Stk± is a kG submodule of StkG and that the only non-trivial kG -submodules of k[P1 (Fq )] are kv , Stk+ , Stk− and StkG . Conclude that k[G /B] is indecomposable and determine the structure of the successive quotients Radi (k[P1 (Fq )])/ Radi+1 (k[P1 (Fq )]), for i 0. 9.3. In this and only this exercise, we suppose that divides q + 1. Let H be the algebra of endomorphisms of k[G /B]. Show that H k[X ]/X 2 , where X is an indeterminant. Deduce a new proof of the fact that k[G /B] is indecomposable. ∼

9.4*. Calculate Dk : K0 (kG ) −→ K0 (kG ) according to the values of . Recall that Dk is defined in Exercise 8.3. (Hint: Use Exercise 8.3 and the fact that decO G ◦DK = Dk ◦ decO G .)

Chapter 10

Equal Characteristic

Hypothesis. In this and only this chapter we suppose that = p. In this chapter we study the representations of the group G in equal, or natural, characteristic. A significant part of this chapter will be dedicated to the construction of the simple kG -modules. This classical construction generalises to the case of finite reductive groups. It turns out that the simple kG -modules are the restrictions of simple “rational representations” of the algebraic group G = SL2 (F). Having obtained this description, the determination of the decomposition matrices is straightforward. We then determine the (very simple) partition into blocks as well as the Brauer correspondents. There is only one block with trivial defect (which corresponds to the Steinberg character StG ) and the two other O-blocks (which both have U as their defect group — recall that the normaliser of U is B) are only distinguished by the action of the centre Z . As the group U is abelian, Broué’s conjecture predicts an equivalence of derived categories between the O-blocks of G and their Brauer correspondent. This result was shown by Okuyama [Oku1], [Oku2], for the principal block and by Yoshii [Yo] for the nonprincipal one (with defect group U) but the proof is too difficult to be included in this book. To finish off, if q = p, then U is cyclic and we determine the Brauer trees of the two blocks with defect group U. R EMARK – The methods used in this chapter are totally different from those employed in the rest of this book. In particular, we use neither the geometry of the Drinfeld curve, nor any cohomology theory with coefficients modulo p. All methods are entirely algebraic.

C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_10, © Springer-Verlag London Limited 2011

109

110

10 Equal Characteristic

Terminology, notation. In this and only this chapter we will denote by G the group SL2 (F) and B (respectively T, respectively U) the subgroup of G formed by upper triangular matrices (respectively diagonal, respectively upper unitriangular matrices). Note that B = T U. We will call a rational representation of G a morphism of algebraic groups G → GLF (V ), where V is a finite dimensional F-vector space. In this case we will often simply say that V is a rational G-module, in which case the morphism is implicit. We denote by K0 (G) the Grothendieck group of the category of rational representations of G. We will denote by ε : T → F× , diag(a, a−1 ) → a, so that n (ε )n∈Z is the set of rational representations of T. Similarly, we identify K0 (T) with the group algebra Z[Z] written exponentially: K0 (T) = ⊕n∈Z Zε n . We will denote by CarG : K0 (G) → K0 (T) the morphism induced by restriction. If V is a rational G-module, we denote by [V ]G its class in K0 (G) and we will write CarG V instead of CarG [V ]G in order to simplify notation.

10.1. Simple Modules 10.1.1. Standard or Weyl Modules Denote by V2 the natural rational G-module of dimension 2, that is to say, V2 = F2 equipped with the standard action of G = SL2 (F). If n is a natural number, we denote by Δ(n) the rational G-module defined by Δ(n) = Symn (V2 ), where Symn (V2 ) denotes the n-th symmetric power of the vector space V2 . These modules are called standard modules or Weyl modules. It is easy to verify that (10.1.1)

dimF Δ(n) = n + 1

and that (10.1.2)

Δ(0) is the trivial G-module.

10.1 Simple Modules

111

We denote by (x, y ) the canonical basis of V2 , so that ab ab · x = ax + cy and · y = bx + dy . cd cd We have,

n

Δ(n) = ⊕ Fx n−i y i . i=0

As

t · x n−i y i

=

ε n−2i (t)x n−i y i ,

we obtain the following formula: n

(10.1.3)

CarG Δ(n) =

∑ ε n−2i .

i=0

We will use the following preliminary results in the construction of the simple rational G-modules. Lemma 10.1.4. Δ(n)U = Fx n . If moreover 0 n q − 1, then Δ(n)U = Fx n . Proof. It is clear that x n ∈ Δ(n)U ⊆ Δ(n)U . To show that Δ(n)U ⊆ Fx n , it is sufficient, after replacing q by a power if necessary, to suppose that 0 n q −1 and to show that Δ(n)U ⊆ Fx n . This will also show the second assertion. Now let f be an element of Δ(n)U . Let us write n

f =

∑ λi x n−i y i .

i=0

Let ξ ∈ Fq . Then u(ξ ) · f = f . Now, u(ξ ) · f =

n

∑ λi x n−i (y + ξ x)i =

i=0

n

∑

i i−j n−j j ∑ ξ λi x y . i=j j

j=0

n

Comparing coefficients of x n , we therefore have, for all ξ ∈ Fq , n

∑ λi ξ i = λ0 .

i=0

As a consequence, the polynomial ∑ni=1 λi X i admits q distinct roots and is of degree q − 1. It is therefore zero, which shows that λ1 = · · · = λn = 0, and hence f ∈ Fx n . Corollary 10.1.5. If V is a non-zero G-submodule of Δ(n), then V contains x n . If 0 n q − 1 and if V is a non-zero G -submodule of ResG G Δ(n), then V contains x n . Proof. After replacing q by a power, we may suppose that 0 n q − 1: it is then enough to prove the second assertion. Let V be a non-zero G U submodule of ResG G Δ(n). Then V = 0 because U is a p-group [CuRe, Theon U rem 5.24]. Therefore x ∈ V ⊆ V by Lemma 10.1.4.

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10 Equal Characteristic

To complete this subsection we will give another construction of Δ(n), using the algebra F[G] of regular functions on G. Denote by ε˜ : B → F× the extension of ε to B which is trivial on U. If n ∈ Z, set Δ (n) = {f ∈ F[G] | ∀ (g , b) ∈ G × B, f (gb) = ε˜(b)−n f (g )}. Proposition 10.1.6. If Δ (n) = 0, then n 0. In this case, the G-modules Δ(n) and Δ (n) are isomorphic. Proof. Equip V2 with the structure of a rational (G, B)-bimodule, letting B act on the right via v · b = ε˜(b)v for all b ∈ B and v ∈ V2 . We denote by V2∗ the dual of V2 , and let (x ∗ , y ∗ ) denote the dual basis of (x, y ). Then V2∗ is, by duality, also equipped with the structure of a rational (G, B)-bimodule. We identify the symmetric algebra Sym(V2 ) with the algebra F[V2∗ ] of regular (i.e. polynomial) functions on V2∗ . This algebra inherits the structure of a (G, B)-bimodule. Via this action we may describe Δ(n) as follows, when n 0: (∗) Set

Δ(n) = {f ∈ Sym(V2 ) | ∀ b ∈ B, f · b = ε˜(b)n f }.

ϕ : Sym(V2 ) −→ F[G] f −→ (g → f (g · y ∗ )).

If g , h, γ ∈ G and if f ∈ F[G], we set (g · f · h)(γ ) = f (g −1 γ h−1 ). This equips F[G] with the structure of a (G, G)-bimodule. One may easily verify that ϕ is a morphism of (G, B)-bimodules and that (∗∗) Set

Δ (n) = {f ∈ F[G] | ∀ b ∈ B, f · b = ε˜(b)n f }. F[G]ρ (U) = {f ∈ F[G] | ∀ u ∈ U, f · u = f }.

We will need the following result. Lemma 10.1.7. ϕ is injective and its image is exactly F[G]ρ (U) . Proof (of Lemma 10.1.7). Let f ∈ Sym(V2 ) be such that ϕ (f ) = 0. Then f is zero on the G-orbit of y ∗ . This orbit is dense in V2∗ , therefore f = 0. This shows the injectivity ϕ . On the other hand, the (G, B)-equivariance of ϕ implies that its image is contained in F[G]ρ (U) . For the other implication, we choose f ∈ F[G]ρ (U) and must show that f is in the image of ϕ . We denote by fˆ the map defined on V2∗ \ {0} by fˆ(v ∗ ) = f (g ) when g ∈ G is such that v ∗ = g · y ∗ . As f ∈ F[G]ρ (U) and U is the stabiliser of y ∗ in G, the function fˆ is well-defined. We will show that it is a regular function on V2∗ \ {0}.

10.1 Simple Modules

113

To this end, write V2∗ \ {0} = Ux ∪˙ Uy , where Uv = {v ∗ ∈ V2∗ | v ∗ (v ) = 0} for all v ∈ V2 . It is sufficient to show that the restriction of fˆ to Uv is regular. Fix g ∈ G such that g · y = v . Set B− = T U− , where U− is the subgroup of G formed by the lower unitriangular matrices. Then B− stabilises the line Fy and the map ρv : B− −→ Uv b −→ gb · y ∗ is clearly an isomorphism of varieties. Moreover, if v ∗ ∈ Uv , then fˆ(v ∗ ) = f (g ρv−1 (v ∗ )), and so fˆ is regular on Uv . We have therefore shown that fˆ is a regular function on V2∗ \ {0}, and therefore fˆ is the restriction of a unique regular function f˜ on V2∗ which, by construction, verifies ϕ (f˜) = f . The proposition then follows immediately from Lemma 10.1.7 and the equalities (∗) and (∗∗). R EMARK – The isomorphism F[G]ρ (U) Sym(V2 ) is in fact an algebraic consequence of the existence of a natural isomorphism of quasi-affine varieties G/U V2∗ \ {0}, g U → g · y ∗ .

10.1.2. Simple Modules We denote by L(n) the G-submodule of Δ(n) generated by x n . Denote by Fp : G −→ G ab ap b p −→ cd cp dp the split Frobenius endomorphism of G over Fp . If i is a non-negative integer and if V is a rational G-module, we denote by V (i) the rational G-module with the same underlying space, but on which the element g ∈ G acts as Fpi (g ) acts on V . For example V (0) = V . If v ∈ V , we will denote by v (i) the corresponding element of the G-module V (i) . If g ∈ G, then g · v (i) = (Fpi (g ) · v )(i) . If m is a non-zero natural number, we will denote by (mi )i 0 the unique sequence of elements of {0, 1, 2, ... , p − 1} such that m=

∑ mi p i .

i 0

114

10 Equal Characteristic

Note that the sequence (mi )i 0 becomes zero after a certain point. Let I (n) = {m ∈ N | ∀ i 0, mi ni }. Note that I (n) ⊆ {0, 1, 2, ... , n} and that 0, n ∈ I (n). Moreover, m ∈ I (n) if and only if n − m ∈ I (n). The following theorem describes the simple rational G-modules as well as the simple FG -modules. Theorem 10.1.8. With notation as above we have: (a) L(n) = ⊕m∈I (n) Fx n−m y m . In particular, if 0 n p − 1, then L(n) = Δ(n). n (a’) If 0 n q − 1, then L(n) is the G -submodule of ResG G Δ(n) generated by x . (b) L(n) is the unique simple G-submodule of Δ(n). (b’) If 0 n q − 1, then ResG G L(n) is the unique simple FG -submodule of ResG G Δ(n). (c) L(n) ⊗ L(ni )(i) = ⊗ Δ(ni )(i) . i 0

i 0

(d) If 0 m, n, then L(m) L(n) if and only if m = n. G (d’)If 0 m, n q − 1, then ResG G L(m) ResG L(n) if and only if m = n. (e) The family (L(n))n 0 is a set of representatives of the isomorphism classes of simple rational G-modules. (e’) The family (ResG G L(n))0 n q−1 is a set of representatives of the isomorphism classes of simple FG -modules. Proof. We begin by proving (a) and (a’). Let us temporarily denote by V (n) the F-subvector space of Δ(n) defined by V (n) =

⊕ Fx n−m y m .

m∈I (n)

Let g =

ab cd

∈ G and let m ∈ I (n). Then

g · x n−m y m = (ax + cy )n−m (bx + dy )m =

∏ (ax + cy )p (ni −mi ) (bx + dy )p mi . i

i

i 0

As a consequence, (∗)

g · x n−m y m =

∏ (ap x p i

i

+ c p y p )ni −mi (bp x p + d p y p )mi . i

i

i

i

i

i

i 0

Expanding this product, we obtain a linear combination of monomials of the form i i i i ∏ x p (ni −mi −si ) y p si x p (mi −ti ) y p ti , i 0

where 0 si ni −mi and 0 ti mi . Set m = ∑i 0 (si +ti )p i . Then m ∈ I (n) and

10.1 Simple Modules

115

∏ x p (ni −mi −si ) y p si x p (mi −ti ) y p ti = x n−m y m , i

i

i

i

i 0

which shows that indeed g · x n−m y m ∈ V (n). Hence V (n) is a G-submodule of Δ(n) containing x n and therefore V (n) contains L(n). To complete the proof of (a) and (a’), we may suppose that 0 n q − 1 (after possibly replacing q by a power). Now set u− : Fq+ −→ G 10 ξ −→ ξ 1 and denote by U − the image of the morphism of groups u− . It will be sufficient for us to show that (?)

If 0 n q − 1, then V (n) is the U − -submodule of Δ(n) generated by x n .

Now, if ξ ∈ Fq , then it follows from (∗) that ni − n ξ m x n−m y m . u (ξ ) · x = ∑ ∏ m i m∈I (n) i 0 If m ∈ I (n), then ∏i 0

ni mi

= 0, therefore it is sufficient for us to show that

(ξ m )m∈I (n),ξ ∈Fq

the matrix is of rank |I (n)|, that is to say that the rows of this matrix are linearly independent. Now, as 0 n q − 1, these rows are complete rows of the matrix (ξ m )0 m q−1,ξ ∈Fq which is a square Vandemonde matrix, and hence is clearly invertible. The claims (b) and (b’) follow immediately from Corollary 10.1.5 and from (a’). We now turn to (c). If m ∈ I (n), we will denote by vm = ⊗ (x ni −mi y mi )(i) ∈ ⊗ L(ni )(i) = ⊗ Δ(ni )(i) . i 0

i 0

i 0

The formula (∗) shows that the F-linear map L(n) → ⊗i 0 L(ni )(i) which sends x n−m y m to vm (for all m ∈ I (n)) and is in fact a morphism of G-modules. Moreover it is an isomorphism. To show (d) and (d’) we may (after replacing q by a power), suppose G that 0 m n q − 1. Now, if ResG G L(m) ResG L(n), then the FT -modules U U L(m) and L(n) are isomorphic. But T acts on L(m)U = Fx m via the charT m T n m acter ResT T ε . Hence, ResT ε = ResT ε and therefore m ≡ n mod q − 1 be× cause ε is injective on T Fq . As 0 m, n q − 1, this can only happen if m = n or if m = 0 and n = q − 1. Now dimF L(0) = 1 = q = dimF L(q − 1). Therefore m = n.

116

10 Equal Characteristic

(e) We denote by F(n) the vector space F on which B acts via the linear character ε˜n . Then (F(n))n∈Z is a family of representatives of the isomorphism classes of simple rational B-modules. Indeed, if S is a simple rational B-module, then S U = 0 (as U is a unipotent group [Bor, Theorems 4.4 and 4.8]) and is B-stable, therefore S = S U is a simple rational T-module. Let V be a simple rational G-module. By the previous considerations there exists n ∈ Z and a surjective B-equivariant morphism ψ : ResG BV → F(−n). Let ψ˜ : V −→ F[G] v −→ (g → ψ (g −1 · v )). ˜ is a morphism of G-modules (where F[G] is equipped with the strucThen ψ ture of a G-module defined by (g · f )(x) = f (g −1 x) for all g , x ∈ G and ˜ is contained in f ∈ F[G]). As ψ is a morphism of B-modules, the image of ψ Δ (n) = {f ∈ F[G] | ∀ (g , b) ∈ G × B, f (gb) = ε˜−n (b)f (g )}. ˜ : V → Δ (n) is a non-zero morphism, and is therefore injective Moreover, ψ as V is simple. As Δ(n) Δ (n) (see Proposition 10.1.6), it follows from (b) that V L(n). The final statement (e’) now follows immediately from (d’) and from the fact that, by Proposition B.3.2(b) and Exercises 1.7 and 1.10, we have | Irr kG | = q. C OMMENTARY – In the case of a general connected reductive group, the classification of the simple rational modules by their highest weight is due to Borel and Weil. In the case of our group G = SL2 (F), this corresponds to part (e) of Theorem 10.1.8. On the other hand, the description (in part (c) of Theorem 10.1.8) of the simple modules as tensor products of simple modules twisted by the action of the Frobenius also generalises to all connected reductive groups: this is the Steinberg tensor product theorem. For further details concerning the representations of reductive groups in positive characteristic the reader may consult the book of Jantzen [Jan]. A generalisation of part (e’) of Theorem 10.1.8 gives a parametrisation of the simple representations of finite reductive groups in equal characteristic: see, for example, [Cartier, §6]. E XAMPLE 10.1.9 – If r 0, then Δ(p r − 1) = L(p r − 1). Indeed, pr − 1 =

r −1

∑ (p − 1)pi

i=0

and therefore I (p r − 1) = {0, 1, 2, ... , p r − 1}.

