Arithmetic Groups

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789 James E. Humphreys

Arithmetic Groups

Springer-Verlag Berlin Heidelberg New York 1980

Author James E. Humphreys Department of Mathematics & Statistics G R C Tower University of Massachusetts Amherst, M A 01003 USA

A M S Subject Classifications (1980): 10 D07, 20 G 25, 20 G 30, 2 0 G 35, 20H05, 22E40 ISBN 3-540-09972-7 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09972-7 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in Publication Data. Humphreys, James E. Arithmetic groups. (Lecture notes in mathematics ; 789) Bibliography: p. Includes index. 1. Linear algebraic groups. 2. Lie groups. I. Title. 1LSeries: Lecture notes in mathematics (Berlin) ; 789. QA3.L28. no. 789. [QA171].510s [5t2'.2] 80-12922 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

PREFACE An arithmetic

group is (approximately)

Lie group defined by arithmetic GL(n,Z)

in GL(n,~),

SL(n,Z)

variety of contextss equivalence

in SL(n,~).

modular

of quadratic

functions,

forms,

these notes I have attempted of the underlying those

themes,

just mentioned.

algebraic

groups

properties

a discrete

analysis,

locally symmetric

illustrated

spaces,

integral

etc.

by specific

groups such as

While no special knowledge

of Lie groups

these particular

here is new.

But by adopting

approach I hope to make the literature

and Matsumoto

[1~)appear

somewhat less formidable.

Chapters i - III formulate

(notably Borel

and discrete

[1], cf. Weil [2~ and Goldstein

taken over local and global fields

Here the

e.g.,

arithmetic

mation. course

determined

interpretations.

These introductory

in number theory,

Chapters

Here one encounters "Siegel sets")

"reduction theory"

approximations

finite presentability system)

are not intended

so the proofs

for GL(n,Z)

of GL(n,Z)

linear and special linear

in the spirit of Borel [5].

and deduces,

or SL(n,E).

domains

The BN-pair

are used heavily here.

to adelic and p-adic groups.

the approach

[I~ to the Congruence

Chapter VI recounts

(in the special

(called

for example,

also a brief introduction

of Matsumoto

to be a first

of a few well known theorems

to fundamental

in GL(n,~)

and Iwasawa decomposition

Finally,

In

domains have nice

Another basic theme is strong approxi-

chapters

IV and V deal with general

emphasizing

domain corresponds

by an integral basis of K over Q.

of adeles or ideles such fundamental

are just sketched. groups,

O K of a number field K inside

of K