Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory

291 Pages·1994·5.627 MB·English

by  T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

Encyclopaedia of
Mathematical Sciences
Volume 55
Editor-in-Chief: R. V. Gamkrelidze
A. N. Parshin I.R. Shafarevich (Eds.)
Algebraic Geometry IV
Linear Algebraic Groups
Invariant Theory
Springer-Verlag Berlin Heidelberg GmbH
Consulting Editors of the Series:
A. A. Agrachev, A. A. Gonchar, E. F. Mishchenko,
N.M. Ostianu, Y.P. Sakharova, A.B. Zhishchenko
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, VoI. 55, Aigebraicheskaya geometriya 4
Publisher VINITI, Moscow 1989
Mathematics Subject Classification (1991 ):
20Cxx,1IE57,20Gxx
ISBN 978-3-642-08119-4
Library of Congress Cataloging-in-Publication Data
Algebraicheskaia geometriia 4. English
Algebraic geometry IV: linear algebraic groups, invariant theory / A. N. Parshin, 1. R. Shafarevich (eds.).
p. cm. - (Encyclopaedia of mathematical sciences; v. 55)
Includes bibliographical references and index.
ISBN 978-3-642-08119-4 ISBN 978-3-662-03073-8 (eBook)
DOI 10.1007/978-3-662-03073-8
1. Linear algebraic groups. 2. Invariants. 1. Parshin, A. N.
Il. Shafarevich, 1. R. (lgor' Rostislavovich). 1923 -
III. Title. IV. Title: Algebraic geometry 4. V. Title: Algebraic geometry four. VI. Series.
QA 179.A4413 1994 512'.2-dc20 93-13928
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© Springer-Verlag Berlin Heidelberg 1994
Originally published by Springer-Verlag Berlin Heidelberg New York in 1994
Softcover reprint of the hardcover 1 st edition 1994
Typesetting: Asco Trade Typesetting Ltd .. Hong Kong
SPIN: 10013453 41/3140 -5 432 I 0-Printed on acid-free paper
List of Editors, Authors and Translators
Editor-in-Chief
R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute,
ul. Vavilova 42, 117966 Moscow, Institute for Scientific Information (VINITI),
ul. Usievicha 20a, 125219 Moscow, Russia
Consulting Editors
A. N. Parshin, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow,
Russia
1. R. Shafarevich, Steklov Mathematical Institute, ul. Vavilova 42,
117966 Moscow, Russia
Authors
V. L. Popov, Chair of Algebra, Department of Mathematics, MIEM, Bolshoj
Vuzovskij Per. 3/12,109028 Moscow, Russia
T. A. Springer, Mathematisch Instituut, Rijksuniversiteit Utrecht, Postbus 80.010,
3508 T A Utrecht, The Netherlands
E. B. Vinberg, Chair of Algebra, Moscow State University, 119899 Moscow,
Russia
Translator of Part II
G. A. Kandall, 230 Hill Street, Hamden, CT 06514-1522, USA
Contents
I. Linear Algebraic Groups
T. A. Springer
1
II. Invariant Theory
V. L. Popov, E. B. Vinberg
123
Author Index
279
Subject Index
282
I. Linear Algebraic Groups
T.A. Springer
Contents
Introduction ................................................... 4
Historical Comments ........................................... 4
Chapter 1. Linear Algebraic Groups over an Algebraically
Closed Field ............................................... 7
§ 1. Recollections from Algebraic Geometry ........................ 7
1.1. Affine Varieties ......................................... 7
1.2. Morphisms ............................................ 7
1.3. Some Topological Properties ............................. 8
1.4. Tangent Spaces ................... . . . . . . . . . . . . . . . . . . . . . 8
1.5. Properties of Morphisms ................................ 8
1.6. Non-Affine Varieties .................................... 8
§ 2. Linear Algebraic Groups, Basic Definitions and Properties ....... 9
2.1. The Definition of a Linear Algebraic Group ................ 9
2.2. Some Basic Facts ....................................... 13
2.3. G-Spaces .............................................. 14
2.4. The Lie Algebra of an Algebraic Group .................... 16
2.5. Quotients .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
§ 3. Structural Properties of Linear Algebraic Groups ............... 19
3.1. Jordan Decomposition and Related Results ................ 20
3.2. Diagonalizable Groups and Tori .......................... 21
3.3. One-Dimensional Connected Groups ...................... 23
3.4. Connected Solvable Groups .............................. 23
3.5. Parabolic Subgroups and Borel Subgroups ................. 25
3.6. Radicals, Semi-simple and Reductive Groups ............... 28
§ 4. Reductive Groups .......................................... 28
4.1. Groups of Rank One .................................... 28
4.2. The Root Datum and the Root System .................... 30
4.3. Basic Properties of Reductive Groups ..................... 33
4.4. Existence and Uniqueness Theorems for Reductive Groups ... 37
4.5. Classification of Quasi-simple Linear Algebraic Groups ...... 39
4.6. Representation Theory .................................. 42
2 T.A. Springer
Chapter 2. Linear Algebraic Groups over Arbitrary Ground Fields 48
§ 1. Recollections from Algebraic Geometry ........................ 48
1.1. F-Structures on Affine Varieties ........................... 48
1.2. F-Structures on Arbitrary Varieties ........................ 49
1.3. Forms ................................................. 50
1.4. Restriction of the Ground Field ........................... 51
§ 2. F -Groups, Basic Properties ................................... 52
2.1. Generalities About F-Groups ............................. 52
2.2. Quotients .............................................. 54
2.3. Forms ................................................. 55