10.1 Simple Modules

117

10.1.3. The Grothendieck Ring of G Theorem 10.1.8 shows that K0 (G) = ⊕ Z [L(n)]G .

(10.1.10)

n0

On the other hand, (10.1.11)

∑

CarG L(n) =

ε n−2m .

m∈I (n)

ε0 εn = ε n + ε −n

Set

if n = 0, if n > 0.

Then (10.1.12)

∑

CarG L(n) =

ε n−2m ∈ ε n +

m∈I (n) m n/2

∑

0 m
Zε m .

0 −1 ∈ G normalises T and therefore acts on the 1 0 Grothendieck group K0 (T). One may verify that Now the element s =

K0 (T)s = ⊕ Zε n . n0

This shows that (10.1.13)

The map CarG : K0 (G) → K0 (T) is injective with image K0 (T)s .

Similarly, by 10.1.3, we have (10.1.14)

∑

CarG Δ(n) =

ε n−2m ∈ ε n +

0 m n/2

∑

0 m
Zε m .

Which shows that ([Δ(n)]G )n 0 is also a Z-basis of the Grothendieck group K0 (G). If m, n 0, we denote by Δm,n = [Δ(n) : L(m)] the multiplicity of L(m) as a factor in a Jordan-Hölder series of Δ(n). These integers Δm,n are uniquely determined by the equation (10.1.15)

CarG Δ(n) =

∑

m0

Δm,n CarG L(m).

In particular, taking into account the formulas 10.1.12 and 10.1.14, we have (10.1.16)

Δm,n ∈ {0, 1}

118

and (10.1.17)

10 Equal Characteristic

⎧ ⎪ ⎨1 if m = n, Δm,n = 0 if m > n, ⎪ ⎩ 0 if m ≡ n mod 2.

Further to these equalities, it is not difficult to completely determine the infinite matrix Δ = (Δm,n )m,n 0 . For this we will need to define recursively a set E (n) as follows: ⎧ {0} if 0 n p − 1, ⎪ ⎪ ⎪ ⎨ n − n0 if n p and n0 = p − 1, E (n) = pE n −p n n −n −p ⎪ ⎪ 0 0 ˙ ⎪ ⎩pE if n p and n0 p − 2. n0 + 1 + pE p p The decomposition matrix is described in the next proposition. Proposition 10.1.18. Let m, n 0. Then Δm,n ∈ {0, 1} and Δm,n = 1 if and only if m ∈ n − 2E (n). Proof. Let us first remark that: If r ∈ E (n), then n − 2r 0. This follows from an elementary recurrence. We will now show the proposition by induction on n. The crucial step is given by the following equality, in which we set Δ(−1) = 0 to simplify notation. We claim that, if n 0, then

(∗)

[Δ(n)]G = [Δ(n0 )]G · [Δ

n−n0 p

(1) ]G

+[Δ(p − n0 − 2)]G · [Δ

n−n0 −p p

(1) ]G .

We now show (∗). Write n = n0 + pn . We then have (n − n0 )/p = n and CarG Δ(n0 ) ⊗F Δ(n )(1) =

n0 n

∑ ∑ ε n0 −2i ε p(n −2j)

i=0 j=0 n0 n

=

∑ ∑ ε n−2(i+pj) .

i=0 j=0

Moreover

10.2 Simple kG -Modules and Decomposition Matrices

CarG Δ(p − n0 − 2) ⊗F Δ(n − 1)

(1)

119

p−n0 −2 n −1

∑ ∑ ε p−n0 −2−2i ε p(n −1−2j)

=

i=0

j=0

p−1 n −1

∑ ∑ ε n−2(i+pj) .

=

i=n0 +1 j=0

To show (∗), it is enough to remark that {0, 1, 2, ... , n} = {i + pj | 0 i n0 and 0 j n } ∪˙ {i + pj | n0 + 1 i p − 1 and 0 j n − 1}. We now return to the proof of Proposition 10.1.18. This result is immediate if n p − 1. Arguing by induction, we may suppose that n p and that the result holds for all n < n. By (∗) and the induction hypothesis, we have (as L(i) ⊗ L(j)(1) = L(i + pj) if 0 i p − 1) [Δ(n)]G = [L(n0 ) ⊗F

∑

m∈E (n )

L(n − 2m)(1) ]G

+ [L(p − n0 − 2) ⊗F

=

∑

m∈E (n −1)

L(n − 1 − 2m)(1) ]G

⎧ ⎪ ∑ [L(n − 2pm)]G ⎪ ⎪ ⎨m∈E (n )

if n0 = p − 1,

⎪ ⎪ ⎪ ⎩

if n0 p − 2.

∑

m∈E (n −1)

[L(n − 2(n0 + 1 + pm))]G

The proof of the proposition is now complete.

E XAMPLE 10.1.19 – If i, j ∈ {0, 1, ... , p − 1}, then [L(pj + i)]G if i = p − 1 or j = 0, [Δ(pj + i)]G = [L(pj + i)]G + [L(pj − i − 2)]G if i p − 2 and j 1.

10.2. Simple kG -Modules and Decomposition Matrices This section is primarily concerned with determining the decomposition matrix of the finite group G modulo p.

10.2.1. Simple kG -Modules Recall that k ⊆ F as k is of characteristic p. On the other hand, O contains the elements of the form ξ + ξ −1 , where ξ is a (q + 1)-th root of unity (see the

120

10 Equal Characteristic

character table in Table 5.4). Therefore k contains all elements of the form ξ + ξ −1 , as ξ traverses μq+1 . By virtue of Exercise 1.1, this implies that Fq ⊆ k ⊆ F. The FG -module ResG G Δ(n) is the extension of scalars to F of a kG -module which we will denote by Δq (n): in fact, Δq (n) = Symn (V2k ), where V2k = k 2 is the natural representation of G = SL2 (Fq ) ⊂ GL2 (k). Similarly, Fp restricts to an automorphism of the finite group G and we may define Δq (n)(i) as previously, and similarly for Lq (n) and Lq (n)(i) . By Theorem 10.1.8(e’), the morphism (10.2.1)

{0, 1, 2, ... , q − 1} −→ Irr kG n −→ [Lq (n)]kG

is bijective. On the other hand, by 10.1.17, we have (10.2.2)

[Δq (n) : Lq (m)] = Δm,n = [Δ(n) : L(m)]

for all m, n ∈ {0, 1, 2, ... , q − 1}, which shows that

(10.2.3) [Δq (n)]kG 0 n q−1 is a Z-basis of K0 (kG ). To finish this subsection, note the following result. Lemma 10.2.4. The simple kG -module Lq (q − 1) is projective. If M is an OG module such that [KM ]KG = StG , then kM Lq (q − 1). Proof. By the assertion (?) in the proof of Theorem 10.1.8, Lq (q − 1) = kU − · x q−1 . As dimk Lq (q − 1) = q by 10.1.1 and Theorem 10.1.8(c), we deduce an isoG − morphism of kU − -modules ResG U − Lq (q − 1) kU . Hence ResU − Lq (q − 1) is − a projective kU -module, hence Lq (q − 1) is a projective kG -module (as U − is a Sylow p-subgroup of G ). On the other hand, Lq (q − 1)B = kx q−1 = 0, and therefore there exists an injection k → ResG B Lq (q − 1) (where k denotes the trivial kB-module). Frobenius reciprocity then implies that there exists a non-zero morphism IndG B k → Lq (q − 1). But Lq (q − 1) is simple and hence this morphism is surjective. The last assertion of the lemma follows.

10.2 Simple kG -Modules and Decomposition Matrices

121

10.2.2. Decomposition Matrices Recall that Dec(OG ) is the transpose of the matrix decO G : K0 (KG ) → K0 (kG ) in the bases Irr KG and Irr kG . Taking into account 10.2.3, we denote by DecΔ (OG ) the transpose of the matrix of the morphism decO G in the bases Irr KG and (Δq (n))0 n q−1 . If we set Δ[q] the matrix (Δm,n )0 m,n q−1 (that is to say, the matrix obtained from the infinite matrix Δ by taking only the first q rows and columns), then (10.2.5)

Dec(OG ) = DecΔ (OG ) tΔ[q].

As we have already determined the matrix Δ[q] in Proposition 10.1.18 it is enough to determine the matrix DecΔ (OG ). To this end, note that the canonical surjection O × → k × admits a unique section which we denote by κ : k × → O × (see 6.1.1). For H ∈ {T , T } let

εH : H −→ O × t −→ κ (ε (t)). We identify K0 (kG ) with the lattice of (virtual) Brauer characters [CuRe, Definition 17.4 and Proposition 17.14]. Now a Brauer character β of G (which is defined on the p-regular elements of G , that is to say the set of semi-simple elements) is totally determined by the restrictions ResG T β and ResG β .

T For example, it follows from 10.1.3 that (10.2.6)

ResG H [Δq (n)]kG =

n

∑ εHn−2i

i=0

for all 0 n q − 1. On the other hand, if H ∈ {T , T }, then Irr H = {εHi | 0 i |H| − 1}. Examining the character table of G , we may easily verify that, if j ∈ Z, then (10.2.7)

j j −j ResG H R(εT ) = δH=T (εH + εH ) + 2

∑

εHi

0 i |H|−1 i≡j mod 2

and (10.2.8)

j −j

j ResG H R (εT ) = −δH=T (εH + εH ) + 2

∑

0 i |H|−1 i≡j mod 2

εHi .

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10 Equal Characteristic

Here, δH=T is equal to 0 if H = T and 1 if H = T (and, similarly, δH=T = 1 − δH=T ). We may easily deduce (again using the character table of G ) the form of the matrix DecΔ (OG ): Proposition 10.2.9. To simply notation set Δq (−1) = Lq (−1) = 0. Then: (a) (b) (c) (d)

decO G 1G = [Δq (0)]kG = [Lq (0)]kG . decO G StG = [Δq (q − 1)]kG = [Lq (q − 1)]kG . If 0 i q − 1, then decO G R(εTi ) = [Δq (i)]kG + [Δq (q − 1 − i)]kG . If 1 i q, then decO G R (εTi ) = [Δq (i − 2)]kG + [Δq (q − 1 − i)]kG . (q−1)/2

(e) decO G R± (εT (f)

) = [Δq ((q − 1)/2)]kG .

(q+1)/2 decO G R ± (εT

)=

[Δq ((q − 3)/2)]kG .

10.3. Blocks 10.3.1. Blocks and Brauer Correspondents ∼

Recall that d denotes the natural isomorphism of groups Fq× → T . We set 1 e+ = (d(1) + d(−1)), 2

G e+ = e+ − eStG

and

1 G e− = e− = (d(1) − d(−1)). 2

Then it is straightforward to verify that (10.3.1)

G G + e− + eStG 1 = e+

is a decomposition of 1 as a sum of pairwise orthogonal central idempotents of OG . The following proposition is even better: G , e G and e Proposition 10.3.2. The central idempotents e+ StG are primitive. −

Proof. Denote by 1+ (respectively 1− ) the trivial (respectively non-trivial) ˜χ the unique linear character of Z linear character of Z . If χ ∈ Irr G , we set ω ˜χ (z)χ (g ) for all (g , z) ∈ G × Z . Let such that χ (zg ) = ω ˜χ = 1+ } \ {StG }, B+ = {χ ∈ Irr G | ω ˜ χ = 1− } B− = {χ ∈ Irr G | ω and

BSt = {StG }.

One may easily verify, thanks to the table of central characters in Table 6.1, that B+ , B− and BSt are the p-blocks of G , which shows the result.

10.3 Blocks

123

We set G , A+ = OGe+

G A− = OGe−

and ASt = OGeStG .

Note that A+ is the principal O-block of G and that ASt is a block with trivial defect. The straightforward calculation of the Brauer correspondence is left to the reader. Proposition 10.3.3. The group U is a defect group of A+ and A− , and OBe? is a Brauer correspondent of A? (for ? ∈ {+, −}).

Proposition 10.3.4. We have Irr kA+ = {[Lq (2n)]kG | 0 n (q − 3)/2}, Irr kA− = {[Lq (2n + 1)]kG | 0 n (q − 3)/2} Irr kASt = {[Lq (q − 1)]kG }.

and

Broué’s conjecture was shown in this case by Okuyama [Oku1], [Oku2] for the principal block and by Yoshii [Yo] for the nonpricipal one. Theorem 10.3.5 (Okuyama, Yoshii). There exist Rickard equivalences Db (A+ ) Db (OBe+ )

and Db (A− ) Db (OBe− ).

10.3.2. Brauer Trees* As Brauer trees are only defined in the case of cyclic defect group we make the following assumption which guarantees that the group U, which is the defect group of A+ and A− , is cyclic. Hypothesis. In this and only this subsection we suppose that = p = q. It follows from this hypothesis that Δ[q] = Δ[p] = Ip , and therefore that Dec(OG ) = DecΔ (OG )

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10 Equal Characteristic

(see 10.2.5). Let R0 = 1G , Ri = R(εTi ) (if 1 i (p − 1)/2) and Ri = R (εTi ) (if 1 i (p + 1)/2). Up to the planar embedding, the Brauer tree of a block with cyclic defect group is completely determined by its decomposition matrix. We conclude from Proposition 10.2.9 that the Brauer trees of A+ and A− are as follows. TA+

R0 i

R2

i

R2 i

R4

i

p p p

χ+ i

+ χexc y

TA−

R1

i

R1 i

R3

i

R3 i

p p p

χ− i

− χexc y.

where

χ+ =

R(p−1)/2

R(p−3)/2

and, similarly, R(p−3)/2 − χ =

R(p−1)/2

R(p−1)/2 + = and χexc

R(p+1)/2 if p ≡ 3 mod 4, if p ≡ 1 mod 4,

if p ≡ 1 if p ≡ 3

mod 4, − and χexc = mod 4,

if p ≡ 1 mod 4, if p ≡ 3 mod 4,

R(p+1)/2

if p ≡ 1 mod 4,

R(p−1)/2

if p ≡ 3 mod 4.

We can also calculate the Brauer trees for the Brauer correspondents A+ and A− of A+ and A− respectively. The exceptional characters of A+ (reB B B B ). Using the spectively A− ) are χ+,+ and χ+,− (respectively χ−,+ and χ−,− character table of the group B, we obtain the following Brauer trees.

εT2

i S

ε4 iT S

TA

+

εT0

i

εTp−3

p

p S Sy p C p C p C i CC iε p−5 T

10.3 Blocks

125

εT3

i S

ε5 iT S

TA

−

εT1

p

p S S p i y C p C p C CC i iε p−4 εTp−2 T

By virtue of Brauer’s theorem (see Appendix C), we may conclude that, if ? ∈ {+, −}, then (10.3.6)

the blocks A? and A? are not Morita equivalent.

R EMARK – The blocks A+ and A− are Morita equivalent because they have the same Brauer tree (Brauer’s theorem) while the O-algebras A+ and A− are isomorphic (by restriction of the isomorphism OB → OB, b → εT (b)b).

Exercises 10.1* (Carter-Lusztig). Denote by K1 (kG ) the Grothendieck group of the category of projective kG -modules, which we may view as a sublattice (and even an ideal) of K0 (KG ). Show that K1 (kG ) = K0 (KG ) · StG . 10.2. If α : T → k × is a linear character, we denote by kα the kT -module defined as follows: the underlying k-vector space is k itself and an element t ∈ T acts as multiplication by α (t). Show that, if α = α −1 , then the kG modules k[G /U] ⊗kT kα and k[G /U] ⊗kT kα −1 are not isomorphic. Compare with Exercise 7.4. 10.3*. The purpose of this exercise is to calculate certain subalgebras of invariants. (a) Let

F[G]U×U = {f ∈ F[G] | ∀ (u, v ) ∈ U × U, u · f · v = f }

and denote by ϕ : G → F the regular function defined by ab = c. ϕ cd

126

10 Equal Characteristic

Show that F[G]U×U is generated by ϕ . (b) Let − F[G]U ×U = {f ∈ F[G] | ∀ (u, v ) ∈ U− × U, u · f · v = f } and denote by resT : F[G] → F[T] the restriction morphism. Show that resT − induces an isomorphism F[G]U ×U F[T]. (c) Let F[G]γ (G) = {f ∈ F[G] | ∀ g ∈ G, g · f · g −1 = f }. Show that resT induces an isomorphism F[G]γ (G) F[T]s and that F[G]γ (G) is generated by the trace function Tr : G → F. (d) Let F[G]ρ (T) = {f ∈ F[G] | ∀ t ∈ T, f · t = f }. ab If g = ∈ G, we set cd f1 (g ) = ab,

f2 (g ) = cd,

d1 (g ) = ad

and

d2 (g ) = bc.

