Mathematical Surveys

and Monographs

Volume 180

American Mathematical Society

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

Martin W. LiebeckGary M. Seitz

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

http://dx.doi.org/10.1090/surv/180

Mathematical Surveys

and Monographs

Volume 180

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

Martin W. Liebeck Gary M. Seitz

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Ralph L. Cohen, ChairJordan S. Ellenberg

Michael A. SingerBenjamin Sudakov

Michael I. Weinstein

2010 Mathematics Subject Classification. Primary 20G15, 20G40, 20G41, 20E45, 20E32,17B45, 17B08.

For additional information and updates on this book, visitwww.ams.org/bookpages/surv-180

Library of Congress Cataloging-in-Publication Data

Liebeck, M. W. (Martin W.), 1954–Unipotent and nilpotent classes in simple algebraic groups and lie algebras / Martin W. Liebeck,

Gary M. Seitz.p. cm. — (Mathematical surveys and monographs ; v. 180)

Includes bibliographical references and index.ISBN 978-0-8218-6920-8 (alk. paper)1. Linear algebraic groups. 2. Lie algebras. I. Seitz, Gary M., 1943– II. Title.

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10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12

For Ann and Sheila

Contents

Preface xi

Chapter 1. Introduction 1

Chapter 2. Preliminaries 92.1. Notation 92.2. Subgroups 132.3. Automorphisms and Lie algebras 152.4. Frobenius morphisms and the Lang-Steinberg theorem 192.5. Nilpotent and unipotent elements 202.6. Distinguished parabolic subgroups 242.7. Centralizers of nilpotent elements 282.8. Distinguished elements in characteristic zero 34

Chapter 3. Classical groups in good characteristic 393.1. Preliminary lemmas 403.2. Proof of Theorem 3.1 433.3. Additional information 49

Chapter 4. Classical groups in bad characteristic: statement of results 59

Chapter 5. Nilpotent elements: the symplectic and orthogonal cases, p = 2 655.1. Indecomposables 655.2. Distinguished nilpotent elements 675.3. Distinguished normal form 695.4. Centralizers of nilpotent elements I: connected centralizers 715.5. Centralizers of nilpotent elements II: component groups 785.6. Orthogonal groups of odd dimension 865.7. Splitting 895.8. Labellings of some nilpotent classes 90

Chapter 6. Unipotent elements in symplectic and orthogonal groups, p = 2 916.1. Indecomposables 916.2. Distinguished unipotent elements 926.3. A map from unipotents to nilpotents in SO(V ) 936.4. A lemma on representations 946.5. Centralizers of unipotents I: connected centralizers 966.6. Centralizers of unipotents II: component groups 1016.7. Splitting 1096.8. Orthogonal groups of odd dimension 1106.9. Labellings of some unipotent classes 110

vii

viii CONTENTS

Chapter 7. Finite classical groups 1137.1. Good characteristic 1137.2. Bad characteristic 116

Chapter 8. Tables of examples in low dimensions 119

Chapter 9. Exceptional groups: statement of results for nilpotent elements 129

Chapter 10. Parabolic subgroups and labellings 13310.1. T -labellings and associated parabolic subgroups 13310.2. Labellings of some distinguished classes in classical groups 136

Chapter 11. Reductive subgroups 139

Chapter 12. Annihilator spaces of nilpotent elements 15312.1. Lemmas on representation theory 15312.2. Annihilator spaces of distinguished nilpotent elements 15612.3. Further results 165

Chapter 13. Standard distinguished nilpotent elements 16913.1. Distinguished nilpotent elements corresponding to distinguished

parabolic subgroups 16913.2. Component groups 18113.3. Subgroups J,R 201

Chapter 14. Exceptional distinguished nilpotent elements 203

Chapter 15. Nilpotent classes and centralizers in E8 21915.1. Preliminary lemmas 22015.2. Proof of Theorem 15.1, I: strategy 22515.3. Proof of Theorem 15.1, II: calculation of the centralizers 22815.4. Proof of Theorem 15.1, III: completeness of the list 258

Chapter 16. Nilpotent elements in the other exceptional types 26316.1. The cases where (G0, p) 6= (F4, 2) or (G2, 3) 26316.2. The case (G0, p) = (G2, 3) 26716.3. The case (G0, p) = (F4, 2) 26816.4. Dual pairs 276