2.4. Restriction of the Ground Field ........................... 56
§3. Tori ....................................................... 57
3.1. F-Tori ................................................. 57
3.2. F-Tori in F-Groups ..................................... 59
3.3. Split Tori in F-Groups ................................... 60
§4. Solvable Groups ............................................ 61
4.1. Solvable Groups ........................................ 61
4.2. Sections ............................................... 62
4.3. Elementary Unipotent Groups ............................ 63
4.4. Properties of Split Solvable Groups ....................... 64
4.5. Basic Results About Solvable F-Groups .................... 64
§ 5. Reductive Groups ........................................... 65
5.1. Split Reductive Groups .................................. 65
5.2. Parabolic Subgroups .................................... 66
5.3. The Small Root System .................................. 68
5.4. The Groups G(F) ....................................... 72
5.5. The Spherical Tits Building of a Reductive F-Group ......... 74
§ 6. Classification of Reductive F-Groups .......................... 75
6.1. Isomorphism Theorem .................................. 75
6.2. Existence .............................................. 77
6.3. Representation Theory of F-Groups ....................... 84
Chapter 3. Special Fields ........................................ 86
§ 1. Lie Algebras of Algebraic Groups in Characteristic Zero ......... 86
1.1. Algebraic Subalgebras ................................... 86
§ 2. Algebraic Groups and Lie Groups ............................ 88
2.1. Locally Compact Fields .................................. 88
2.2. Real Lie Groups ........................................ 91
§ 3. Linear Algebraic Groups over Finite Fields ..................... 94
3.1. Lang's Theorem and its Consequences ..................... 94
3.2. Finite Groups of Lie Type ................................ 97
3.3. Representations of Finite Groups of Lie Type ............... 99
§4. Linear Algebraic Groups over Fields with a Valuation ........... 101
4.1. The Apartment and Affine Dynkin Diagram ................ 101
I. Linear Algebraic Groups 3
4.2. The Affine Building ..................................... 104
4.3. Tits System, Decompositions ............................ 107
4.4. Local Fields ........................................... 108
§ 5. Global Fields .............................................. 109
5.1. Adele Groups ......................................... 109
5.2. Reduction Theory ...................................... 112
5.3. Finiteness Results ...................................... 115
5.4. Galois Cohomology .................................... 118
References 118
4 T.A. Springer
In trod uction
A linear algebraic group over an algebraically closed field k is a subgroup
ofa group GLn(k) of invertible n x n-matrices with entries in k, whose elements
are precisely the solutions of a set of polynomial equations in the matrix coordi
nates. The present article contains a review of the theory of linear algebraic
groups.
To develop the theory one needs tools from algebraic geometry. The reader
is assumed to have some familiarity with that subject.
Chapter 1 of the article reviews the basic facts from the theory of linear
algebraic groups over an algebraically closed field k. This theory culminates in
a classification of simple linear algebraic groups. I have tried to explain carefully
the fundamental notions and results, to illustrate them with concrete examples,
and to give some idea of the methods of proof.
There are several monographs about the material of this chapter ([B2], [Hu],
[Sp3]), where the interested reader can find more details about this material.
Chapter 2 discusses the relative theory, where a field of definition comes
into play. This is, roughly, a subfield F of k such that the polynomial equations
of the first line can be taken to have coefficients in F. This relative theory is
required, for example, if one wishes to deal with arithmetical questions involv
ing algebraic groups.
At the moment there do not exist monographs covering this theory, which
makes it less accesible. I have tried to present a coherent picture, following the
same lines as in Chapter 1.
In Chapter 3 special features are discussed of the relative theory, for particu
lar fields of definition F, notably finite, local and global fields. The aim of the
chapter is to show how the theory of algebraic groups is used in questions about
such special fields.
There is a great abundance of material. Because of limitations of space I
have sometimes been quite sketchy.1
The references at the end of the article do not have the pretension of being
complete. But I hope that with the help of them a reader will be able to trace in
the literature further details, of he wishes to do so.
A reference in the article to I, 2.3.4 (resp. 2.3.4) refers to no. 2.3.4 of Chapter
1 (resp. of the same Chapter).
Historical Comments
By way of introduction to the subject of linear algebraic groups there follows
a brief review of anterior developments which have been incorporated, in some
1 This article was written in 1988. Today (in 1993) I would perhaps have written some parts differently.
But I have not tried to rewrite the article. I only made some necessary adjustments.