Show that F[G]ρ (T) is generated by f1 ,f2 , d1 and d2 and that the kernel of the canonical morphism F[G]ρ (T) F[F1 , F2 , D1 , D2 ] −→ P −→ P(f1 , f2 , d1 , d2 ) is generated by F1 F2 − D1 D2 and D1 − D2 − 1 (here, F1 , F2 , D1 and D2 are indeterminates).

Part IV

Complements

The goals of these two final chapters are totally opposite. In Chapter 11, we consider the results obtained for SL2 (Fq ) when q ∈ {3, 5, 7} and point out various curious facts that occur in these small cases. We will study morphisms to SL2 (Fr ) as well as encountering exceptional reflection groups of ranks 2 and 3, the Klein curve and certain exceptional isomorphisms. In Chapter 12, we give a very brief summary (without proof) of DeligneLusztig theory, whose goal is the study of representations of arbitrary finite reductive groups using geometric methods. We explain to what extent the results that we have seen for SL2 (Fq ) may be interpreted as a special case of the general theory.

Chapter 11

Special Cases

In this chapter we will make explicit certain exotic properties of the groups SL2 (Fq ) when q = 3, 5 or 7. These include exceptional isomorphisms, inclusions as subgroups of SL2 (Fq ), and realisations as subgroups of reflection groups. For a recollection of definitions, results about reflection groups, see the Appendix C. the Notation. We will denote by Z the centre of G = SL2 (Fq ), Z centre of G = GL2 (Fq ), PGL2 (Fq ) = G /Z the projective general linear group and PSL2 (Fq ) = G /Z the projective special linear group. If n is a non-zero natural number, we denote by Sn the symmetric group of degree n (i.e. the group of permutations of the set {1, 2, ... , n}) and εn : Sn → {1, −1} the sign homomorphism. The alternating group of degree n (the kernel of the sign homomorphism) will be denoted An .

11.1. Preliminaries The linear character α0 of Fq× , of order 2, is the Legendre character: α0 (z) is equal to 1 if z is a square and is −1 otherwise. In particular, the linear char factorises to give a character of G /Z which we will denote acter α0 ◦ det of G by det0 : PGL2 (Fq ) → {1, −1}. One may easily verify that (11.1.1)

PSL2 (Fq ) = Ker(det0 ).

The action of PGL2 (Fq ) on the finite projective line P1 (Fq ) induces, after enumerating its elements 1 up to q + 1, a homomorphism of groups

C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_11, © Springer-Verlag London Limited 2011

129

130

11 Special Cases

φq : PGL2 (Fq ) −→ Sq+1 . The character det0 may be expressed with the help of the sign homomorphism on Sq+1 : (11.1.2)

det0 = εq+1 ◦ φq .

denotes the subgroup of G formed by upper triangular Proof. Recall that B matrices. As G = B ∪ Bs B, the group G is generated by the conjugates of B. then As a consequence, it is enough to show that, if g ∈ B,

α0 (det(g )) = εq+1 ◦ φq (¯ g ), where g¯ denotes the image of g in PGL2 (Fq ). Write ab g= . 0c Let ∞ = [1; 0] ∈ P1 (Fq ). Then g · ∞ = ∞ and, if z ∈ Fq , then g · [z; 1] = [(az + b)/c; 1]. Denote by σ : Fq → Fq , z → (az + b)/c. It suffices to show that the sign of the permutation σ is equal to α0 (ac). To this end, fix a total order on Fq and set A=

∏

(z − z) ∈ Fq× .

z,z ∈Fq z
Then the sign ε of σ is defined by

εA =

∏

(σ (z ) − σ (z)).

z,z ∈Fq z
It follows that, in Fq× , we have ε = (a/c)q(q−1)/2 = (ac)q(q−1)/2 c −q(q−1) = (ac)(q−1)/2 . Hence ε = α0 (ac) as expected. Hence the morphism φq induces by restriction a morphism

φq : PSL2 (Fq ) −→ Aq+1 . Recall that (11.1.3)

φq and φq are injective

and (11.1.4)

|PGL2 (Fq )| = q(q 2 − 1)

and |PSL2 (Fq )| =

q(q 2 − 1) . 2

11.2 The Case when q = 3

131

11.2. The Case when q = 3 Hypothesis. In this and only this section we suppose that q = 3.

11.2.1. Structure Recall that (11.2.1)

| SL2 (F3 )| = 24,

|PGL2 (F3 )| = 24 and

|PSL2 (F3 )| = 12,

which shows that the morphisms φ3 and φ3 defined in the previous sections are isomorphisms, that is to say PGL2 (F3 ) S4

(11.2.2)

and PSL2 (F3 ) A4 .

Therefore SL2 (F3 ) is a non-trivial central extension of A4 by Z/2Z (which is unique up to isomorphism) while PGL2 (F3 ) is a non-trivial central extension of S4 (there are three such extensions as H 2 (S4 , Z/2Z) Z/2Z × Z/2Z). We now show the following result, stated in Theorem 1.2.4. Proposition 11.2.3. We have G = N U and N = D(G ). The group N is isomorphic to the quaternionic group of order 8. The only non-trivial normal subgroups of G are Z and N . Proof. We have |N | = 8 and |G | = 24, and therefore N is a Sylow 2-subgroup of G . It is therefore the inverse image, under φ3 , of the Sylow 2-subgroup of A4 . Now the Sylow 2-subgroup of A4 is normal in A4 (and even in S4 ), therefore N is normal in G . On the other hand, as N ∩ U = {I2 } and |N | · |U| = |G |, we have G = N U. This shows in particular that D(G ) ⊆ N . The equality N = D(G ) and the two last assertions of the proposition are left as an (easy) exercise. Let I=

0 −1 , 1 0

Then

J=

1 1 1 −1

and

K=

−1 1 . 1 1

N = {±I2 , ±I , ±J, ±K }

and the multiplication is given by I 2 = J 2 = K 2 = −I2 ,

IJ = −JI = K ,

JK = −KJ = I

and KI = −IK = J,

which shows again that N is the quaternionic group of order 8. To simplify notation let

132

11 Special Cases

11 . u = u+ = 01 One may easily verify that U = u,

u 3 = I2 ,

u

u

I = J,

J =K

and

u

K = I.

11.2.2. Character Table The character table 5.4 specialises as follows. √ Let i denote an element of μ4 of order 4 (and note that q0 = −3), let j = (−1 + −3)/2 (so that j is a third root of unity in K ) and let i ∧ be a linear character of order 4 of μ4 . The character table of SL2 (F3 ) is given in Table 11.1. Table 11.1 Character table of SL2 (F3 )

g

I2

− I2

d (i)

u+

u−

− u+

− u−

| ClG (g )|

1

1

6

4

4

4

4

o(g )

1

2

4

3

3

6

6

CG (g )

G

G

T

ZU

ZU

ZU

ZU

1G

1

1

1

1

1

1

1

R+ (θ0 )

1

1

1

j

j2

j

j2

R− (θ0 )

1

1

1

j2

j

j2

j

R (i ∧ )

2

−2

0

−1

−1

1

1

R+ (α0 )

2

−2

0

−j 2

−j

j2

j

R− (α0 )

2

−2

0

−j

−j 2

j

j2

StG

3

3

−1

0

0

0

0

11.2 The Case when q = 3

133

11.2.3. The Group SL2 (F3 ) as a Subgroup of SL2 (F ) Amongst the three characters of degree 2 of G , only R (i ∧ ) has rational values. Here we study the possible fields of definition of a representation admitting this character. For this, we commence by calculating the FrobeniusSchur indicator of R (i ∧ ): 1 R (i ∧ )(g 2 ) |G | g∑ ∈G =

1 (2 + 2 + 6 × (−2) + 4 × (−1) + 4 × (−1) + 4 × (−1) + 4 × (−1)) = −1. 24

As a consequence (see [Isa, Corollary 4.15]), R (i ∧ ) is not the character of a real representation, even though it has rational (and hence real) values. Proposition 11.2.4. Let R be a ring in which 2 is invertible and −1 is a sum of two squares. Then exists an injective homomorphism ρ : SL2 (F3 ) → SL2 (R) there −1 0 and such that, if g ∈ SL2 (F3 ), then sending −I2 to 0 −1 Tr(ρ (g )) = R (i ∧ )(g ) · 1R ∈ R. Proof. Choose a and b in R such that a2 + b 2 = −1. Write 0 −1 a b −b a ρ (I ) = , ρ (J) = , ρ (K ) = 1 0 b −a a b and

ρ (u) =

1 b−a−1 1−a−b . 2 −1 − a − b a − b − 1

One verifies easily that det ρ (u) = 1, Tr ρ (u) = −1 (and therefore ρ (u)2 + ρ (u) + 1 = 0), that −1 0 , ρ (I )2 = ρ (J)2 = ρ (K )2 = 0 −1 that

ρ (I )ρ (J) = −ρ (J)ρ (I ) = ρ (K ),

and that ρ (u)

ρ (I ) = ρ (J),

ρ (u)

ρ (I ) = ρ (J) and

ρ (u)

ρ (I ) = ρ (J).

It follows that we can extend ρ to a morphism of groups

ρ : SL2 (F3 ) −→ SL2 (R)

134

11 Special Cases

satisfying the conditions of the proposition. Corollary 11.2.5. Let R be a principal ideal domain of characteristic different from 2 and let K be its field of fractions. We assume that −1 is the sum of two squares in K. Then there exists an injective morphism of groups ρ : SL2 (F3 ) → SL2 (R) −1 0 and such that, if g ∈ SL2 (F3 ), then sending −I2 to 0 −1 Tr(ρ (g )) = R (i ∧ )(g ) · 1R ∈ R. Proof. By Proposition 11.2.4, there exists morphism of groups an injective −1 0 ρ : SL2 (F3 ) → SL2 (K) sending −I2 to . The proposition follows 0 −1 from Proposition B.3.1 (as R is a principal ideal domain). The corollary shows that, if K is a field of characteristic different from 2 in which −1 is a sum of two squares, then there exists a faithful irreducible representation SL2 (F3 ) over K of dimension 2 admitting R (i ∧ ) as character. In fact the converse is also true (see Exercise 11.4). E XAMPLE 11.2.6 – Denote by ζn a primitive nth root of unity in C. Then Z[ζ3 ], Z[ζ4 ] and Z[ζ5 ] satisfy the hypotheses of Corollary 11.2.5. Indeed, −1 = ζ32 + (ζ32 )2 = ζ42 + 02 = (ζ5 + ζ5−1 )2 + (ζ52 − ζ5−2 )2 . E XAMPLE 11.2.7 – In a finite field every element is the sum of two squares. This is immediate in characteristic 2. If K is a finite field of characteristic different from 2 and cardinality r , then, if we let C = {a2 | a ∈ K}, we have |C | = (r + 1)/2. Therefore, if z ∈ K, the sets C and z − C cannot be disjoint. Therefore C + C = K and, in particular, −1 is the sum of two squares. As a consequence, for all odd prime numbers , −1 is the sum of two squares in F and, by Hensel’s lemma, −1 is the sum of two squares in Z . We then obtain two injective group homomorphisms SL2 (F3 ) → SL2 (Z ) and ρ3→ : SL2 (F3 ) → SL2 (F ). We may then deduce a new proof of Theorem 1.4.3(d): if ≡ ±3 mod 8, then S = ρ3→ (N ) is a Sylow 2-subgroup of SL2 (F ). Therefore, the image of ρ3→ is contained in NSL2 (F ) (S), which shows that |NSL2 (F ) (S)/S| 3.

11.2.4. The Group SL2 (F3 ) as a Reflection Group of Rank 2 A review of reflection groups is contained in Appendix C. Let V be a Cvector space of dimension 2 and let ρ : G → GLC (V ) be an irreducible representation of G whose character does not take on only rational values. If u is an element of G of order 3, and if we denote by ζ , ζ the eigenvalues of ρ (u), then ζ and ζ are third roots of unity such that ζ + ζ ∈ {−j, −j 2 }.

11.2 The Case when q = 3

135

Consequently, {ζ , ζ } = {1, j} or {1, j 2 }. In either case, ρ (u) is a reflection. As G is generated by elements of 3, we obtain: (11.2.8)

ρ (G ) G is a reflection group of rank 2.

Because N is the derived group of G , we have ρ (N ) ⊆ SLC (V ). As a consequence, Ref(ρ (G )) consists of ρ (u), where u belongs to the set of elements of order 3 in G . Hence |Ref(G )| = 8. If we denote by (d1 , d2 ) the sequence of degrees of ρ (G ) (see the ShephardTodd-Chevalley theorem in Appendix C), then d1 + d2 − 2 = |Ref(ρ (G ))| = 8 and

d1 d2 = |G | = 24,

and so d1 = 4 and d2 = 6. The group ρ (G ) is an exceptional complex reflection group denoted G4 in the classification of Shephard and Todd [ShTo] (see also the recent book of Lehrer and Taylor [Leh]). The group G therefore admits a presentation 11 3 3 and b G =< a, b | a = b = 1, aba = bab >, sending, for example, a to 01 1 0 . to −1 1

11.2.5. The Group PSL2 (F3 ) and the Isometries of the Tetrahedron Denote by Q[P1 (F3 )] the representation over Q of G with character IndG B 1B . Denote by H the hyperplane of Q[P1 (F3 )] equal to H ={

∑

δ ∈P1 (F3 )

aδ δ |

∑

δ ∈P1 (F3 )

aδ = 0}.

Then H is G -stable and the corresponding representation G → GLQ (H ) admits StG as character, and factorises to give an injection

τ3 : PSL2 (F3 ) → GLQ (H ). Let us equip Q[P1 (F3 )] with a scalar product such that the set P1 (F3 ) provides an orthonormal basis. Denote by π : Q[P1 (F3 )] → H the orthogonal projection and consider Δ = {π (δ ) | δ ∈ P1 (F3 )}.

136

11 Special Cases

Then Δ is a regular tetrahedron in H upon which PSL2 (F3 ) acts (via τ3 ) as isometries. One may easily verify that PSL2 (F3 ) is the group of oriented isometries of the tetrahedron Δ.

11.3. The Case when q = 5 Hypothesis. In this and only this section we suppose that q = 5.

11.3.1. Structure We have (11.3.1)

| SL2 (F5 )| = 120,

|PGL2 (F5 )| = 120 and |PSL2 (F5 )| = 60.

By Theorem 1.2.4, PSL2 (F5 ) is simple (of order 60), therefore (11.3.2)

PSL2 (F5 ) A5 .

This isomorphism may be seen as follows. Firstly, note that |N| = 8 and |G | = 8 × 15, and therefore N is a Sylow 2-subgroup of G . As q ≡ 5 mod 8, we have |NG (N)/N| = 3, which shows that NG (N) is of index 5 in G . The action of G on G /NG (N) (or, equivalently, the action of G on its five Sylow 2-subgroups) induces a morphism G → S5 having kernel Z and whose image, of order 60, is necessarily equal to the alternating group A5 . We can also see NG (N) as being the image of SL2 (F3 ) under the morphism ρ3→5 of Example 11.2.7. It follows that G = SL2 (F5 ) is a non-trivial central extension of A5 (which is unique up to isomorphism). Note that the isomorphism PSL2 (F5 ) A5 can be seen in other ways (see Exercise 11.1).

11.3.2. Character Table The character table 5.4 specialises as follows. Let i denote an element of F5× such that i 2 = −1, j an element of μ6 such that j 2 +j +1 = 0 (i.e. j is of order 3), i ∧ a linear character of F5× of order 4, j ∧ a linear character of μ6 of order 3 and denote by −j ∧ the linear character of order 6 of μ6 equal to θ0 j ∧ . Then, setting √ √ 1+ 5 1− 5 ∗ and ω = , ω= 2 2 the character table of SL2 (F5 ) is given in Table 11.2.