Chapter 17. Exceptional groups: statement of results for unipotent elements 281

Chapter 18. Corresponding unipotent and nilpotent elements 287

Chapter 19. Distinguished unipotent elements 29919.1. The proof of Theorem 19.1 30119.2. The proof of Theorem 19.2 310

Chapter 20. Non-distinguished unipotent classes 31720.1. The case G = E6, E7, E8 31720.2. The case G = F4, p 6= 2 32920.3. The case G = F4, p = 2 33020.4. The case G = G2 33220.5. Proofs of the results 17.1 - 17.10 332

CONTENTS ix

Chapter 21. Proofs of Theorems 1, 2 and Corollaries 3 - 8 341

Chapter 22. Tables of nilpotent and unipotent classes in the exceptionalgroups 351

22.1. Classes and centralizers in exceptional algebraic groups 35122.2. Unipotent classes and centralizers in finite exceptional groups 35122.3. The dual pairs J,R 352

Bibliography 373

Glossary of symbols 377

Index 379

Preface

This book concerns the theory of unipotent elements in simple algebraic groupsover algebraically closed or finite fields, and nilpotent elements in the correspond-ing simple Lie algebras. These topics have been an important area of study fordecades, with applications to representation theory, character theory, the subgroupstructure of algebraic groups and finite groups, and the classification of the finitesimple groups. Even detailed information on centralizers is important. For exam-ple, information regarding the component groups of centralizers is useful in studyingrepresentations of Weyl groups.

There is a great deal of literature on unipotent and nilpotent elements, andmany beautiful general results have been proved. In addition to the general theory,there are many situations where precise information on conjugacy classes is ofgreat importance, such as class representatives and precise centralizers. Here theliterature is less satisfactory.

More than anything else, our reason for writing this book is that we believethat the information on centralizers is of sufficient importance that it deservesa single source, where results are presented completely in all characteristics, andwith consistent notation. In particular the detailed tables of results for exceptionalalgebraic and finite groups in Chapter 22 should be easily understandable andusable by readers, and likewise tables for some low-dimensional classical groups inChapter 8.

This is our aim and our approach to this, while using ideas from the literature, isin many parts new. Our results go beyond what is currently known in several ways.For example, the literature on centralizers of unipotent and nilpotent elementsin classical groups and Lie algebras in characteristic 2 is not complete, and weobtain complete information. We establish a number of new structural results oncentralizers, their embeddings in certain parabolic subgroups, and how the reductivepart of the centralizer is embedded in the ambient group.

The book is divided into 22 chapters. The first is an introduction to the topicand overview of the results in the book, and the second contains a number ofbasic results on algebraic groups that will be used throughout; some of these arestandard, others less so, but proofs are provided in most cases. Our results forclassical groups are proved in Chapters 3–6. Chapter 3 concerns the case wherethe characteristic of the underlying field is “good” (meaning that it is not 2 forsymplectic and orthogonal groups), and the analysis is fairly short and elementary.This is not the case for characteristic 2, covered in Chapters 4,5 and 6. Here ourapproach is for the most part new, as are many of the results, and takes substantialeffort. In Chapter 7, these results are applied to give corresponding results on

xi

xii PREFACE

classes and centralizers in finite classical groups, and some tables illustrating ourresults for various classical groups of dimension up to 10 are given in Chapter 8.

The remainder of the book, Chapters 9–22, is devoted to the exceptional groupsG2, F4, E6, E7 and E8. A key feature of our approach is that we first focus on theclasses and centralizers of nilpotent elements, and then use these results to dealwith the unipotent elements. This approach has the advantage that our theory fornilpotent elements e has a number of structural features that are not present forunipotent elements, such as the existence of a naturally defined 1-dimensional torusacting on the 1-space spanned by e, and an associated parabolic subgroup, whichturns out to contain the centralizer of e. The main results for nilpotent elements arestated in Chapter 9, and proved in the following seven chapters. Unipotent elementsare then handled in Chapters 17–20. Finally, Chapter 21 contains proofs of someof our general results on the structure and embedding of centralizers, together withvarious corollaries of our main results; and Chapter 22 has detailed tables of classesand centralizers in the exceptional algebraic groups, and also in the associated finitegroups of Lie type.

