PROCEEDING S O F SYMPOSI A IN PUR E MATHEMATIC S

Volum e 9

Algebrai c Group s an d

Discontinuou s Subgroup s

Arman d Bore l an d Georg e D . Mostow , Editor s

AMERICA N MATHEMATICA L SOCIET Y

PROVIDENCE , RHOD E ISLAN D

http://dx.doi.org/10.1090/pspum/009

PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT THE UNIVERSITY OF COLORADO BOULDER, COLORADO JULY 5 - AUGUST 6, 1965

Prepared by the American Mathematical Society under National Science Foundation Grant GP-3983

and Office of Naval Research Grant Nonr(G)00055-65

International Standard Serial Number 0082-0717 International Standard Book Number 0-8218-1409-5

Library of Congress Catalog Number 66-18581

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Foreword

This book is an outgrowth of the twelfth Summer Mathematical Institute of the American Mathematical Society, which was devoted to Algebraic Groups and Discontinuous Subgroups. The Institute was held at the University of Colorado in Boulder from July 5 to August 6, 1965, and was financed by the National Science Foundation and the Office of Naval Research. The present volume consists of the Institute lecture notes, in part slightly revised, and of a few papers written somewhat later.

From the beginning, it was understood that a comprehensive exposition of the arithmetic aspects of algebraic groups should be a central aim of the Institute. In order to survey effectively the topics chosen for discussion, some important parts of the theory of Lie groups and algebraic groups had to be omitted, and the program was concentrated around five major themes: linear algebraic groups and arithmetic groups, adeles and arithmetic properties of algebraic groups, auto-morphic functions and spectral decomposition of L2-spaees of coset spaces, holomorphic automorphic functions on bounded symmetric domains and moduli problems, vector valued cohomology and deformation of discrete subgroups. The lectures fulfilled diverse needs, and accordingly the papers in this book are intended to serve various purposes: to supply background material, to present the current status of a topic, to describe some basic methods, to give an exposition of more or less known material for which there is no convenient reference, and to present new results. It is hoped that this collection of papers will facilitate access to the subject and foster further progress.

A. Borel G. D. Mostow

in

Contents

I. Algebraic Groups, Arithmetic Groups

Linear Algebraic Groups 3 BY ARMAND BOREL

Reduction Theory for Arithmetic Groups 20 BY ARMAND BOREL

Rationality Properties of Linear Algebraic Groups 26 BY ARMAND BOREL AND T. A. SPRINGER

Classification of Algebraic Semisimple Groups 33 BY J. TITS

p-adic Groups 63 BY FRANCOIS BRUHAT

Generalized Tits System (Bruhat Decomposition) on p-Adic Semisimple Groups 71

BY NAGAYOSHI IWAHORI

On Rational Points on Projective Varieties Defined Over a Complete Valuation Field 84

BY TSUNEO TAMAGAWA

Groups Over Z 90 BY BERTRAM KOSTANT

Subgroups of Finite Index in Certain Arithmetic Groups 99 BY H. MATSUMOTO

The Problem of the Maximality of Arithmetic Groups 104 BY NELO D. ALLAN

II. Arithmetic Properties of Algebraic Groups. Adele Groups

Ad&es 113 BY TSUNEO TAMAGAWA

On Tamagawa Numbers 122 BY TAKASHI ONO

The Siegel Formula for Orthogonal Groups. I 133 BY J. G. M. MARS

The Siegel Formula for Orthogonal Groups. II 138 BY J. G. M. MARS

The Volume of the Fundamental Domain for Some Arithmetical Subgroups of Chevalley Groups 143

BY R. P. LANGLANDS V

VI CONTENTS

Galois Cohomology of Linear Algebraic Groups 149 BY T. A. SPRINGER

Hasse Principle for Hl of Simply Connected Groups 159 BY MARTIN KNESER

Nonabelian H2 in Galois Cohomology 164 BY T. A. SPRINGER

Inseperable Galois Cohomology 183 BY PIERRE CARTIER

Strong Approximation 187 BY MARTIN KNESER

III. Automorphic Functions and Decomposition of L2(G/r)

Introduction to Automorphic Forms 199 BY ARMAND BOREL

The Decomposition of L2(G/T) for T = SL(2, Z) 211 BY R. GODEMENT

The Spectral Decomposition of Cusp-Forms 225 BY R. GODEMENT

Eisenstein Series 235 BY R. P. LANGLANDS

Dimension of Spaces of Automorphic Forms 253 BY R. P. LANGLANDS

Spherical Functions and Ramanujan Conjecture 258 BY ICHIRO SATAKE

Algebraic Curves Mod p and Arithmetic Groups 265 BY YASUTAKA IHARA

Discrete Subgroups of PL(2, kp) 272 BY YASUTAKA IHARA

IV. Bounded Symmetric Domains, Holomorphic Automorphic Forms, Moduli

On Compactifications of Orbit Spaces of Arithmetic Discontinuous Groups Acting on Bounded Symmetric Domains 281

BY WALTER L. BAILY, JR.

Fourier-Jacobi Series 296 BY WALTER L. BAILY, JR.

On the Desingularization of Satake Compactifications 301 BY JUN-ICHI IGUSA

Classical Theory of ^-functions 306 BY WALTER L. BAILY, JR.

Moduli of Abelian Varieties and Number Theory 312 BY GORO SHIMURA

CONTENTS Vii

Hecke's Polynomial as a Generalized Congruence Artin L-function 333 BY MICHIO KUGA

Fiber Varieties over a Symmetric Space Whose Fibers Are Abelian Varieties 338

BY MICHIO KUGA Families of Abelian Varieties 347

BY DAVID MUMFORD Symplectic Representations of Algebraic Groups 352

BY ICHIRO SATAKE The Modular Groups of Hilbert and Siegel 358

BY WILLIAM F. HAMMOND Quantum Mechanical Commutation Relations and Theta Functions 361

BY PIERRE CARTIER

V. Quotients of Symmetric Spaces. Deformations Cohomologies of Vector-values Forms on Compact, Locally Symmetric

Riemann Manifolds 387 BY SHINGO MURAKAMI

On Deformations of Lattices in Lie Groups 400 BY HOWARD GARLAND

On Deformations of Discrete Groups in the Noncompact Case 405 BY HOWARD GARLAND

On the Conjugacy of Subgroups of Semisimple Groups 413 BY G. D. MOSTOW

Index 421 Authors 425

Index

abelian variety, 312,313,315,316, 318,338,345

addle ring, 114 adelized

group, 115 variety, 114

adjoint group, 38,42 representation, 5

admissible pair, 72 PEL-structure, 319

affine algebraic group, 4,5 or extended Dynkin diagram, 34 Weyl group, 35,68,74

algebraic matrix group, 3 torus, 7 transformation group, 5

algebras L with a positive involu­tion, 319

anisotropic over K, 7 K-torus over, 7 kernels, 39 reductive groups, 13 reductive kernel, 39 semisimple kernel

antihermitian form, p-, 317 arithmetic groups, 20,99,104 automorphic

factor, 387 form, 200,201,204,326,391

automorphism group, 38 automorphy

canonical factor, 202,388,391,392 factor, 201-203

J3iV-pair, 71 Borel subgroup, 11 boundary components, 287 bounded

subgroup, 63 symmetric domain, 319,320

canonical automorphy factor, 202,388,391,392 numbering, 286

Cayley transforms, natural compacti-fications, 285

center, 38 character, 6 Chevalley groups, 143,148 class

field, 322 numbers, 188 of o*~lattices, 323

classical groups, 66 cocycle of r, 49 cohomologous, innerly, 49 cohomology

exact sequences in noncommu-tative, 153

Galois, 164,190,192 noncommutative, 151,164

commensurability subgroup of r, 20 compactification, 281,292,284 complex multiplication, 320,322 condition (Hi), 352 conjecture

Ramanujan -Peterson, 264 and spherical functions, 258

conjugacy, Frobenius classes, 267 connected, simply

algebraic group, 154 covering, 42 group, 68 universal covering, 38

cusp-forms, 222,226,230,233,331 spectral decomposition, 225

cylindrical sets, 296

d-cohomology, 398 groups, 394

of vector valued forms, 388 decomposition

Jordan, 6,26 spectral cusp-forms. 225

421

422 INDEX

deformation of Cr-structure, 407 desingularization, problem of, 301-303 dimension, fields of, 154 discrete

series, 262 subgroups, 26S, 272

divisor, basic, 314 duality, Tate-Nakayama, 127 Dynkin diagram, 33-35, 37,41,46,53

Eisenstein -Poincare series, 284,285,291 series, 146,206,208, 209,214,215, 221,235, 236,247,251, 252

-Siegel series, 140 element, i?-regular, 414 exact sequences, 173

in noncommutative cohomology, 153 Euler product, 327

fibre system of abelian varieties, 323 variety, 326,327

field of dimension, 154 moduli, 282,321

Fock representation, 381,363 Fourier-Jacobi series, 288, 296 Frobenius

conjugacy classes, 267 reciprocity law, 372

fundamental set, 21

r-automorphic right spherical function, 261

Galois cohomology, 164,190,192 generalized Tits system, 71, 74, 79

harmonic theory, 392 Hasse

principle for semisimple simply connected groups, 155

zeta-function, 312 Hecke

algebra, 79 ring, 78, 79, 326 operators, 326,327

Hermitian mapping

quasi-, 288 (V-),297

modular group, 283 Hilbert

modular forms, 358-360

group, 358 -Siegel modular group, 282 variety, 358

Hodge group of an abelian variety, 348 of a complex torus, 347

homogeneous, principal space of a linear algebraic group, 149

index, 39 inner and outer form, 39 innerly cohomologous, 49 involution

of an algebra, 313 opposition, 36

isogenous, (strictly) over k or k-isogenous, 34,42

isogeny, 34,313

Jordan decomposition, 6,26

k anistropic torus over, 7 or ^-isogenous, strictly over, 34,42 -primary, 353 -rank, 13 representation strongly rational over, 18

roots, 13 split torus, 7 Weyl group relative to, 13

L-functions, 333,334 Laplace transform, 212 lattice, 400 Lefschetz imbedding theorem, 307,308 Lie algebra of an algebraic group, 5 linear algebraic groups, 3 locally trivial, 389

map connecting, 171 G, 406,408

maximal family of polarized abelian varieties, 315

modular form, 308 function, 212 group, Hilbert-Siegel, 282 imbeddings, 358-360

moduli variety, 324

for PEL-structures, 321 modulus, supersingular, 265

INDEX

moronism of algebraic groups, 5

Nakayama-Tate duality, 127 natural compactification, 284, 285

and Cayley transforms, 285 Navier-Stokes equations, 16,45,47,49 nilpotent, 9

solvable algebraic group, 8 noncommutative

cohomology, 151,164 exact sequences in cohomology, 153

normal PEL-type, 321,322

1-cocycle of r, 49 opposition involution, 36 orbits, distinguished, 39 orthogonal group, special, 107

parabolic standard £-subgroups, 14 subgroup, 9,38,41

partial Cayley transformations, 286 desingularizations theorem, 302,303

PEL -structure, 347,318

admissible, 319 moduli for, 321

-type, 318 normal, 321,322

Peterson-Ramanujan conjecture, 264 Poincare

-Eisenstein series, 284,285,291 series, 204,205

polarization, 306,314 polarized

abelian variety, 314 normally, 282

positive, 313 involution, 316

principal homogeneous space of a linear algebraic group

matrices, 306 series, 211,261

quaternion algebra totally definite, 319 totally indefinite, 319

p-antihermitian form, 317 p-compactification, Satake, 413,415 radical (of an algebraic group), 9

Ramanujan, conjecture and spherical func­tions, 258

-Peterson conjecture, 264 rank

of an algebraic group, 10 M3

rational boundary components, 284,289,290 points, 184 representation, 6

reciprocity law, Frobenius, 372 reductive

algebraic group, 9 anistropic

groups, 13 kernel, 39

regular element, 30 relative Weyl group, 40 representation

adjoint, 5 induced, 370 infinitesimal, 370 rational, 6 special, 275 strongly rational over kt 18

Riemann conditions, 298,306 form, 313

rigid at °°, 407 rigidity, 418 ring of Thetanullwerthe, 309 root

M 3 system, 11,12

s-semiautomorphism, 176 Satake p-compactification, 413,415 schemes, semisimple group, 99 Schrodinger representation, 361 semiautomorphism, s-, 176 semisimple

algebraic group, 9 anistropic kernel, 39 element, 414,416 group schemes, 99

sequences, exact, 173 Siegel

domain, 22-24,296 -Eisenstein series, 140 formula, 133,136,141,143 -Hilbert modular group, 282 modular form, 358-360 sets, 297 variety, 358

424

simple, 313 almost, 33

singularity theorem, 302,303 solvable

nilpotent algebraic group, 8 split group, 9

spherical r-automorphic right function, 261 zonal function, 258

split groups, 66 fc-torus, 7 solvable group, 9

structure theorem, 272 supersingular, 266,268

modulus, 265 supplementary series, 261 Sylow subgroup, 76 symmetric domain, 339

bounded, 319,320 symplectic

pair, 339 representations, 358

Tamagawa number, 124,127,128,133 Tate, Nakayama-, duality, 127 tensor, twisted product, 79

^-function, 311 Thetanullwerthe, 308

ring of, 309 Tits system, 68,69,72

(BN-pair), 71 generalized, 71,74, 79 saturated, 72

torus, 68 algebraic, 7 anistropic over A, 7 split*-, 7

trace formula, 329 transform, Laplace, 212

INDEX

transformation group, algebraic, 5 truncated

Siegel sets, 290 domains, 297

twisted tensor product, 79 highly, 194

twisting, 151

U -function, 291 unbounded realizations, 285 unimodular group, 123 unipotent, 122

algebraic group, 8 radical (of an algebraic group), 9

Unitary, special group, 107

variety, adelized, 114 vertices, 34

weight group, 36 Weil's conjecture, 123,331 Weyl chamber, 12,14,203,249

chamber, 12,14,203,249 commutations relations, 366 group, 11,12,14,15,16,33,35,40, 68,71,73,194,235,414

affine, 35,68, 74 extended, 194 generalized, 71 relative, 40 relative to k, 13,416

Zariski dense, 11,26 density, 104,105

Zeta function, 329,331 Hasse, 312

Zonal spherical function. 258

Authors

This is a list of the authors of the various papers in the book, together with their permanent addresses.

Dr. Nelo D. Allan, Department of Mathematics, The University of Chicago, Chicago, Illinois, and Department of Mathematics, DePaul University, Chicago, Illinois.

Professor Walter L. Baily, Jr., Department of Mathematics, University of Chicago, Chicago, Illinois.

Professor Armand Borel, School of Mathematics, The Institute for Advanced Study, Princeton, New Jersey.

Professor Francois Bruhat, 80 Boulevard Pasteur, Paris XV°, France. Professor Pierre Cartier, Department of Mathematics, University of Strasbourg,

Strasbourg, France. Professor Howard Garland, School of Mathematics, The Institute for Advanced Study,

Princeton, New Jersey. Professor R. Godement, Institut Henri Poincare, Rue Pierre Curie, Paris, Ve, France. William F. Hammond, 5007 Falls Road Terrace, Baltimore, Maryland 21210. Professor Jun-ichi Igusa, 911 Breezewick Road, Towson, Maryland. Professor Yasutaka Ihara, School of Mathematics, The Institute for Advanced Study,

Princeton, New Jersey. Professor Nagayoshi Iwahori, Department of Mathematics, University of California,

Berkeley, California Martin L. Kneser, Merkelstrasse 39, 34 Gottingen, Germany. Professor Bertram Kostant, Department of Mathematics, Massachusetts Institute of

Technology, Cambridge, Massachusetts 02139. Professor Michio Kuga, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo, Japan. Dr. Robert P. Langlands, Fine Hall, Princeton University, Princeton, New Jersey. Professor J. G. M. Mars, Mathematique Institut, Boothstraat 17, Utrecht, The Netherlands. Professor Hideya Matsumoto, Institut Henri Poincare, 11, Rue Pierre Curie, Paris 5e, France. Professor George D. Mostow, Department of Mathematics, Yale University, New Haven,

Connecticut. Professor David Mumford, Department of Mathematics, Harvard University, Cambridge,

Massachusetts 02138. Professor Shingo Murakami, Department of Mathematics, Osaka University, Osaka, Japan. Professor Takashi Ono, Department of Mathematics, University of Pennsylvania,

Philadelphia, Pennsylvania 19104. Professor Ichiro Satake, Department of Mathematics, University of Chicago, Chicago,

Illinois 60637. 425

426 AUTHORS

Professor Goro Shimura, Department of Mathematics, Princeton University, Princeton, New Jersey.

Professor T. A. Springer, Mathematisch Institut, Boothstraat 17, Utrecht, The Netherlands. Professor Tsuneo Tamagawa, Department of Mathematics, Yale University, New

Haven, Connecticut. Professor Jacques L. Tits, Mathematisches Institut der UniversitAt, Wegelerstrasse 10,

53 Bonn, Germany.