11.3 The Case when q = 5

137

Table 11.2 Character table of SL2 (F5 ) d(i) d (j) d (−j) u+

g

I2

− I2

u−

− u+

− u−

| ClG (g )|

1

1

30

20

20

12

12

12

12

o(g )

1

2

4

3

6

5

5

10

10

CG (g )

G

G

T

T

T

ZU

ZU

ZU

ZU

1G

1

1

1

1

1

1

1

1

1

R+ (θ0 )

2

−2

0

−1

1

−ω ∗

−ω

ω∗

ω

R− (θ0 )

2

−2

0

−1

1

−ω

−ω ∗

ω

ω∗

R+ (α0 )

3

3

−1

0

0

ω

ω∗

ω

ω∗

R− (α0 )

3

3

−1

0

0

ω∗

ω

ω∗

ω

R (j ∧ )

4

4

0

1

1

−1

−1

−1

−1

R (−j ∧ )

4

−4

0

1

−1

1

1

−1

−1

StG

5

5

1

−1

−1

0

0

0

0

R(i ∧ )

6

−6

0

0

0

1

1

−1

−1

11.3.3. The Group SL2 (F5 ) as a Subgroup of SL2 (Fr ) Let ζ5 be the fifth root of unity in K equal to χ+ (1) (recall that the linear χ+ : F5+ → K × was fixed in Section 5.2). We then have, by definition character √ of 5 (see §5.2.3), √ −1 + 5 . ζ5 + ζ5−1 = 2 We remark that G admits two irreducible representations of degree 2 and in both representations the centre Z acts non-trivially. The following proposition shows that we can realise these representations over the field Q(ζ5 ). Proposition 11.3.3. There exists an (injective) homomorphism of groups ρ± : SL2 (F5 ) → GL2 (Q(ζ5 )) with character R± (θ0 ). Proof. Denote by m the Schur indicator of R± (θ0 ) over Q(ζ5 ) (see [Isa, Definition 10.1]). The proposition amounts to showing that m = 1 (see [Isa, Corollary 10.2 (e)]). But R± (θ0 ) appears with multiplicity 1 in IndG U χ+ . As χ+ has

138

11 Special Cases

values in Q(ζ5 ) and has Schur index 1 over this field, it results from [Isa, Lemma 10.4] that m = 1. R EMARK 11.3.4 – The field Q(ζ5 ) is the smallest field over which a representation of SL2 (F5 ) admits R± (θ0 ) as character. Indeed, it is impossible to realise such a representation over Q, because the character has values in √ Q(√5) = Q(ζ5 + ζ5−1 ). Nor is it possible to realise such a representation over Q( 5) because this field is contained in R and the Frobenius-Schur indicator √ of R± (θ0 ) is −1 (as may be easily calculated). Now Q and Q( 5) are the only proper subfields of Q(ζ5 ), and the minimality of Q(ζ5 ) follows. Because the ring Z[ζ5 ] is a principal ideal domain, it follows from Proposition 11.3.3 and B.3.1 that there exists a morphism ρ± : SL2 (F5 ) → GL2 (Z[ζ5 ]) such that Tr(ρ± (g )) = R± (θ0 )(g ) for all g ∈ SL2 (F5 ). Because SL2 (F5 ) is perfect we have (11.3.5)

Im ρ± ⊆ SL2 (Z[ζ5 ]).

If is a prime number (possibly equal to p = 5) and if λ is a maximal ideal : of Z[ζ5 ] containing , we denote by ρλ the reduction modulo λ of ρ+

ρλ : SL2 (F5 ) −→ SL2 (Z[ζ5 ]/λ ). Proposition 11.3.6. The representation ρλ : SL2 (F5 ) −→ SL2 (Z[ζ5 ]/λ ) is (absolutely) irreducible. Moreover: (a) If = 2, then Ker ρλ = {±I2 }. (b) If is odd, then ρλ is injective. Proof. The only normal subgroups of SL2 (F5 ) are {I2 }, {±I2 } and SL2 (F5 ). As Tr(ρλ (d (j))) = 1 = 0, we obtain that Ker ρλ = SL2 (F5 ). −1 0 , which completes the proof of (a) On the other hand, ρλ (−I2 ) = 0 −1 and (b). Now, if the representation were not (absolutely) irreducible, the image of ρλ would be contained, up to conjugation, in the subgroup of SL2 (Fr ) formed by upper triangular matrices (for some r 1). In particular, the image would be solvable, which is not the case. √ the ring of integers of Q( 5) (recall that R5 = Z[(1 + √ Denote by R5 −1 5)/2] = Z[ζ5 + ζ5 ]). As R+ (θ0 ) takes values in R5 , we have Tr(ρλ (g )) ∈ R5 /(λ ∩ R5 ) for all g ∈ G . Consequently, it results from [Isa, Theorem 9.14(b)] (with this result following from Wedderburn’s theorem that all finite division rings are commutative) that, after conjugating by an element of GL2 (Z[ζ5 ]/λ ), we obtain an (absolutely) irreducible representation

ρ5→ : SL2 (F5 ) −→ SL2 (R5 /(λ ∩ R5 )). We can then deduce from Proposition 11.3.6 the following results.

11.3 The Case when q = 5

139

Corollary 11.3.7. With the above notation, we have: (a) If = 2, then λ ∩ R5 = 2R5 , Ker ρ5→ = {±I2 }, R5 /(λ ∩ R5 ) = F4 and ρ5→ induces an isomorphism PSL2 (F5 ) SL2 (F4 ). (b) If > 2, then ρ5→ is injective. √ (c) If = 5, then ∩ R5 = 5R5 , R5 /(λ ∩ R5 ) = F5 and ρ5→ is an isomorphism SL2 (F5 ) SL2 (F5 ). (d) If ≡ ±1 mod 10, then R5 /(λ ∩ R5 ) = F and therefore SL2 (F5 ) is isomorphic (via ρ5→ ) to a subgroup of SL2 (F ). (e) If ≡ ±3 mod 10, then R5 /(λ ∩ R5 ) = F2 and therefore SL2 (F5 ) is isomorphic (via ρ5→ ) to a subgroup of SL2 (F2 ). Proof. It is enough to determine the structure of the field R5 /(λ ∩ R5 ),√that is to say, to decide whether X 2 − X − 1 (the minimal polynomial of (1 + 5)/2) admits a root in F or not. If ∈ {2, 5} this is straightforward, and if ∈ {2, 5} we must determine whether 5 is a square modulo . By the law of quadratic reciprocity, this is equivalent to determining if is a square modulo 5. The statements of the corollary then follow. E XAMPLE 11.3.8 – Proposition 11.3.7(e) shows that SL2 (F5 ) is isomorphic to a subgroup H of SL2 (F9 ). As H is of index 6, the left action of SL2 (F9 ) on SL2 (F9 )/H induces a homomorphism SL2 (F9 ) → S6 . As SL2 (F9 ) is equal to its derived group and PSL2 (F9 ) is simple (see Theorem 1.2.4), one may compare cardinalities to conclude that the above morphism induces an isomorphism ∼ PSL2 (F9 ) → A6 . E XAMPLE 11.3.9 – Proposition 11.3.7(d) shows that SL2 (F5 ) is isomorphic to a subgroup H of SL2 (F11 ). As H is of index 11, the left action of SL2 (F11 ) on SL2 (F11 )/H yields a homomorphism SL2 (F11 ) → S11 . As SL2 (F11 ) is equal to its derived group and PSL2 (F11 ) is simple (see Theorem 1.2.4), this morphism induces an injection PSL2 (F11 ) → A11 with image a transitive subgroup. This morphism is “exceptional” by comparison with the natural action of PSL2 (F11 ) on the set of 12 elements of P1 (F11 ). One may show that the image of PSL2 (F11 ) → A11 is contained in the Mathieu group M11 (a simple subgroup of A11 of order 7920). This image, of index 12, then allows one to construct a transitive action of the Mathieu group M11 on a set of order 12.

11.3.4. The Group SL2 (F5 ) × Z/5Z as a Reflection Group of Rank 2 : SL (F ) → SL (Q(ζ )) Let us reconsider the irreducible representation ρ+ 2 5 2 5 C and denote by μ5 the subgroup of GL2 (Q(ζ5 )) given by scalar transforma . We have μ C ∩ W + = {1}. We may tions of order 1 or 5. Set W + = Im ρ+ 5 therefore consider

140

11 Special Cases

W = μ5C × W + ⊆ GL2 (Q(ζ5 )). We have (11.3.10)

D(W ) = W + .

Proposition 11.3.11. W is a subgroup of GL2 (Q(ζ5 )) generated by reflections. All its reflections are of order 5 and |Ref(W )| = 48. Furthermore, D(W ) = W + and Z(W ) = μ 10 . The sequence of degrees of W is (20, 30) and W is the group denoted G16 in the Shephard-Todd classification. Proof. Amongst the elements of order 5 of W + , denote by E5+ the set of elements whose eigenvalues are ζ52 and ζ5−2 (the others have eigenvalues ζ5 and ζ5−1 ). Set R5 = {ζ5−2 g | g ∈ E5+ }. Then the elements of R5 are reflections of order 5 (with eigenvalues 1 and ζ5 , and determinant ζ5 ). We will show that W is generated by R5 .

(∗)

Indeed, if W denotes the subgroup of W generated by R5 and if π+ : W → W + denotes the canonical projection, then π+ (W ) is the subgroup of W + generated by E5+ . As E5+ is stable under conjugation, π+ (W ) is a normal subgroup of W + generated by elements of order 5, and hence is equal to W + SL2 (F5 ) (see Theorem 1.2.4). In particular, 120 divides |W |, and W is of index dividing 5 in W . On the other hand the homomorphism det : W → μ 5 is surjective, with kernel W + ∩ W . Therefore W + ∩ W is a normal subgroup of W + of index dividing 5. Hence W + ∩ W = W + by Theorem 1.2.4, that is to say W + ⊆ W and |W /W + | = 5. Hence W = W , which shows (∗). The equalities D(W ) = W + and Z(W ) = μ 10 now follow easily. Note that the order of a reflection is equal to the order of its determinant, which shows that all reflections are of order 5. Now consider f : Ref(W ) → W + , g → det(g )2 g . The image f is indeed contained in W + as det(g )5 = 1. On the other hand, if f (g ) = f (g ), then g = λ g with λ ∈ μ 5 , which is only possible if λ = 1 or λ = det(g )−1 (because g and g are reflections of order 5). Hence the fibres of f have cardinality at most 2. Moreover, the image of f is contained in the set E5 of elements of order 5 of W + . We conclude that, |Ref(W )| 2 · |E5 | = 48. If we denote by (d1 , d2 ) the sequence of degrees of W , then, by the Shephard-Todd-Chevalley theorem (see Appendix C), we have d1 d2 = |W | = 600,

d1 + d2 − 2 48 and

10 divides d1 and d2 .

This forces (d1 , d2 ) = (20, 30) and |Ref(W )| = 48, as claimed.

11.3 The Case when q = 5

141

E XAMPLE 11.3.12 – In an analogous manner we can construct other reflection groups having W + as derived group. Let us view W + as a subgroup of SL2 (Q(μ 60 )). Given an even divisor n of 60 let Wn = W + · μ n . Then one can show, by similar arguments to those employed in the proof of Proposition 11.3.11 (with some modifications . . . ), that Wn is generated by reflections if and only if n ∈ {4, 6, 10, 12, 20, 30, 60}. (n)

(n)

If this is the case, denote by (d1 , d2 ) the sequence of degrees. These degrees together with the name of Wn in the Shephard-Todd classification are given in Table 11.3. Table 11.3 Reflection groups having SL2 (F5 ) as derived group (n)

(n)

n

|Wn | |Ref(Wn )| (d1 , d2 ) Z(Wn ) Standard name

4

240

30

(12, 20)

μ4

G22

6

360

40

(12, 30)

μ6

G20

10

600

48

(20, 30)

μ 10

G16

12

720

70

(12, 60)

μ 12

G21

20 1200

78

(20, 60)

μ 20

G17

30 1800

88

(30, 60)

μ 30

G18

60 3600

118

(60, 60)

μ 60

G19

11.3.5. The Group PSL2 (F5 ), the Dodecahedron and the Icosahedron The two irreducible representations of SL2 (F5 ) of degree 3 can be realised over the real numbers, as may be easily shown by calculating the FrobeniusSchur indicator (which is 1 in both cases). Both representations contain −I2 in their kernel, and therefore factorise to yield a representation of PSL2 (F5 ) with image contained in the special linear group (as SL2 (F5 ) is equal to its derived group). Let us choose a homomorphism

142

11 Special Cases

ρ : PSL2 (F5 ) −→ SLR (V ), where V is an R-vector space of dimension 3. We may fix an invariant scalar product , on V , so that ρ gives a homomorphism to the group of oriented isometries ρ : PSL2 (F5 ) −→ SOR (V , , ). Denote by E3 (respectively E3 ) the set of elements of order 3 (respectively 5) of PSL2 (F5 ). We have |E3 | = 20 and

|E5 | = 24.

Every non-trivial element of SOR (V , , ) is the rotation around a welldefined axis, and therefore the intersection of this axis with the unit sphere S consists of two elements. Let Δ3 = {v ∈ S | ∃ g ∈ E3 , g · v = v } and

Δ5 = {v ∈ S | ∃ g ∈ E5 , g · v = v }.

If v ∈ Δ3 (respectively v ∈ Δ5 ), there exist two (respectively four) elements of E3 (respectively E5 ) which fix it. In particular, |Δ3 | =

2 · 20 = 20 and 2

|Δ5 | =

2 · 24 = 12. 4

Proposition 11.3.13. The group PSL2 (F5 ) acts transitively on Δ3 and Δ5 . Proof. Let v ∈ Δ3 . Denote by H the stabiliser, in PSL2 (F5 ), of v and let n = |H|. Then H is isomorphic to a finite subgroup of the group of oriented isometries of the orthogonal to v , which is of dimension 2. Hence H is cyclic. Moreover, by construction, 3 divides n. The possible orders of the elements of PSL2 (F5 ) are 1, 2, 3, 4 or 5 and hence n = 3. Consequently, the orbit of v under PSL2 (F5 ) has cardinality 20 = |Δ3 |. The result follows. The transitivity of the action on Δ5 is shown in the same way. From Proposition 11.3.13 we may easily conclude the following corollary. Corollary 11.3.14. Δ3 (respectively Δ5 ) is a regular dodecahedron (respectively an icosahedron) and PSL2 (F5 ) is its group of oriented isometries.

11.4. The Case when q = 7 Hypothesis. In this and only this section we suppose that q = 7.

11.4 The Case when q = 7

143

11.4.1. Structure We have (11.4.1)

|PGL2 (F7 )| = 336 and |PSL2 (F7 )| = 168.

| SL2 (F7 )| = 336,

Recall that PSL2 (F7 ) is simple. However, contrary to the case of the groups PSL2 (F3 ) and PSL2 (F5 ), PSL2 (F7 ) is not isomorphic to an alternating group. We will show below (see Proposition 11.4.4) that (11.4.2)

PSL2 (F7 ) GL3 (F2 ).

11.4.2. Character Table The character table 5.4 specialises as follows. Let j denote an element of μ6 of order 3, j ∧ a linear character of order 3 of μ6 , i a square root of −1 in μ8 , i ∧ a linear character of order 4 of√μ8 , ζ8 an element of μ8 of order 8, ζ8∧ a linear character of μ8 of order 8 and 2 = ζ8∧ (ζ8 ) + ζ8∧ (ζ8 )−1 . We let √ √ 1 + −7 1 − −7 and ϖ ∗ = . ϖ= 2 2 The character table of SL2 (F7 ) is given in Table 11.4.

11.4.3. The Isomorphism Between the Groups PSL2 (F7 ) and GL3 (F2 ). By Table 11.4, the group SL2 (F7 ) admits two irreducible representations of dimension 3 over K on which −I2 acts trivially. These representations may be realised in the -adic cohomology of the variety Y (with coefficients in a sufficiently large extension of Q ) for all = 7. In particular, we can take = 2, so that K is a finite extension of Q2 . Denote by O the ring of integers of K . Proposition B.3.1, together with the fact that O is a principal ideal domain, shows that there exists an injective homomorphism

ρ : PSL2 (F7 ) → GL3 (O) such that (11.4.3)

g )) = R+ (θ0 )(g ) Tr(ρ (¯

144

11 Special Cases Table 11.4 Character table of SL2 (F7 ) d(j) d(−j) d (i) d (ζ8 ) d (ζ83 ) u+

g

I2

− I2

u−

− u+

− u−

| ClG (g )|

1

1

56

56

42

42

42

24

24

24

24

o(g )

1

2

3

6

4

8

8

7

7

14

14

CG (g )

G

G

T

T

T

T

T

ZU

ZU

ZU

ZU

1G

1

1

1

1

1

1

1

1

1

1

1

R+ (θ0 )

3

3

0

0

−1

1

1

−ϖ ∗

−ϖ

−ϖ ∗

−ϖ

R− (θ0 )

3

3

0

0

−1

1

1

−ϖ

−ϖ ∗

−ϖ

−ϖ ∗

R+ (α0 )

4

−4

1

−1

0

0

0

ϖ

ϖ∗

−ϖ

−ϖ ∗

R− (α0 )

4

−4

1

−1

0

0

0

ϖ∗

ϖ

−ϖ ∗

−ϖ

R (i ∧ )

6

6

0

0

2

0

0

−1

−1

−1

−1

R (ζ8∧ )

6

−6

0

0

0

√

√ − 2

−1

−1

1

1

R (ζ8∧3 )

6

−6

0

0

0

√ − 2

√

−1

−1

1

1

StG

7

7

1

1

−1

−1

−1

0

0

0

0

R(j ∧ )

8

8

−1

−1

0

0

0

1

1

1

1

R(j ∧ )

8

−8

−1

1

0

0

0

1

1

−1

−1

2

2

for all g ∈ SL2 (F7 ) (we denote by g¯ the image of g in PSL2 (F7 )). Recall that l denotes the maximal ideal of O. We denote by

ρl : PSL2 (F7 ) → GL3 (O/l) the reduction√modulo l of ρ . root of the polynomial X 2 −X +2 which splits modNow (1+ −7)/2 is a√ ulo 2 and therefore (1 + −7)/2 ∈ Z2 . We conclude from 11.4.3 and the character table 11.4 that Tr(ρl (g )) ∈ F2 for all g ∈ PSL2 (F7 ). Furthermore, by [Isa, Theorem 9.14(b)], after conjugating ρl by a suitable element of GL3 (O/λ ), we obtain a homomorphism

ρ7→2 : PSL2 (F7 ) −→ GL3 (F2 ).

11.4 The Case when q = 7

145

Proposition 11.4.4. The homomorphism ρ7→2 : PSL2 (F7 ) −→ GL3 (F2 ) is an isomorphism of groups. Proof. As |PSL2 (F7 )| = | GL3 (F2 )| = 168, it is enough to show that ρ7→2 is injective. Because PSL2 (F7 ) is simple (see Theorem 1.2.4), it is enough to show that Ker ρ7→2 = PSL2 (F7 ). But, by 11.4.3 and the character table 11.4, there exists an element g ∈ PSL2 (F7 ) such that Tr(ρ7→2 (g )) = 0. Therefore ρ7→2 (g ) is not the identify matrix, which completes the proof of the proposition. R EMARK 11.4.5 – The group GL3 (F2 ) has a “natural” action on the set of 7 elements given by the projective plane P2 (F2 ). On the other hand, from the point of view of PSL2 (F7 ), the most “natural” action is on the set of eight elements given by the projective line P1 (F7 ).

11.4.4. The Group PSL2 (F7 ) × Z/2Z as a Reflection Group of Rank 3 Let us fix an irreducible representation ρ : PSL2 (F7 ) → GL3 (C) with character R+ (θ0 ). Denote by W + the image of ρ . As ρ is injective and PSL2 (F7 ) is equal to its derived group, we have W + ⊆ SL3 (C). Denote by μ2C the subgroup of GL3 (C) given by scalar transformations of order 1 or 2 and set W = W + × μ2C PSL2 (F7 ) × Z/2Z. Then (11.4.6)

|W | = 336.

Let I denote the set of elements of W + of order 2. We have (11.4.7)

|I | = 21.

Set R = {−s | s ∈ I }. Then (11.4.8)

|R| = 21.

Proposition 11.4.9. The elements of R are reflections, the group W is generated by R and R = Ref(W ). The sequence of degrees of W is (4, 6, 14). The group W is isomorphic to the complex reflection group G24 in the Shephard-Todd classification. Proof. An element of order 2 in SL3 (C) necessarily has characteristic polynomial (X + 1)2 (X − 1). Hence, if we multiply such an element by −1 we obtain a reflection (of order 2). Hence the elements of R are reflections. Let W = R. Denote by π : W → W + the canonical projection. Then π (W ) is a subgroup of W + containing I and is therefore normal and

146

11 Special Cases

non-trivial. The simplicity of PSL2 (F7 ) implies that π (W ) = W + . Hence |W | ∈ {168, 336}. Now, if |W | = 168, then W PSL2 (F7 ) and therefore the morphism det is non-trivial on PSL2 (F7 ), which is impossible. Hence |W | = 336, and R generates W . The group W is therefore generated by its reflections and R ⊆ Ref(W ). On the other hand, if s ∈ Ref(W ), then det(s) = −1, hence −s ∈ W + and −s is an involution. We have therefore shown that Ref(W ) = R. Now, denote by (d1 , d2 , d3 ) the sequence of degrees of W . Then, by the Shephard-Todd-Chevalley theorem, d1 , d2 and d3 are even (as |Z(W )| = 2), d1 + d2 + d3 − 3 = |Ref(W )| = 21 and d1 d2 d3 = |W | = 336. It is straightforward to see that this forces (d1 , d2 , d3 ) = (4, 6, 14). The last assertion is easily verified. Let us identify the algebra of polynomial functions on C3 with C[X , Y , Z ]. Proposition 11.4.9 shows that there exists a homogeneous polynomial P ∈ C[X , Y , Z ]W of degree 4. This homogeneous polynomial defines a plane projective curve C = {[x; y ; z] ∈ P2 (C) | P(x, y , z) = 0}. One may verify that this plane projective curve is smooth. As the curve is of degree d = 4, it has genus g = (d − 1)(d − 2)/2 = 3 [Har, Exercise I.7.2(b) and Proposition IV.1.1]. Now, by a theorem of Hurwitz [Har, Exercise IV.2.5], we have | Aut C| 84 · (g − 1) = 168. As the group PSL2 (F7 ) acts faithfully on on this curve, we deduce that (11.4.10)

Aut C PSL2 (F7 ) GL3 (F2 ).

Hence this curve provides an example in which the bound of Hurwitz is attained. One may verify that in fact C is the Klein quartic (defined, for example, in [Har, Exercise IV.5.7]).

Exercises 11.1. Suppose that q = 5. After numbering from 1 to 6 the points of P1 (F5 ) → S6 , which we de on P1 (F5 ) induces a homomorphism G the action of G note by φ . Set Γ = Im φ PGL2 (F5 ). (a) Let n be a non-zero natural number and let H be a subgroup of Sn of index n. Denote by φ0 : Sn → Sn the morphism induced by the action of Sn on Sn /H (after enumerating the elements of Sn /H from 1 up to n). (a1) Show that φ0 is an automorphism. (Hint: If n = 4, use the simplicity of An and for n = 4, note that A4 does not have a subgroup of order 6.)

11.4 The Case when q = 7

147

(a2)Show that, if n is the number corresponding to H in Sn /H, then φ0 (H) is the stabiliser of n in Sn . Deduce that H Sn−1 . (a3) Show that if H is a transitive subgroup of Sn , then φ0 is not an inner automorphism. (b) Deduce from (b) that Γ S5 and that S6 has a non-inner automorphism. (c) Deduce a new proof that PSL2 (F5 ) A5 (see 11.3.2). 11.2*. Let Γ be a finite group. Show that: (a) If Γ is simple of order 60, then Γ PSL2 (F5 ) A5 . (b∗ ) If Γ is simple of order 168, then Γ PSL2 (F7 ) GL3 (F2 ). (c∗ ) If Γ is simple of order 360, then Γ PSL2 (F9 ) A6 . (d∗∗ )If Γ is simple of order 504, then Γ PSL2 (F8 ) = SL2 (F8 ). (e∗∗ ) If Γ is simple of order 660, then Γ PSL2 (F11 ). (f∗ ) If Γ is simple of order 1000, then Γ is isomorphic to PSL2 (Fq ), for some q ∈ {5, 7, 8, 9, 11}. 11.3. Consider

τ : SL F3+ 3 ) −→ 2 (F ab −→ (a2 + c 2 )(ab + cd). cd

Show that τ is a homomorphism of groups and that, if x ∈ F3 , then τ (u(x)) = x. Show also that N = Ker τ . 11.4. Let Q8 be the quaternionic group of order 8 and let K be a field of characteristic zero. Suppose that there exists an irreducible representation of Q8 over K of dimension 2. Show that −1 is a sum of two squares in K. (Hint: Set Q8 = {±1, ±I , ±J, ±K } (using standard notation) and fix an irreducible representation ρ : Q8 → GL2 (K). Show that may always sup we 0 −1 ab and that, if we set ρ (J) = , then the relations pose that ρ (I ) = 1 0 cd ρ (J)2 = ρ (I )2 and (ρ (I )ρ (J))2 = ρ (I )2 force d = −a, b = c and a2 + b2 = −1.) R EMARK – Denote by H the “standard” algebra of quaternions over a field K of characteristic different from 2. That is H = K ⊕ KI ⊕ KJ ⊕ KK , where I 2 = J 2 = K 2 = −1, IJ = −JI = K , JK = −KJ = I and KI = −IK = J. Then H is split over K (that is, isomorphic to Mat2 (K)) if and only if −1 is a sum of two squares in K. This result appears similar to that shown in Exercise 11.4 above. In fact, the two results are equivalent if we view the algebra H as a quotient of the group algebra KQ8 by the ideal generated by z + 1, where z is the unique non-trivial central element of Q8 . 11.5*. Let K be a field of characteristic different from 2. Show that SL2 (F5 ) admits an irreducible faithful representation of dimension 2 over K if and only if 5 is a square in K and −1 is a sum of two squares in K.

148

11 Special Cases

11.6 (McKay correspondence). Let Γ be a finite subgroup of SL2 (C) and χnat the character of the natural representation Γ → SL2 (G). Define a graph as follows: the set of vertices of G˜Γ is Irr Γ and two characters χ and χ are connected by an edge if χ is an irreducible factor of χ χnat . We denote by GΓ the full subgraph of G˜Γ obtained by removing the vertex 1Γ . ∗ ). (a) Show that χnat is real valued (use Exercise 1.13 to show that χnat = χnat (b) Deduce that, if χ , χ ∈ Irr Γ, then χ is an irreducible factor of χ χnat if and only if χ is an irreducible factor of χ χnat .

5 ) the subgroup of SL2 (C) given by the im 4 (respectively A Denote by A age of SL2 (F3 ) (respectively SL2 (F5 )) under the irreducible representation of 4 the dimension 2 with character R (i ∧ ) (respectively R+ (θ0 )). Denote by S 4 in SL2 (C). For background on Dynkin diagrams and root normaliser of A systems, see [Bou, Chapter VI]. 4. (c) Calculate the character table of S ˜ ˜ ) is a graph of type (d) Show that G˜A (respectively G 4 4 , respectively GA S 5 E˜6 (respectively E˜7 , respectively E˜8 ). Here, E˜? denotes the affine Dynkin diagram associated to the Dynkin diagram of type E? . (e) Show that GA 4 (respectively GS 4 , respectively GA 5 ) is a Dynkin diagram of type E6 (respectively E7 , respectively E8 ). (f) Generalise the above to all finite subgroups of SL2 (C). 11.7. Show that the group of outer automorphisms of A6 is isomorphic to Z/2Z × Z/2Z. (Hint: Use Example 11.3.8 and Exercise 1.14.)

Chapter 12

Deligne-Lusztig Theory: an Overview*

This chapter gives a very succinct overview of Deligne-Lusztig theory. We will recall some of the principal results of the theory (including the parametrisation of characters and partition into blocks) with the goal of connecting this general theory with what we have seen for SL2 (Fq ). This chapter requires some knowledge of algebraic groups, for which we refer the reader to [Bor] or [DiMi]. For more details on the subjects covered in this chapter the reader is referred to the books [Lu1], [Carter], [DiMi] or [CaEn]. We also recommend the magnificent and foundational article on the subject, written by Deligne and Lusztig in 1976 [DeLu]. Hypotheses and notation. In this and only this chapter we fix a reductive group G defined over a field F as well as a Frobenius endomorphism F : G → G which equips G with a rational structure over the finite field Fq . Denote by B an F -stable Borel subgroup of G, T an F -stable maximal torus of B and U the unipotent radical of B. The Weyl group of G relative to T will be denoted W ; by definition, W = NG (T). Furthermore, we return to the assumption that is a prime number different from p. The finite reductive group is the group GF of fixed points of F over G: GF = {g ∈ G | F (g ) = g }. The object of Deligne-Lusztig theory is the study of ordinary and modular representations (in unequal characteristic) of the finite group GF . Recall one of the fundamental theorems concerning algebraic groups defined over a finite field.

C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8_12, © Springer-Verlag London Limited 2011

149

150

12 Deligne-Lusztig Theory: an Overview*

Lang’s theorem. If H is a connected algebraic group and if F : H → H is a Frobenius endomorphism of H, then the morphism H → H, h → h−1 F (h) is an unramified Galois covering with group HF ; in particular, it is surjective.

12.1. Deligne-Lusztig Induction If w ∈ W , we fix a representative w˙ of w in NG (T) and denote by wF : T → T, t → w˙ F (t)w˙ −1 . Then wF is a Frobenius endomorphism T. The DeligneLusztig variety is the variety Y(w˙ ) = {g U ∈ G/U | g −1 F (g ) ∈ Uw˙ U}. The action of GF by left translations on G/U stabilises Y(w˙ ) as does the action of TwF by right translations on G/U (recall that T normalises U). Hence Y(w˙ ) comes equipped with the structure of a (GF , TwF )-variety and its cohomology groups Hci (Y(w˙ )) inherit the structure of a (K GF , K TwF )-bimodule. We define Rw : K0 (K TwF ) −→ [M ]K TwF −→ and

∗R w

: K0 (K GF ) −→ [M ]K GF −→

∑ (−1)

i

i 0

K0 (K GF ) i [Hc (Y(w˙ )) ⊗K TwF

M ]K GF

K0 (K TwF ) ∑ (−1) [Hci (Y(w˙ ))∗ ⊗K GF M ]K GF i

i 0

∗R w

The morphisms Rw and are called Deligne-Lusztig induction and restriction respectively. One may easily verify (thanks to Lang’s Theorem, see Exercise 12.1) that these morphisms depend only on w and not on the choice of a representative w˙ . They are adjoint (for the standard scalar product on the Grothendieck groups). If w ∈ W and if θ is a linear character of TwF , the (virtual) character Rw (θ ) of GF is called a Deligne-Lusztig character. One of the first fundamental results of this theory is the Mackey formula [DeLu, Theorem 6.8]. Mackey formula. Let w and w be two elements of W and let θ and θ be two linear characters of TwF and Tw F respectively. Then (12.1.1)

Rw (θ ), Rw (θ ) GF = |{x ∈ W | xwF (x)−1 = w and θ = θ ◦ x}|.

R EMARK – Note that two Deligne-Lusztig characters can be orthogonal and yet have common irreducible factors. Indeed, such characters are virtual characters.

12.1 Deligne-Lusztig Induction

151

The next corollary follows from Mackey formula by computing the norm of the difference of the two virtual characters involved. Corollary 12.1.2. If x and w are two elements of W , then Rx −1 wF (x) = Rw ◦ x ∧ , where x ∧ : (Tx

−1 wF (x)

∼

)∧ → (TwF )∧ , θ → θ ◦ x −1 .

The second fundamental result is the following [DeLu, Corollary 7.7]. Theorem 12.1.3. If γ is an irreducible character of GF , then there exists w ∈ W and a linear character θ of TwF such that Rw (θ ), γ GF = 0. It follows that, in order to parametrise the irreducible characters of GF , it is “enough” to decompose the Deligne-Lusztig characters. This enormously difficult work was completed by Lusztig in 1984 (for groups with connected centre, which constitutes the largest part of the work). An important notion used to achieve this parametrisation is that of geometric conjugacy. In order to define this, let us introduce an integer n0 1 such that the Frobenius endomorphisms (w˙ F )n0 and F n0 of G agree for all w ∈ W and induce the identify on W (such an n0 always exists). Denote by Nw : T F t

n0

−→ −→ t

TwF wF t · · · (wF )n0 −1 t

the norm map. Let ∇(G, F ) denote the set of couples (w , θ ), where w ∈ W , and θ is a linear character of TwF . If (w , θ ) and (w , θ ) are two elements of ∇(G, F ), we say that (w , θ ) and (w , θ ) are geometrically conjugate if the characters n θ ◦ Nw and θ ◦ Nw of TF 0 are conjugate under W . If this is the case we write (w , θ ) ≈ (w , θ ). A geometric series is an equivalence class for the relation ≈. The proof of the following theorem may be found in [DeLu, Theorem 6.1]. Theorem 12.1.4. Let (w , θ ) and (w , θ ) be two elements of ∇(G, F ) such that Rw (θ ) and Rw (θ ) contain a common irreducible factor. Then (w , θ ) ≈ (w , θ ). If S is a geometric series, we set E (GF , S ) = {γ ∈ Irr GF | ∃ (w , θ ) ∈ S , Rw (θ ), γ GF = 0}. The set of characters E (GF , S ) is called a Lusztig series. Combining Theorems 12.1.3 and 12.1.4, we obtain (12.1.5)

Irr GF =

˙ S ∈∇(G,F )/≈

E (GF , S ).

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12 Deligne-Lusztig Theory: an Overview*

Once one has obtained this initial description, “all that remains to do” is to parametrise the characters in a given Lusztig series. This was completed by Lusztig in a number of very long articles and a book [Lu2]. If (w , θ ) ∈ ∇(G, F ), denote by W (w , θ ) the stabiliser, in the group W , of the n linear character θ ◦ Nw of TF 0 . Then W (w , θ ) is a wF -stable subgroup of W . If the centre of G is connected, then W (w , θ ) is a subgroup of W generated by reflections. Theorem 12.1.6 (Lusztig). Let S be a geometric series and let (w , θ ) ∈ S . If the centre of G is connected, then E (GF , S ) is in bijection with a subset which only depends on the pair (W (w , θ ), wF ), and not on the group G or the cardinality q of the finite field. E XAMPLE 12.1.7 – If W (w , θ ) = 1, then |E (GF , S )| = 1 and E (GF , S ) = {ε (w )Rw (θ )}, where ε (w ) is the sign of w . Via the bijection of Theorem 12.1.6, Lusztig gives formulas for the degrees of the characters, for their multiplicities in all Deligne-Lusztig characters etc. He also deduces an analogous result for groups with non-connected centre, but for this one needs further techniques (see [Lu3]) which we will not recall here. In the group G = SL2 (Fq ) the non-connectedness of the centre is responsible, for example, both for the decomposition of the characters R(α0 ) and R (θ0 ) and for the existence of quasi-isolated blocks. We call a unipotent character of GF any element of E (GF , S1 ), where S1 is the geometric series {(w , 1) | w ∈ W }. In this case W (w , 1) = W and we conclude from Theorem 12.1.6 that (12.1.8)

|E (GF , S1 )| only depends on the pair (W , F ), not on q.

E XAMPLE 12.1.9 – Let us suppose that G = SL2 (F) and that F : G → G is the natural split Frobenius endomorphism q q ab a b F = q q . c d cd Then GF is our favourite finite group G = SL2 (Fq ). Moreover, W = {1,¯s }, where ¯s denotes the class of s in NG (T)/T. We have TF = T

and

Y(1) = GF /UF = G /U

therefore Hc∗ (Y(1)) = K [GF /UF ] by Theorem A.2.1(c), which shows that the Deligne-Lusztig induction R1 is nothing but the Harish-Chandra induction R defined in Section 3.2.

12.1 Deligne-Lusztig Induction

153

Let us now turn to to the Deligne-Lusztig induction map associated to s. On the other hand, it is a well-known geometrical fact that the morphism of varieties 2 G/U −→ A (F) \ {(0, 0)} ab −→ (a, c) cd is an isomorphism. Via this isomorphism we have TsF μq+1

and

Y(s) Y,

the isomorphism being compatible with the actions of G , μq+1 and F . As a consequence, Rs = −R . The map R is therefore none other (up to a sign) than the Deligne-Lusztig induction associated to the non-trivial element of the Weyl group. The reader may verity that the above Mackey formula 12.1.1 is then a condensed version of the various Mackey formulas (3.2.4, 4.1.7 and 4.2.1) obtained for our group SL2 (Fq ). Moreover, E (G , S1 ) = {1, StG }, which allows one to verify 12.1.8. Furthermore, (1, 1) ≡ (s, 1), (1, α0 ) ≡ (s, θ0 ) and W if θ 2 = 1, W (w , θ ) = 1 if θ 2 = 1. One may then verify the above claims concerning Lusztig series. We finish this section by recalling some geometric properties of the varieties Y(w˙ ). Proposition 12.1.10. Let w ∈ W . Then: (a) The variety Y(w˙ ) is quasi-affine, smooth, and purely of dimension l(w ) = dim Bw B/B. (b) The group TwF acts freely on Y(w˙ ). (c) The stabilisers, in GF , of elements of Y(w˙ ) are p-groups. R EMARK – It is conjectured that Deligne-Lusztig varieties are always affine (and not only quasi-affine) but for the moment this is only known when q is large enough [DeLu, Theorem 9.7] and in certain particular cases. The smallest examples where it is not known if the varieties Y(w˙ ) are affine occur in the group of type G2 , when q = 2 and l(w ) = 3.

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12 Deligne-Lusztig Theory: an Overview*

12.2. Modular Representations Recall that is a prime number different from p. Because of its use of -adic cohomology and geometric methods, DeligneLusztig theory is particularly well adapted to the study of representations in unequal characteristic. Here we will recall some facts which illustrate this phenomenon.

12.2.1. Blocks If (w , θ ) ∈ ∇(G, F ), we denote by θ the -part of θ . A pair (w , θ ) is called -regular if θ = θ . We denote by ∇ (G, F ) the set of -regular elements of ∇(G, F ). If S is a geometric series, we denote by S = {(w , θ ) | (w , θ ) ∈ S }. Then S is also a geometric series. A geometric series S is called -regular if S = S . If S is an -regular geometric series, we denote by ˙

E (GF , S ) =

E (GF , S ).

S ∈∇(G,F )/≈ S =S

The following result is due to Broué and Michel [BrMi, Theorem 2.2]. Theorem 12.2.1 (Broué-Michel). If S is an -regular geometric series, then E (GF , S ) is a union of -blocks. In other words, if we set ()

eS =

∑

eχ ,

χ ∈E (GF ,S )

then Theorem 12.2.1 of Broué and Michel may be translated as (12.2.2)

()

eS ∈ OGF .

12.2 Modular Representations

155

12.2.2. Modular Deligne-Lusztig Induction If w ∈ W , we denote by Rw and ∗Rw the functors of Deligne-Lusztig induction and restriction: Rw : Db (OTwF ) −→ Db (OGF ) M −→ RΓc (Y(w˙ ), O) ⊗O TwF M Rw : Db (OGF ) −→ Db (OTwF ) M −→ RΓc (Y(w˙ ), O)∗ ⊗O GF M. These functors are well-defined and adjoint to one another. Indeed, the complex of bimodules RΓc (Y(w˙ ), O) is perfect as a left and right module by virtue of Proposition 12.1.10 and Theorem A.1.5. If Λ ∈ {K , O, k}, we denote by ΛRw and Λ ∗Rw the extension of scalars of the functors Rw and ∗Rw respectively. For example, the functor K Rw (respectively K ∗Rw ) induces the morphism Rw (respectively ∗Rw ) between Grothendieck groups. The Morita equivalence of Theorems 8.1.1 and 8.1.4 may be generalised to the case of arbitrary finite reductive groups in the following way [Bro, Theorem 3.3]. Theorem 12.2.3 (Broué). Let (w , θ ) ∈ ∇ (G, F ) such that W (w , θ ) = 1 and let S be the geometric series which contains it. Then the cohomology group l(w ) () () Hc (Y(w˙ ), O) induces a Morita equivalence between OTwF eθ and OGF eS . ()

In Theorem 12.2.3, eθ denotes the primitive central idempotent of OTwF ()

such that θ ∈ Irr K TwF eθ .

12.2.3. The Geometric Version of Broué’s Conjecture We will conclude this overview with a geometric version of Broué’s conjecture (the general conjecture is given in Appendix B). For this, we make the following hypotheses, which allow us to considerably simplify the exposition.

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12 Deligne-Lusztig Theory: an Overview*

Hypotheses and notation. In this and only this subsection we suppose that: (1) F acts trivially on W (if G is semi-simple, this is equivalent to saying that F is a split Frobenius endomorphism of G). (2) does not divide |W |. (3) Denote by d the order of q modulo : we suppose that d is a regular number for W in the sense of Springer [Spr, §4].

Broué’s conjecture (geometric version). There exists an element w ∈ W of order d and regular in the sense of Springer (see [Spr, §4]), as well as a complex of (OGF , ONGw˙ F (T))-bimodules Cw such that: GF ×(N w˙ F (T))opp

G (a) ResGF ×(TwF )opp )

Cw D RΓc (Y(w˙ ), O).

(b) Cw induces a Rickard equivalence between the principal blocks of GF and NGw˙ F (T).

R EMARK – Note that, under the previous assumptions, if S denotes a Sylow -subgroup of GF , then NGF (S) is isomorphic to NGw˙ F (T). This version has been shown in very few cases. The case where divides q − 1 was shown by Puig (see, for example, [CaEn, Theorem 23.12]): Theorem 12.2.4 (Puig). If d = 1 (that is to say if divides q − 1), then the geometric version of Broué’s conjecture is true. In this case the induced equivalence is even a Morita equivalence, because the Deligne-Lusztig variety in question is of dimension zero. The next result was proved by Rouquier and the author [BoRo, Theorem 4.6]. Theorem 12.2.5. If GF = GLn (Fq ), SLn (Fq ) or PGLn (Fq ) and if d = n, then the geometric version of Broué’s conjecture is true. E XAMPLE – In the case where GF = SL2 (Fq ), we have already seen Puig’s theorem (see Corollary 8.3.3) and Theorem 12.2.5 (see Corollary 8.3.7). Very recently, Olivier Dudas has shown in his thesis [Du] the final result of this chapter. Theorem 12.2.6 (Dudas). If G is of type Bn , Cn and d = 2n, then the geometric version of Broué’s conjecture is true.

12.2 Modular Representations

157

Exercises 12.1. Let w˙ and w ¨ be two representatives of the same element w ∈ W in ¨ ) are isomorphic. NG (T). Show that the (GF , TwF )-varieties Y(w˙ ) and Y(w Deduce that the morphisms Rw and ∗Rw (as well as the functors Rw and ∗R ) do not depend on a choice of representative w ˙ of w . (Hint: Use Lang’s w theorem.) 12.2. Verify the assertions of Example 12.1.9. Use the Lang map G → G, g → g −1 F (g ) and Lang’s theorem to show that Y/G A1 (F).

Appendix A

-Adic Cohomology

The étale topology, the fundamentals of which were developed in the Séminaire de Géométrie Algébrique du Bois-Marie in the 1960’s (see, for example, [SGA1], [SGA4], [SGA4 12 ]) allows one to associate functorially to any algebraic variety a complex and corresponding cohomology groups (with coefficients in Λ, which can be K , O or O/ln , for n 1) enjoying numerous remarkable properties. In this appendix we recall those properties which are essential to our goal of studying the representations of SL2 (Fq ) (functoriality, perfectness, exact sequences, bounds, Künneth formula, Poincaré duality, traces, the Lefschetz fixed-point theorem, . . . ). We refer the reader to [SGA4 12 ] for all results stated without reference.

A.1. Properties of the Complex* The complex of -adic cohomology inherits an action of the monoid of endomorphisms of the variety. Even better, it can be realised as a complex of modules for this monoid (see the works of Rickard [Ric3] and Rouquier [Rou3]). Therefore, in this appendix we work under the following hypotheses. Let V be a quasi-projective algebraic variety defined over F and let Γ be a monoid acting on V via endomorphisms. We also set Λ to be one of K , O or O/ln (for n 1). E XAMPLE – We can take V = Y (the Drinfeld curve) and Γ = G × (μq+1 F mon ). By [Rou3], there exists a bounded complex RΓc (V, Λ) of ΛΓ-modules whose cohomology groups, which we denote Hci (V, Λ), are, as ΛΓ-modules, the cohomology groups with compact support of the variety V (with coefficients in the constant sheaf Λ). To simplify notation we denote by Hci (V) the K Γ-module Hci (V, K ). C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8, © Springer-Verlag London Limited 2011

159

160

A -Adic Cohomology

Recall that (A.1.1)

RΓc (V, Λ) Λ ⊗LO RΓc (V, O).

Here, ⊗LO denotes the derived functor of the tensor product. In particular, as K is O-flat, (A.1.2)

RΓc (V) K ⊗O RΓc (V, O)

and (A.1.3)

Hci (V, K ) K ⊗O Hci (V, O).

Moreover, by [SGA4 12 , Arcata, III, §3], (A.1.4)

if V is a smooth curve, then Hci (V, O) is torsion-free.

Another essential property is given in the following proposition. Theorem A.1.5 (Rickard, Rouquier). Suppose that Γ is a finite group. Then: (a) RΓc (V, Λ) is homotopic to a bounded complex of ΛΓ-modules of finite type. (b) If the stabilisers of points of V are of order invertible in Λ, then RΓc (V, Λ) is homotopic to a bounded complex of projective ΛΓ-modules.

A.2. Properties of the Cohomology Groups A.2.1. General Properties We begin by recalling certain properties of cohomology groups with coefficients in the general ring Λ. Theorem A.2.1. Let d = dim V and let I (V) denote the set of irreducible components of V of dimension d. Then: (a) Hci (V, Λ) is a Λ-module of finite type. (b) Hci (V, Λ) = 0 if i < 0 or if i > 2d. If V is affine and purely of dimension d, then, moreover, Hci (V) = 0 if i < d. (c) If Γ is a group we have an isomorphism of ΛΓ-modules Hc2d (V) Λ[I (V)]. (d) If U is a Γ-stable open subvariety of V with closed complement Z, then we have a long exact sequence of ΛΓ-modules · · · −→ Hci (U, Λ) −→ Hci (V, Λ) −→ Hci (Z, Λ) −→ Hci+1 (U, Λ) −→ · · · (e) If Γ is contained in a connected algebraic group acting regularly on V, then Γ acts trivially on Hci (V, Λ).

A.2 Properties of the Cohomology Groups

Λ (f) Hci (Ad (F), Λ) = 0

161

if i = 2d, otherwise.

Proof. (a), (b), (c), (d) and (f) are shown in [SGA4 12 ]. (e) is shown in [DeLu, Proposition 6.4].

A.2.2. Cohomology with Coefficients in K We will now state some properties which hold when Λ = K . Künneth formula. If V is another variety on which Γ acts, then we have an isomorphism of K Γ-modules (A.2.2)

r

Hcr (V × V ) ⊕ Hci (V) ⊗K Hcr −i (V ) i=0

for all r 0. If Δ is a finite subgroup of Γ which is normalised by Γ (i.e. γ Δ = Δγ for all γ ∈ Γ), then the quotient monoid Γ/Δ acts on the quotient variety V/Δ and we have an isomorphism of Λ(Γ/Δ)-modules (A.2.3)

Hci (V/Δ) Hci (V)Δ .

Poincaré duality. If V is irreducible, projective and smooth of dimension d, then we have, for all i ∈ {0, 1, 2, ... , 2d}, a perfect Γ-equivariant duality (A.2.4)

Hci (V) × Hc2d−i (V) −→ Hc2d (V).

R EMARK – Note that, under the above hypotheses, dimK Hc2d (V) = 1 (see Theorem A.2.1 (c)). However, it is possible for Γ not to act trivially on Hc2d (V). For example, a Frobenius endomorphism over Fq acts as multiplication by q d (see Theorem A.2.7(b) below).

A.2.3. The Euler Characteristic It is often useful to consider the Euler characteristic of V. If one takes into account the action of the monoid Γ, the Euler characteristic becomes not just a number, but an element in the Grothendieck group K0 (K Γ) of the category of K Γ-modules of finite dimension over K defined by Hc∗ (V) =

∑ (−1)i [Hci (V)]Γ .

i 0

162

A -Adic Cohomology

We use the notation Hc∗ (V)Γ if we wish to emphasise the monoid Γ. If γ ∈ End(V) or Γ, we denote by Tr∗V (γ ) the alternating sum Tr∗V (γ ) =

∑ (−1)i

i 0

Tr(γ , Hci (V)).

The function Tr∗V : End(V) −→ K is called the Lefschetz character of V. Recall that, if γ is an automorphism of V of finite order, then (A.2.5)

Tr∗V (γ ) ∈ Z.

Theorem A.2.6. Let d = dim V. Then: (a) If U is an open Γ-stable subvariety of V and if Z denotes its closed complement, then Hc∗ (V) = Hc∗ (U) + Hc∗ (Z). In particular, Tr∗V = Tr∗U + Tr∗Z . (b) If Δ is a finite subgroup of Γ normalised by Γ, then Hc∗ (V/Δ)Γ/Δ = Hc∗ (V)Δ . (c) If V is an algebraic variety upon which Γ also acts, then Hc∗ (V ×V ) = Hc∗ (V)⊗ Hc∗ (V ). (d) If s and u are two invertible elements of Γ such that su = us, s is of order prime to p and u is of order a power of p, then Tr∗V (su) = Tr∗Vs (u). (e) If Γ is a finite group and S is a torus acting on V and commuting with the action of Γ, then Hc∗ (V) = Hc∗ (VS ). Proof. (a), (b) and (c) may be found in [SGA4 12 ], while (d) is shown in [DeLu, Theorem 3.2] and (e) is shown, for example, in [DiMi, Proposition 10.15].

A.2.4. Action of a Frobenius Endomorphism Suppose that V is defined over the finite field Fq , with associated Frobenius endomorphism F . Then F acts on the cohomology groups Hci (V). We assemble here some classical results concerning this action. Theorem A.2.7. We have: (a) Tr∗V (F ) = |VF | (Lefschetz fixed-point theorem). (b) If V is irreducible of dimension d, then F acts on Hc2d (V) as multiplication by qd . (c) The eigenvalues of F on Hci (V) are algebraic integers of the form ω q j/2 , where j is a natural number such that 0 j i, and ω is an algebraic number, all of whose complex conjugates are of norm 1. In particular, F is an automorphism of the K -vector space Hci (V). (d) If V is projective and smooth, then the eigenvalues of F on Hci (V) are algebraic integers, all of whose complex conjugates have norm q i/2 .

A.3 Examples

163

Proof. (a) and (b) are shown in [SGA4 12 ]. (c) and (d) have been shown twice by Deligne [De1], [De2]: these results constitute the last difficulty in resolving the celebrated Weil conjectures, which are analogues for algebraic varieties of the Riemann hypothesis for number fields.