It will be apparent even from this brief discussion that in this book we arefocussing almost exclusively on the classification and centralizer structure of unipo-tent and nilpotent classes. There are many other issues concerning these classeswhich are of great interest in algebraic group theory, algebraic geometry and rep-resentation theory. We shall not touch upon these subjects directly, although anumber of proofs do require a certain amount of representation theory.

This book does not contain an introduction to the theory of algebraic groups;neither does it contain definitions and basic properties of the simple groups. Never-theless, we have written it with the intention of being comprehensible to graduatestudents and researchers who have a basic knowledge of these topics.

We would like to thank Tim Burness for reading the manuscript and suggestingmany corrections, Ross Lawther for double-checking some of the calculations inChapter 13, and Donna Testerman and Bob Guralnick for helpful comments.

Martin Liebeck and Gary Seitz

Author addresses:

Department of Mathematics, Imperial College, London SW7 2AZ, England

email: m.liebeck@imperial.ac.uk

Department of Mathematics, University of Oregon, Eugene, Oregon 97403, USA

email: seitz@uoregon.edu

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Glossary of symbols

Ãi, subsystem Ai of short roots, 11

A�n(q), D�n(q), E

�6(q), 19

Altn, alternating group of degree n, 12

Aut(G), automorphism group of G, 17

BG, Borel subgroup, 11

C3(a1), 129, 269

CG(T, e) = CG(T ) ∩ CG(e), 5CV (T, e), 158

CV (e), annihilator space of e, 153

dτ , differential of τ , 41

D(m), indecomposable module, 86

Dn(ai), distinguished class in Dn, 58, 90,

110

Dih2n, dihedral group of order 2n, 12

e, nilpotent element, 3

e, u corresponding nilpotent and unipotentelements, 287

eα, root vector in L(G), 11

ec1...cr , notation for ec1α1+...+crαr , 11

eij..., notation for eαi+αj+..., 11

fα = e−α, 11G(q), finite group of Lie type, 1

Gσ , fixed point group of σ in G, 114

Gτ , fixed point group of τ , 9

hα(c), element of maximal torus TG, 11

Inndiag(G(q)), 352

Ji, Jordan block, 39

K, algebraically closed field, 1

L(G), Lie algebra of G, 1

L(G)(q), Lie algebra over Fq , 22L(G)i, L(Q)i, 136

L(G)≥i, L(Q)≥i, 136

L(Q)(i), 12

L(Q)k, 31

L(Q)≥k, 31

L(Q(≥i)), 12[m; l], a χ-function, 59

M1/M2/ · · · , notation for a module, 12P , parabolic subgroup, 4

P−, opposite parabolic, 11Pij..., parabolic subgroup, 25

Q(≥i)/Q(≥i+1), ith level of Q, 12Q≥2, 4Q≥k, 31, 136Ru(X), unipotent radical of X, 9

sx, image of s under x, 13

sα, reflection in the root α, 11SLn(K), 9

SOn(K), On(K), 9

S2V , symmetric square of V , 42Sx, fixed points of x in S, 13

Sp2n(K), 9Symn, symmetric group of degree n, 12

T , 1-dimensional torus, 4

T -labelling, 133T (G)1, tangent space at the identity, 35

TG, maximal torus of G, 9

TX(λ), tilting module of high weight λ, 11u, unipotent element, 4

Uα, root subgroup, 11

Ui, connected unipotent group ofdimension i, 9

Uc1...cr , notation for Uc1α1+...+crαr , 11

Uij..., notation for Uαi+αj+..., 11V ↓ Y , restriction of V to Y , 13V (m),W (m), indecomposables for u, 59, 91

V (m),W (m),Wl(m), indecomposables fore, 59, 65, 66

VX(λ) (or just λ), irreducible KX-moduleof high weight λ, 11

W (G), Weyl group of G, 11

WX(λ), Weyl module of high weight λ, 11X.Y , extension of X by Y , 12

Zp, cyclic group of order p, 5

∆-module, 142∆(λ;µ), 141

Π(G), system of fundamental roots, 9Σ(G), root system, 9αij..., notation for αi + αj + ..., 11χV , χ-function, 59κ, map from unipotents to nilpotents, 94

