UNIVERSITY LECTURE SERIES VOLUME 66

American Mathematical Society

Lectures on Chevalley GroupsRobert Steinberg

Lectures on Chevalley Groups

https://doi.org/10.1090//ulect/066

Lectures on Chevalley GroupsRobert Steinberg

Notes prepared by John Faulkner and Robert Wilson

UNIVERSITY LECTURE SERIES VOLUME 66

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Jordan S. EllenbergWilliam P. Minicozzi II (Chair)

Robert GuralnickTatiana Toro

2010 Mathematics Subject Classification. Primary 20G15; Secondary 14Lxx.

For additional information and updates on this book, visitwww.ams.org/bookpages/ulect-66

Library of Congress Cataloging-in-Publication Data

Names: Steinberg, Robert, 1922- | Steinberg, Robert, 1922- Works. 1997.Title: Lectures on Chevalley groups / Robert Steinberg ; notes prepared by John Faulkner and

Robert Wilson.Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Univer-

sity lecture series ; volume 66 | Part of: Robert Steinberg, collected papers (Providence, R.I. :American Mathematical Society, 1997). | Includes bibliographical references and index.

Identifiers: LCCN 2016042277 | ISBN 9781470431051 (alk. paper)Subjects: LCSH: Chevalley groups. | AMS: Group theory and generalizations – Linear algebraic

groups and related topics – Linear algebraic groups over arbitrary fields. msc | Algebraicgeometry – Algebraic groups – Algebraic groups. msc

Classification: LCC QA179 .S74 2016 | DDC 512/.5–dc23 LC record available at https://lccn.loc.gov/2016042277

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10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16

Contents

Foreword vii

Acknowledgement ix

Preface xi

Chapter 0. Introduction 1

Chapter 1. A basis for L 5

Chapter 2. A basis for U 11

Chapter 3. The Chevalley Groups 19

Chapter 4. Simplicity of G 33

Chapter 5. Chevalley Groups and Algebraic Groups 37

Chapter 6. Generators and Relations 43

Chapter 7. Central Extensions 47

Chapter 8. Variants of the Bruhat Lemma 61

Chapter 9. The Orders of the Finite Chevalley Groups 77

Chapter 10. Isomorphisms and Automorphisms 85

Chapter 11. Some Twisted Groups 99

Chapter 12. Representations 117

Chapter 13. Representations Continued 129

Chapter 14. Representations Completed 137

Appendix on Finite Reflection Groups 149I. Preliminaries 149II. The Function N 151III. A Fundamental Domain for W 152IV. Generators and relations for W 153V. Appendix 154

Bibliography 155

Index 159

v

Foreword

In 1997 the American Mathematical Society published Robert Steinberg, TheCollected Papers, which contains forty key publications selected by the author. Onenotable omission, however, was his Lectures on Chevalley Groups, which has onlyappeared in mimeograph form, printed by the Yale Mathematics Department. Itwas my uncle’s constant desire to publish a revised version of this work. As theauthor’s namesake, it is a privilege to assist in the completion of this posthumousedition: Robert Steinberg was a devoted uncle and source of inspiration to all ofhis nieces and nephews.

The current volume incorporates a new introductory chapter, corrections con-tributed by astute readers of the original edition, and changes that Robert Stein-berg penciled into his personal copies. Footnotes marked by “(�)” are Englishtranslations of footnotes from the 1975 Russian edition of the work (edited by A.A.Kirillov).

This edition would not be possible without the generous assistance of Christo-pher Drupieski, who painstakingly transcribed the original mimeographed editioninto LATEX, nor without the editorial contributions of my uncle’s dear colleague andfriend Raja V. S. Varadarajan. I would also like to acknowledge the editorial staffof the American Mathematical Society, and especially Sergei Gelfand for creatingthe bibliography and the index, and for translating the editorial footnotes from theRussian edition.

Since its original publication, several books have appeared that may help thereader approach these lectures, including Roger W. Carter, Simple Groups of LieType, Wiley, New York, 1989; James E. Humphreys, Introduction to Lie Algebrasand Representation Theory, Springer, New York, 1972; and V. S. Varadarajan, LieGroups, Lie Algebras, and Their Representations, Springer, New York, 1984.

Robert R. Snapp

vii

Acknowledgement

The notes are dedicated to my wife, Maria. They might also have been ded-icated to the Yale Mathematics Department typing staff whose uniformly highstandards will become apparent to anyone reading the notes. Needless to say, I amgreatly indebted to John Faulkner and Robert Wilson for writing up a major partof the notes. Finally, it is a pleasure to acknowledge the great stimulus derivedfrom my class and from many colleagues, too many to mention by name, duringmy stay at Yale.

Robert SteinbergJune 1968

ix

Preface

These notes presuppose the theory of complex semisimple Lie algebras throughthe classification, as may be found in the books of E. B. Dynkin, N. Jacobson, orJ.-P. Serre, or in the notes of Seminaire Sophus Lie. An appendix dealing with themost frequently needed results about finite reflection groups and root systems hasbeen included. The reader is advised to read this part first, which can be donerather quickly.

xi

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Index

algebra

commuting, 138

algebraic subset, 37

automorphism

Frobenius, 123

Borel subgroup, 38

Cartan decompositioin, 66

Cartan integers, 5

central extension, 47

universal, 47

Chevalley basis, 7

Chevalley group, 19

closed set of roots, 21

complete algebaic variety, 95

covering, 47, 54

universal, 54

degree, 13

diagonal automorphism, 92

field automorphism, 93

flag

incident, 27

standard, 26

graph isomorphism, 92

group

adjoint, 30

fundamental, 30

Janko, 114

Mathieu, 114

McLaughlin, 115

Ree, 105

semisimple, 38

simply connected, 54

Suzuki, 115

Suzuzki, 105

universal, 30

groups

sporadic, 114

highest weight, 15

highest weight vector, 15

homomorphism, 11

of algebraic groups, 37

ideal, 21

index, 100

isomorphism, 37

lattice, 15

matrix algebraic group, 37

maximal torus, 38

monomial, 13

radical, 38

representation

contragredient, 16

linear, 49

projective, 49

root, 5

global, 39

Schur multiplier, 49

simplex, 27

string of roots, 6

subgroup

Borel, 38

monomial, 26

parabolic, 34

unipotent, 21

subgroups

commensurable, 72

system

positive, 149

simple, 149

theorem

Birkhoff-Witt, 11

Lie–Kolchin, 95

universal enveloping algebra, 11

159

160 INDEX

weight, 14fundamental, 28

weight vector, 14Weyl group, 5Weyl’s formula, 126Whitehead group, 56

Zarisky topology, 37

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42 Holger Brenner, Jurgen Herzog, and Orlando Villamayor, Three Lectures onCommutative Algebra, 2008

41 James Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics, 2008

40 Vladimir Pestov, Dynamics of Infinite-dimensional Groups, 2006

39 Oscar Zariski, The Moduli Problem for Plane Branches, 2006

38 Lars V. Ahlfors, Lectures on Quasiconformal Mappings, Second Edition, 2006

37 Alexander Polishchuk and Leonid Positselski, Quadratic Algebras, 2005

36 Matilde Marcolli, Arithmetic Noncommutative Geometry, 2005

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33 Kristian Seip, Interpolation and Sampling in Spaces of Analytic Functions, 2004

32 Paul B. Larson, The Stationary Tower, 2004

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For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/ulectseries/.

Robert Steinberg’s Lectures on Chevalley Groups were delivered and written during the

author’s sabbatical visit to Yale University in the 1967–1968 academic year. The work

presents the status of the theory of Chevalley groups as it was in the mid-1960s. Much

of this material was instrumental in many areas of mathematics, in particular in the

theory of algebraic groups and in the subsequent classifi cation of fi nite groups. This

posthumous edition incorporates additions and corrections prepared by the author

during his retirement, including a new introductory chapter. A bibliography and editorial

notes have also been added.

This is a great unsurpassed introduction to the subject of Chevalley groups that influenced

generations of mathematicians. I would recommend it to anybody whose interests include

group theory.

—Efim Zelmanov, University of California, San Diego

Robert Steinberg’s lectures on Chevalley groups were given at Yale University in 1967. The

notes for the lectures contain a wonderful exposition of the work of Chevalley, as well as

important additions to that work due to Steinberg himself. The theory of Chevalley groups is

of central importance not only for group theory, but also for number theory and theoretical

physics, and is as relevant today as it was in 1967. The publication of these lecture notes in

book form is a very welcome addition to the literature.

—George Lusztig, Massachusetts Institute of Technology

Robert Steinberg gave a course at Yale University in 1967 and the mimeographed notes of that

course have been read by essentially anyone interested in Chevalley groups. In this course,

Steinberg presents the basic constructions of the Chevalley groups over arbitrary fields. He

also presents fundamental material about generators and relations for these groups and

automorphism groups. Twisted variations on the Chevalley groups are also introduced. There

are several chapters on the representation theory of the Chevalley groups (over an arbitrary

field) and for many of the finite twisted groups. Even 50 years later, this book is still one of the

best introductions to the theory of Chevalley groups and should be read by anyone interested

in the field.

—Robert Guralnick, University of Southern California

A Russian translation of this lecture course by Robert Steinberg was published in Russia more

than 40 years ago, but for some mysterious reason has never been published in the original

language. This book is very dear to me. It is not only an important advance in the theory of

algebraic groups, but it has also played a key role in more recent developments of the theory

of Kac-Moody groups. The very different approaches, one by Tits and another by Peterson and

myself, borrowed heavily from this remarkable book.

—Victor Kac, Massachusetts Institute of Technology

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