A.3. Examples A.3.1. The Projective Line We identify A1 (F) with the open subvariety of P1 (F) equal to {[x; y ] ∈ P1 (F) | y = 0} and set ∞ = [1; 0]. By Theorem A.2.1(d), we have an exact sequence 0 −→ Hc0 (A1 (F), Λ) −→ Hc0 (P1 (F), Λ) −→ Hc0 (∞, Λ) −→ Hc1 (A1 (F), Λ) −→ Hc1 (P1 (F), Λ) −→ Hc1 (∞, Λ) −→ Hc2 (A1 (F), Λ) −→ Hc2 (P1 (F), Λ) −→ Hc2 (∞, Λ) −→ 0. Using Theorem A.2.1(f) for d = 0 and d = 1, we obtain an exact sequence 0 −→ 0 −→ Hc0 (P1 (F), Λ) −→ Λ −→ 0 −→ Hc1 (P1 (F), Λ) −→ 0 −→ Λ −→ Hc2 (P1 (F), Λ) −→ 0 −→ 0. We conclude: (A.3.1)

Hci (P1 (F), Λ)

Λ = 0

if i = 0 or 2, otherwise.

On the other hand, the connected group GL2 (F) acts regularly on P1 (F), therefore, by Theorem A.2.1(e), (A.3.2)

GL2 (F) acts trivially on Hci (P1 (F)).

To conclude, we can keep track of the action of the Frobenius endomorphism F : P1 (F) −→ P1 (F), [x; y ] → [x q ; y q ] in the above exact sequences. Using Theorem A.2.7(b), we obtain (A.3.3)

F = 1 on Hc0 (P1 (F))

and F = q on Hc2 (P1 (F)).

In particular, we have verified the Weil conjectures (Theorem A.2.7(c)) and the Lefschetz fixed-point theorem (Theorem A.2.7(a)) which says that

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A -Adic Cohomology

|P1 (Fq )| = q + 1, which is no surprise.

A.3.2. The One-Dimensional Torus Denote by U = A1 (F) \ {0}. By Theorem A.2.1(d), we have an exact sequence 0 −→ Hc0 (U, Λ) −→ Hc0 (A1 (F), Λ) −→ Hc0 (0, Λ) −→ Hc1 (U, Λ) −→ Hc1 (A1 (F), Λ) −→ Hc1 (0, Λ) −→ Hc2 (U, Λ) −→ Hc2 (A1 (F), Λ) −→ Hc2 (0, Λ) −→ 0. Using Theorem A.2.1(f) for d = 0 and d = 1, we therefore obtain an exact sequence 0 −→ Hc0 (U, Λ) −→ 0 −→ Λ −→ Hc1 (U, Λ) −→ 0 −→ 0 −→ Hc2 (U, Λ) −→ Λ −→ 0 −→ 0. We conclude: (A.3.4)

Hci (A1 (F) \ {0}, Λ) =

Λ 0

if i = 1 or 2, otherwise.

On the other hand, the connected group F× acts regularly (by multiplication) on A1 (F) \ {0}, therefore, by Theorem A.2.1(e), (A.3.5)

F× acts trivially on Hci (A1 (F) \ {0}).

To conclude, we can keep track of the action of the Frobenius endomorphism F : A1 (F) −→ A1 (F), x → x q in the above exact sequence. Using Theorem A.2.7(b), we obtain (A.3.6)

F = 1 on Hc1 (A1 (F) \ {0})

and

F = q on Hc2 (A1 (F) \ {0}).

The Lefschetz fixed-point theorem (Theorem A.2.7(a)) tells us that |A1 (Fq ) \ {0}| = q − 1, which is again no surprise.

A.3 Examples

165

Exercises A.1. Determine the cohomology of Pn (F). A.2. Let a1 ,. . . , an be pairwise distinct points of A1 (F). Calculate the cohomology of A1 (F) \ {a1 , ... , an }. A.3. In statement (c) of Theorem A.2.7, the algebraic number ω is not necessarily an algebraic integer. This may occur and when it does it considerably restricts the values of ω . Indeed, show that an algebraic integer (over Z) all of whose complex conjugates are of norm 1 is a root of unity.

Appendix B

Block Theory

Fix a finite group Γ. In this appendix we recall the “essential” results (that is, essential for our purposes) of block theory, where the object of study is the representations of the algebras OΓ and kΓ. As in the rest of this book we assume that the algebras K Γ and kΓ are split for all groups Γ met in this appendix. We denote by OΓ → kΓ, a → a the reduction modulo l, and extend this notation to O-modules. For further details and developments the reader may consult one of the many references on this subject: [Alp], [CuRe, Chapters 2 and 7], [Isa, Chapter 15], [NaTs], [Ser, Part 3], [The].

B.1. Definition Set Λ = O or Λ = k. A Λ-bloc of Γ is an indecomposable direct summand of the (ΛΓ, ΛΓ)-bimodule ΛΓ. If we decompose the (ΛΓ, ΛΓ)-bimodule ΛΓ as a direct sum of indecomposable factors in two different ways ΛΓ = A1 ⊕ · · · ⊕ Ar = A 1 ⊕ · · · ⊕ A s , and write 1 = e1 + · · · + er = e1 + · · · + es , where ei ∈ Ai and ej ∈ A j . Then, the next proposition follows from [The, Corollary 4.2]. Proposition B.1.1. With the above notation we have: (a) r = s and there exists a permutation σ of {1, 2, ... , r } such that A i = Aσ (i) for all i (and therefore ei = eσ (i) ). (b) ei is a primitive central idempotent of ΛΓ and Ai = ΛΓei ; furthermore, Ai is a Λ-algebra with identity ei . (c) If i = j, then ei ej = ej ei = 0. (d) The Λ-algebras ΛΓ and A1 × · · · × Ar are isomorphic.

C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8, © Springer-Verlag London Limited 2011

167

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B Block Theory

The previous proposition shows that the set of Λ-blocks of Γ is welldefined. The following proposition (which follows from results about lifting idempotents [The, Theorems 3.1 and 3.2]) shows that it does not depend too much on Λ. Proposition B.1.2. If A is an O-block of Γ, then A is a k-block of Γ. If e is a primitive central idempotent of OΓ, then e is a primitive central idempotent of kΓ. The reduction modulo l induces a bijection between the O-blocks and k-blocks of Γ. It also induces a bijection between the primitive central idempotents of OΓ and of kΓ. The isomorphism of O-algebras OΓ A1 × · · · × Ar of Proposition B.1.1 induces an isomorphism K Γ KA1 × · · · × KAr and hence induces a partition (B.1.3)

Irr K Γ = Irr(KA1 ) ∪˙ · · · ∪˙ Irr(KAr ).

We call an -block of Γ a subset of Irr K Γ of the form Irr(KA), where A is an O-block of Γ. If χ is an irreducible character of Γ and if a ∈ Z(OΓ), we denote by ωχ (a) the scalar by which a acts on an irreducible representation affording χ as character (by virtue of Schur’s lemma). Then ωχ : Z(OΓ) → O is a morphism of O-algebras which will be called the central character associated ˆ = ∑γ ∈X γ ∈ OΓ. Recall that (Cl to χ . If X is a subset of Γ, we set X Γ (γ ))γ ∈[Γ/∼] is a O-basis of Z(OΓ). Moreover, recall [Isa, Theorem 3.7] that (B.1.4)

χ (γ ) | ClΓ (γ )| ∈ O. ωχ (Cl Γ (γ )) = χ (1)

The following proposition gives a characterisation of the partition into -blocks of Irr K Γ using central characters (see, for example, [Isa, Theorem 15.18]). Proposition B.1.5. Let χ and χ be two irreducible characters of Γ. Then the following are equivalent: (1) χ and χ are in the same -block. (2) ωχ (a) ≡ ωχ (a) mod l for all a ∈ Z(OΓ). χ (γ ) | ClΓ (γ )| χ (γ ) | ClΓ (γ )| ≡ mod l for all γ ∈ Γ. (3) χ (1) χ (1)

B.2. Brauer Correspondents B.2.1. Brauer’s Theorems If D is a subgroup of Γ, we define

B.2 Brauer Correspondents

BrD :

169

Z(kΓ) −→ Z(kNΓ (D)) ∑γ ∈Γ aγ γ −→ ∑γ ∈CΓ (D) aγ γ .

Then [Isa, Lemma 15.32] we have the following. Proposition B.2.1. If D is an -subgroup of Γ, then BrD is a morphism of kalgebras. When D is an -subgroup of Γ, BrD is called the Brauer morphism. If A is an O-block, we call the defect group of A any -subgroup D of Γ such that BrD (A) = 0 and which is maximal with this property. Then [Isa, Lemma 15.33]. Proposition B.2.2. If D and D are two defect groups of an O-block of Γ, then D and D are conjugate in Γ. Proposition B.2.2 justifies the abuse of language committed by referring to the defect group of A. The following proposition relates the -valuation of the degree of a character to that of the order of the defect group of the corresponding -block [Isa, Theorem 15.41]. Proposition B.2.3. If D is the defect group of A and if χ ∈ Irr(KA), then |D| · χ (1) ∈ O. |Γ| Moreover, there exists at least one character χ ∈ Irr(KA) such that |D| · χ (1) ∈ O ×. |Γ| E XAMPLE B.2.4 – Let χ an irreducible character of Γ such that |Γ|/χ (1) ∈ O × . Then eχ ∈ Z(OΓ) and therefore eχ is a primitive central idempotent of OΓ (because it is already primitive in K Γ). As a consequence, χ is alone in its -block. Moreover, Proposition B.2.3 shows that the defect group of OΓeχ is trivial. For a proof of the following theorem the reader is referred to [The, Corollary 37.13] or [Isa, Theorem 15.45]. Brauer’s first main theorem. Let D be an -subgroup of Γ. The map BrD induces a bijection between the set of O-blocks of Γ with defect group D and the set of Oblocks of NΓ (D) with defect group D. If A is an O-block of Γ with defect group D and if A is the O-block of NΓ (D) associated to A by the bijection of Brauer’s first main theorem, we say that A is the Brauer correspondent of A. It is characterised as follows: A is

the unique O-block of NΓ (D) such that BrD (A)A = 0. If we denote by e (respectively e ) the primitive central idempotent of OΓ (respectively ONΓ (D))

170

B Block Theory

such that A = OΓe (respectively A = ONΓ (D)e ), then e is characterised by the property that BrD (e) = e . We call the principal block of Γ the unique O-block A of Γ such that 1Γ ∈ Irr(KA). We denote the principal block of Γ by B0 (Γ). Brauer’s third main theorem. Let S be a Sylow -subgroup of Γ. Then S is the defect group of B0 (Γ) and B0 (NΓ (S)) is the Brauer correspondent of B0 (Γ).

B.2.2. Conjectures One of the central problems in block theory is to relate the representations in an O-block to those of its Brauer correspondent. There exist diverse conjectures in this direction. We will be content state the following. McKay conjecture (global). Denote by Irr (Γ) the set of characters of Γ of degree prime to and let S be a Sylow -subgroup of Γ. Then | Irr (K Γ)| = | Irr K NΓ (S) |. In case the defect group is abelian, Broué proposed the following conjecture. It is of a much more structural nature, but there does not exist a version if the defect group is not abelian. Broué’s conjecture. Let A be a block of Γ with defect group D and let A be its Brauer correspondent. If D is abelian, then the derived categories Db (A) and Db (A ) are equivalent as triangulated categories. This is not the place to enter into an exhaustive description of the numerous variants and refinements of these conjectures. We remark however that if the Sylow -subgroup of Γ is abelian, then Broué’s conjecture implies McKay’s conjecture. Note also that these conjectures (and their variants) have been verified in a very large number of cases.

B.2.3. Equivalences of Categories: Methods In order to obtain Morita equivalences or derived equivalences for our group SL2 (Fq ) we will make use of some general results. Fix A and A as two sums of O-blocks of finite groups Γ and Γ . A fundamental property of blocks of finite groups is that these algebras are symmetric [The, §6]. This fact will considerably simplify our work in what follows. Morita equivalences. Fix an (A, A )-bimodule M. We will recall some criteria used to verify that the functor M ⊗A − : A −mod −→ A−mod is an equiva-

B.2 Brauer Correspondents

171

lence of categories (then called a Morita equivalence between A and A ). The following theorem is shown in [Bro, Theorem 0.2]. Theorem B.2.5 (Broué). Suppose that the bimodule M is projective both as a left A-module and as a right A -module. Then the following properties are equivalent: (1) The functor M ⊗A − : A −mod −→ A−mod is a Morita equivalence. (2) The functor KM ⊗KA − : KA −mod −→ KA−mod is a Morita equivalence. (3) Every irreducible character of KA is an irreducible component of the KG -module KM, and the map Irr KA → Irr KA, [V ]KA → [KM ⊗KA V ]KA is well-defined and bijective. (4) Every irreducible character of KA is a factor of KM and the natural map KA → EndKA (KM)opp is an isomorphism. (5) Every irreducible character of KA is a factor of KM and the natural map A → EndA (M)opp is an isomorphism. Rickard equivalences. Let C be a complex of (A, A )-bimodules. We recall some criteria which may be used to verify that the functor C ⊗LA − : Db (A ) −→ Db (A) is an equivalence of categories (then called a Rickard equivalence between A and A ). Recall first that, if C = M[i] (that is to say, the complex with one nonzero term M in degree −i) and if M induces a Morita equivalence between A and A , then C induces a Rickard equivalence between A and A . We denote by H • (C ) the (A, A )-bimodule ⊕i∈Z H i (C ). Note that, as K is O-flat, we have H • (K C ) = KH • (C ). The following theorem is part of the folklore of the subject, but we have not been able to find a satisfactory reference (note however, that the proof below is very strongly influenced by [Ric4, Theorem 2.1]). Theorem B.2.6. Suppose that C is perfect, both as a complex of left A-modules and of right A -modules, and that HomDb (A) (C , C [i]) = 0 for all i = 0. Then the following properties are equivalent: (1) The functor C ⊗A − is a Rickard equivalence between A and A . (2) All irreducible characters of KA are factors of H • (K C ) and the natural map A → EndDb (A) (C )opp is an isomorphism. Proof. Firstly, it is clear that (1) implies (2). It remains to show that (2) implies (1). Therefore suppose that all irreducible characters of KA occur as factors of H • (K C ) and that the natural map A → EndDb (A) (C )opp is an isomorphism. First step: reduction to the case where A is an O-block. Let e be the central idempotent of OΓ such that A = OΓe and let e = e1 +· · ·+en be a decomposition of

172

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e into a sum of primitive central idempotents. Set Ai = OΓei = Aei . Then the action of ei on C is an element of EndDb (A) (C ), therefore there exists an element ei of A such that the endomorphism induced by right multiplication by ei is equal (in the derived category) to that induced by left multiplication by ei . Moreover, ei is idempotent and central (because this endomorphism commutes with the action of A ). Let A i = A ei . Then n

A = ∏ Ai , i=1

n

A = ∏ A i

n

n

i=1

i=1

and C = ⊕ ei C = ⊕ C ei .

i=1

It is now sufficient to show that ei C = C ei induces a Rickard equivalence between Ai and A i . It is easy to verify that ei C satisfies the same hypotheses as C and so we can (and will) suppose that A is an O-block of OΓ. Second step: adjunction. Firstly, after replacing C by a homotopic complex, we may suppose that C is a complex of (A, A )-bimodules which are projective as left and right modules. As the algebras A and A are symmetric, we have isomorphisms C ∗ C HomCb (A) (C , A) C HomCb (A ) (C , A ). As a consequence, the functors C ⊗A − and C ∗ ⊗A − between the categories of complexes Cb (A) and Cb (A ) are left and right adjoint to each other. The unit and counit of these adjunctions induce natural morphisms α

β

A −→ C ⊗A C ∗ −→ A of complexes of (A, A)-bimodules. Tensoring with C we obtain natural morphisms of complexes of (A, A )-bimodules α

β

C −→ C ⊗A C ∗ ⊗A C −→ C . It follows from general properties of adjoint functors that α (respectively β ) is a split injection (respectively split surjection) [McL, Theorem IV.1.1]. We may therefore write C ⊗A C ∗ ⊗A C C ⊕C0 , where C0 is a complex of (A, A )bimodules projective as left and right modules. By assumption, the natural morphism A → C ∗ ⊗A C is an isomorphism in the derived category (that is a quasi-isomorphism) and therefore C ⊗A C ∗ ⊗A C D C . As a consequence, C0 is acyclic and α and β are quasi-isomorphisms. In particular, β ◦ α is a quasi-isomorphism. On the other hand, β ◦ α is an endomorphism of the (A, A)-bimodule A and therefore there exists an element z in the centre Z(A) of A such that β ◦ α (a) = za for all a ∈ A. Suppose that β ◦ α is not an isomorphism. Then z is an element in the radical of Z(A) (as A is a block, Z(A) is a local ring). In this case β ◦ α cannot be a quasi-isomorphism (as C is non-zero by hypothesis). We have therefore shown that β ◦ α is an isomorphism.

B.3 Decomposition Matrices

173

Third step: conclusion. We may therefore write C ⊗A C ∗ = A ⊕ C1 , where C1 is a complex of (A, A)-bimodules, projective as left and right modules. Now, we have an isomorphism (in the derived category) C ⊗A C ∗ ⊗A C ⊗A C ∗ D C ⊗A A ⊗A C ∗ D C ⊗A C ∗ . Therefore

A D A ⊕ C1 ⊕ C1 ⊕ C1 ⊗A C1 .

Therefore C1 is acyclic, which shows that C ⊗A C ∗ D A and completes the proof of the theorem.

R EMARK B.2.7 – If C induces a Rickard equivalence between A and A , then: (a) (b) (c) (d)

The complex kC induces a Rickard equivalence between kA and kA . The complex K C induces a Rickard equivalence between KA and KA . The centres of A and A are isomorphic. | Irr KA| = | Irr KA | and | Irr kA| = | Irr kA |.

B.3. Decomposition Matrices We begin with a classical result in representation theory [CuRe, Propositions 23.16 and 16.16]. Proposition B.3.1. Let R be a principal ideal domain and A an R-algebra which is of finite type and free as an R-module. Denote by K the field of fractions of R. Let V be a KA-module of finite type. Then there exists an A-stable R-lattice M of V (so that V = K ⊗R V ). Moreover, for all maximal ideals m of R, the class [M/mM ]A/mA in the Grothendieck group K0 (A/mA) does not depend on the choice of the A-submodule M of V such that V = K ⊗R M. If A is an O-algebra which is free and of finite type as an O-module, Proposition B.3.1 shows that we can define a decomposition map decA : K0 (KA) −→ K0 (kA) as follows: if V is a KA-module, we may find in V an A-stable O-lattice M (so that V = KM) and we set decA ([V ]KA ) = [kM ]kA . We denote by Dec(A) the decomposition matrix of A, that is to say, the transpose of the matrix of the map decA in the canonical bases ([V ]KA )V ∈Irr KA and

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B Block Theory

([S ]kA )S∈Irr kA . Its rows are indexed by Irr KA, while its columns are indexed by Irr kA. In the case of group algebras and their blocks, we recall the following results (see [The, Theorems 42.3 and 42.8] or [Isa, Corollary 2.7 and 15.11]). Proposition B.3.2. We have: (a) | Irr K Γ| is equal to the number of conjugacy classes of elements of Γ. (b) | Irr kΓ| is equal to the number of conjugacy classes of -regular elements of Γ. ⊕ decA and Dec(OΓ) = ⊕ Dec(A). (c) decO Γ = A∈{O -blocks of Γ}

A∈{O -blocks of Γ}

E XAMPLE B.3.3 – If A is an O-block of Γ with trivial defect group and if V denotes the unique (up to isomorphism) simple KA-module (see Example B.2.4), then, for all A-stable O-lattices M of V , kM is the unique simple kA-module. As a consequence, Dec(A) = (1). In particular, if Γ is an -group, then all O-blocks of Γ have trivial defect group and hence the decomposition map decO Γ induces a bijection between Irr K Γ and Irr kΓ and the decomposition matrix Dec(OΓ) is the identity.

B.4. Brauer Trees* Hypothesis. We fix an O-block A of Γ with defect group D. In this section only we suppose that the group D is cyclic. For the definitions and results referred to without reference in this section, we refer the reader to [HiLu]. To each block of a group with cyclic defect group is associated a graph, called the Brauer tree (the graph is in fact a tree). We recall its construction. Let A denote the Brauer correspondent of A. The primitive central idem

potent e of kNΓ (D) such that A = kNΓ (D)e is in fact an element of kCΓ (D). We denote by d the number of primitive central idempotents of kCΓ (D) occurring in the decomposition of e . Recall that d divides p − 1. The set Irr KA decomposes as follows: Irr KA = {χ1 , ... , χd } ∪˙ {χλ | λ ∈ Λ}, where the χλ (λ ∈ Λ) are the exceptional characters of KA. We then set χexc = ∑λ ∈Λ χλ and VA = {χ1 , ... , χd , χexc }. The Brauer tree TA of A is then the pointed graph (that is a graph with a distinguished vertex, called the exceptional vertex) defined as follows:

B.4 Brauer Trees*

175

• The set of vertices of TA is VA . The exceptional vertex is χexc . • We join two distinct elements χ and χ in VA if χ + χ is the character of a projective indecomposable A-module. The following facts are classical: (a) TA is a connected tree. (b) If ψ is the character of an indecomposable projective A-module, then there exists two distinct elements χ and χ of VA such that ψ = χ + χ . In particular, the isomorphism classes of projective indecomposable Amodules are in bijection with the edges of the Brauer tree TA . In depictions of the Brauer trees given in this book the exceptional vertex will be represented by a round black circle yand the non-exceptional vertices will be represented by a round white circle i. R EMARK B.4.1 – The Brauer tree of a general finite group is equipped with an extra piece of data, its planar embedding, that is to say its representation in the plane. This data is fundamental, but requires much more information to obtain. As almost all the Brauer trees which occur in this book have only one planar embedding we will not preoccupy ourselves with this issue. A fundamental theorem in the theory of blocks of groups with cyclic defect group is the following. Theorem B.4.2 (Brauer). Let Γ and Γ be two finite groups and let A and A be two O-blocks with cyclic defect groups of the same order. Then A and A are Morita equivalent if and only if the Brauer trees TA and TA are isomorphic (as pointed graphs equipped with a planar embedding). E XAMPLE B.4.3 – Suppose that Γ = A5 , the alternating group of degree 5, and that = 5. Recall that Irr A5 = {χ1 , χ3 , χ3 , χ4 , χ5 }, where the index denotes the degree of the character. If A the principal O-block of Γ, then Irr KA = {χ1 , χ3 , χ3 , χ4 }, with defect group D =< (1, 2, 3, 4, 5) > a Sylow 5-subgroup of Γ and VA = {χ1 , χ4 , χ3 + χ3 }. The Brauer tree TA is therefore TA

y

i

i

On the other hand, the reader may verify that the Brauer tree of the principal block A of the normaliser of D in Γ (that is to say of the Brauer correspondent of A) is TA

i

y

i .

where Irr KA = {1, ε , ψ2 , ψ2 }, and 1 and ε (respectively ψ2 and ψ2 ) are the two characters of KA of degree 1 (respectively 2). The characters of degree 2 are the exceptional characters.

176

B Block Theory

By virtue of Brauer’s Theorem B.4.2, the blocks A and A are not Morita equivalent: in fact, TA and TA are not isomorphic as pointed graphs, even though they are as graphs. However Broué’s conjecture, as verified by Rickard [Ric2], asserts that the derived categories Db (A) and Db (A ) are equivalent as triangulated categories.

Appendix C

Review of Reflection Groups

Fix a field K of characteristic zero as well as a K-vector space V of finite dimension n. If g ∈ GLK (V ), we say that g is a reflection if Ker(g − IdV ) is of codimension 1. If Γ is a finite subgroup of GLK (V ), we denote by Ref(Γ) the set of its reflections. We say that Γ is a reflection group on V if it is generated by the reflections that it contains. The K-algebra of polynomial functions on V will be denoted K[V ]. This algebra inherits an action of GLK (V ). A K-algebra of finite type is called polynomial if it is isomorphic to an algebra K[V ], where V is a finite dimensional K-vector space. If Γ is a finite subgroup of GLn (K), the algebra of invariants K[V ]Γ is a K-algebra of finite type. The Shephard-Todd-Chevalley theorem characterises the reflection groups in terms of their algebras of invariants [Bou, chapitre V, §5]. Shephard-Todd-Chevalley theorem. Let Γ be a finite subgroup of GLK (V ). Then Γ is a reflection group if and only if the invariant algebra K[V ]Γ is polynomial. If this is the case, then there exists n homogeneous and algebraically independent polynomials p1 ,. . . , pn ∈ K[V ] such that K[V ]Γ = K[p1 , ... , pn ]. If di = deg(pi ) and if the pi are chosen so that d1 d2 · · · dn , then: n Tr(γ ) 1 1 = (here, X is an indeterminate). ∑ ∏ |Γ| γ ∈Γ det(1 − γ X ) i=1 1 − X di (b) The sequence (d1 , ... , dn ) does not depend on the choice of the pi . We call it the sequence of degrees of Γ. (c) |Γ| = d1 d2 · · · dn and |Ref(Γ)| = ∑ni=1 (di − 1). (d) If moreover Γ is an irreducible subgroup of GLK (V ), then it is absolutely irreducible, its centre is cyclic and |Z(Γ)| = pgcd(d1 , d2 , ... , dn ).

(a)

C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8, © Springer-Verlag London Limited 2011

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References [Ric4] [Rou1]

[Rou2]

[Rou3] [Sch] [Ser]

[SGA1] [SGA4] [SGA4 12 ] [ShTo] [Spr] [The]

[We] [Yo]

181 J. R ICKARD, Splendid equivalences: derived categories and permutation modules, Proc. L.M.S. 72 (1996), 331-358. R. R OUQUIER, Some examples of Rickard complexes, Proceedings of the Conference on representation theory of groups, algebras, orders, Ann. St. Univ. Ovidius Constantza 4 (1996), 169–173. R. R OUQUIER, The derived category of blocks with cyclic defect groups, dans Derived equivalences for group rings, 199–220, Lecture Notes in Math. 1685, Springer, Berlin, 1998. R. R OUQUIER, Complexes de chaînes étales et courbes de Deligne-Lusztig, J. Algebra 257 (2002), 482–508. I. S CHUR, Untersuchungen über die Darstellung des endlichen Gruppen durch geborchene lineare Substitutionen, J. für Math. 132 (1907). J.-P. S ERRE, Représentations linéaires des groupes finis, Troisième édition, Hermann, Paris, 1978, 182pp. English translation: Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer-Verlag, New YorkHeidelberg, 1977, x + 170pp. A. G ROTHENDIECK, Revêtements étales et groupe fondamental (SGA1), Lecture Notes in Mathematics 224, Springer, 1971. M. A RTIN, A. G ROTHENDIECK ET J.-L. V ERDIER, Théorie des topos et cohomologie étale des schémas (SGA4), Lecture Notes in Mathematics 269, 270, 305, Springer, 1972–1973. P. D ELIGNE, Cohomologie étale (SGA4 21 ), Lecture Notes in Mathematics 569, Springer, 1977. G.C. S HEPHARD & J.A. T ODD, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274–304. T.A. S PRINGER, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198. J. T HÉVENAZ, G -algebras and modular representation theory, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995, xxviii + 470pp. A. W EIL, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508. Y. Y OSHII, Broué’s conjecture for the nonprincipal block of SL(2, q) with full defect, J. Algebra 321 (2009), 2486–2499.

Index

Symbols 1Γ 28 (A, B)−bimod xxi xxi A+ xxi A× 124 A+ , A− 72 Aα , A θ A−mod xxi ¯ a 167 129 An B 3 ˜ B 33 73 bα , bθ B 110, 149 Bα , Bθ 72 xxi Cb (A), Cb (A, B) xxi CΓ (E ) C [i] xxii C C C , C K C , C D C 110 CarG xxi ClΓ (γ ) d 4 5 d D(Γ) xxi xxi Db (A), Db (A, B) D 22 Dec(A) 173 173 decA 129 det0 D,D∗ 94 122 e+ , e− 31 eU 28 eχ , eχΓ ()

154 xxi • Endgr Z G Hc (Y, Z ) eS γE

80

xxii

xxi [E / ∼] xxi |E | E (n) 118 E (GF , S ) 151 E (GF , S ) 154 F 15 113 Fp F(n) 116 F, Fp , Fq 3 29 FM , ∗ FM 112 F[G]ρ (U) F 75 FM , ∗ FM 29 G 3 xxii GLR (M) ˜ G 33 G 110, 149 GF 149 g(C) 22 G 22 xxii GLn (R) 161 Hc∗ (V) Hc• (Y, Z ) 80 171 Hc• (C ) Hci (V, Λ), Hci (V) I (n) 114 Irr A xxi Irr Γ 28 xxii I n , I2 K 28 28 Kα Kb (A), Kb (A, B) K0 (A) xxi K0 (G) 110 L(n) 113 120 Lq (n) 28

159

xxi

C. Bonnafé, Representations of SL2 (Fq ), Algebra and Applications 13, DOI 10.1007/978-0-85729-157-8, © Springer-Verlag London Limited 2011

183

184 l 61 113 (mi )i 0 xxii Matn (R) tM xxii xxii [M ]M xxi [M ], [M ]A N 4 5 N 3 N2 xxi NΓ (E ) 151 Nw xxi o(γ ) O 61 p 3 129 PGL2 (Fq ), PSL2 (Fq ) q 3 57 q0 xxi R M 124 R0 , Ri , Ri xxii RM ˜ R 33 45 R + (θ0 ), R − (θ0 ) 32 R+ (α0 ), R− (α0 ) 31 RK , ∗ RK 150 Rw , ∗ Rw . Ref(Γ) 177 28 regΓ 159 RΓc (V, Λ) R 77 R, ∗R 75 RK , ∗ RK 30 RΛ , ∗ RΛ 75 Rw , ∗ Rw 155 10 S , S xxii SLn (R) s 4 s 5 s ,˜ 32 StG S1 152 S 154 129 Sn T 3 71 T , T ˜ T 33 T 110, 149 TA 174 162 Tr∗V 51 Tr, Tr 3 Tr2 U 3 8 u+ , u− U 110, 149 u 4 113 V (i) , v (i)

Index 110 31 37 110 VA 174 W 149 W (w , θ ) 152 w˙ 150 (x, y ) 111 Y 15 Y(w˙ ) 150 Y 20 Z 6 ˜ Z 34 Z(A) xxi Z(Γ) xxi ≈ 151 ≡ 8 , Γ , 28 α0 32 αB 31 ˜B˜ α 33 χ∗ 28 χ+ 54 χα 65 χθ 66 χα+0 , χα−0 65 B χσB,τ , χ±,± 67 χθ+0 , χθ−0 66 Δ 22 Δ (n) 112 Δ(n) 110 Δq (n) 120 δ,δ∗ 94 Δ[q] 121 Δm,n 117 ε 110 εn 129 ˜ ε 112 [γ , γ ] xxi Γ∧ 28 γ : Y → A1 (F) 17 ∼ γ¯ : Y/G → A1 (F) 17 μn 3 μ (O ), μ (O ), μ ∞ (O ) 63 ∇(G, F ) 151 ∇ (G, F ) 154 ωχ 168 φq 130 φq 130 π : Y → P1 (F) \ P1 (Fq ) 19 ∼ π¯ : Y/μq+1 → P1 (F) \ P1 (Fq ) π0 : Y → P1 (F) 21 V2 VB Vθ [V ]G

19

Index

185 ∼

π¯0 : Y/μq+1 → P1 (F) 21 ψ+ 55 ρ1 46 ρ± 47 θ0 45 θ 154 54 Υ+ , Υ− υ : Y → A1 (F) \ {0} 18 ∼ υ¯ : Y/U → A1 (F) \ {0} 18 A Abhyankar’s conjecture

character 150 induction 37, 77, 150 restriction 150 theory 149 variety 150 Dickson invariants 24 Drinfeld curve 15 dual character 28 module 28 E

24

B Λ-block 167 -block 168 Brauer correspondence 168 correspondent 169 first main theorem 169 morphism 169 third main theorem 170 tree 174 Broué’s conjecture 170 Bruhat decomposition 4

Euler characteristic exceptional character 174 174 vertex

161

F formula Künneth Mackey

161 32, 40, 150

G 151 geometric conjugacy geometric series 151 reflection group 177

C

H

character Deligne-Lusztig 150 category derived xxii homotopy xxii of complexes xxii character central 168 cuspidal 31 dual 28 exceptional 174 Steinberg 32 unipotent 152 conjecture McKay 64 cuspidal, character 31

Harish-Chandra induction 30, 75 restriction 30, 75 Hurwitz bound 23 formula 22

D decomposition map 173 matrix 173 degrees 177 Deligne-Lusztig

I induction Deligne-Lusztig Harish-Chandra

37, 77, 150 30, 75

K Künneth formula

161

L -regular 154 Lang theorem 149 Lusztig series 151 M Mackey formula

32, 40, 150

186

Index

McKay conjecture module dual 28 rational 110 standard 110 Weyl 110 Morita equivalence

64, 170

170

N nilpotent block

module 110 representation 110 Raynaud’s theorem 24 reflection 177 restriction Deligne-Lusztig 150 Harish-Chandra 30, 75 Rickard equivalence 171 S

74

P polynomial algebra 177 principal block 170

semi-simple xxii standard module 110 U

Q quasi-isolated block

74

unipotent

R

W

rational

Weyl module

xxii

110