λi, fundamental dominant weight, 11

ω, semilinear map on L(G), 258σ, Frobenius morphism, 114, 258

σq , q-field morphism, 19∧2V , alternating square of V , 42

377

Index

annihilator space, 153

bad primes, 2

Bala-Carter theory, 3, 35, 39, 40

classical groups, 1, 9, 39

canonical form for nilpotents, 50, 60

canonical form for unipotents, 50, 61, 93

centralizers of nilpotents, 39, 71, 78

centralizers of unipotents, 39, 96, 101

distinguished nilpotent classes, 67, 136

distinguished nilpotent element, 60

distinguished unipotent classes, 43, 61,

92

dual pair J ,R, 46

finite, 113

in bad characteristic, 59

in good characteristic, 39

Lie algebras of, 16

natural module, 39

splitting of centralizers, 77

splitting of classes, 49, 109

tables of examples, 119

unipotent classes, 43, 91

component group, 9

correspondence e→ u, 287

dense double coset, 7

distinguished nilpotent element, 20

annihilator space of, 156

exceptional, 203

standard, 169

distinguished normal form, 39, 49, 60, 61,

69

distinguished parabolic subgroup, 24

in classical groups, 54

distinguished unipotent element, 20, 92,

299

exceptional, 299

standard, 299

dual pair J ,R, 6, 46, 48, 51, 201, 276, 352

dual root system, 18

Dynkin diagram, 1, 10

exceptional graph morphism, 19

exceptional groups, 129

distinguished unipotent classes, 299

dual pair J ,R, 220, 276

exceptional distinguished nilpotents, 203

Levi subgroups, 139

maximal rank subgroups, 139, 142, 143

nilpotent classes, 130, 263

nilpotent elements, 129

non-distinguished unipotent classes, 317

reductive subgroups, 139

regular nilpotent classes, 169

standard distinguished nilpotents, 169

tables of dual pairs, 352

tables of nilpotent classes, 351

tables of unipotent classes, 351

unipotent classes, 281

exceptional unipotent classes, 299

field morphism, 19

finite classical groups, 113

splitting of classes, 115

finite exceptional groups, 284

unipotent classes, 351

finite group of Lie type, 1, 19

twisted groups, 19

Frobenius morphism, 19, 21, 114, 258

q-field morphism, 19

fundamental dominant weight, 11

good characteristic, 2

good primes, 2

graph automorphism

exceptional, 19

standard, 17

height of a root, 12

high weight, 11

ideal, 311

labelled diagram, 133

labelling, 4, 32, 54, 133

Lang-Steinberg theorem, 3, 19, 114

level of Q, 12

level of a root, 12

Levi subgroup, 3, 11, 20

maximal rank subgroup, 11

minimal module, 12, 153

379

380 INDEX

natural module, 39

nilpotent classes

centralizers in E8, 219

distinguished, 20

in E8, 219

in classical groups, 39, 65

in exceptional groups, 219, 263

regular, 60, 90, 169

simply connected groups, 130

splitting of, 89

nilpotent element, 1

annihilator space of, 153

centralizer of, 39, 71, 78, 130

centralizers in E8, 219

distinguished, 20, 60, 169, 203

regular, 60, 90, 169

orders of unipotent elements, 45, 92, 284

parabolic subgroup, 4

distinguished, 24

labelling, 133

opposite, 11, 33

standard, 11

unipotent radical, 11

real element, 6

Ree groups, 19

regular nilpotent class, 60, 90, 169

regular unipotent class, 57, 61, 110, 288

shape of a root, 12

simple algebraic group, 1, 9

simply connected group, 40, 57, 130, 285

spin group, 57

spin module, 148

splitting of centralizers, 77, 283

splitting of classes, 49, 109, 115, 285

Springer correspondence, 21

variations of, 281

Springer map, 2, 21

standard basis, 16, 113

subsystem subgroup, 11

surj-inj property, 154

Suzuki groups, 19

tilting module, 11

unipotent classes

distinguished, 43, 92, 299

exceptional, 299

in classical groups, 43, 91

in exceptional groups, 281

non-distinguished, 317

numbers of, 284

regular, 57, 61, 110, 288

simply connected groups, 40, 57, 285

spin groups, 57

splitting of, 49, 109, 115, 285

standard distinguished, 299

unipotent element, 1

centralizer, 39, 62, 281

distinguished, 3, 20, 92, 299regular, 57, 61, 110, 288

standard distinguished, 299

unipotent radical, 5notation Ru(X), 9

untwisted diagonal subgroup, 226

Weyl group, 11

Weyl module, 11

SURV/180

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This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups.

The main focus is on obtaining full information on class representatives and central-izers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new—for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers.