Introduction to Harmonic Analysis on Reductive P-adic Groups: Based on lectures by Harish-Chandra at The Institute for Advanced Study, 1971-73

382 Pages·1979·18.894 MB·English

by  Allan G. Silberger

1 / / c
•) I_ 'jr,
INTRODUCTION TO
HARMONIC ANALYSIS ON
REDUCTIVE P-ADIC GROUPS
BY
ALLAN J. SILBERG ER
MATHEMATICAL NOTES
PRINCETON UNIVERSITY PRESS

INTRODUCTION TO HARMONIC ANALYSIS
ON REDUCTIVE P-ADIC GROUPS
Based on lectures by Barish-Chandra at
The Institute for Advanced Study, 1971-73
by Allan J. Silberger
Princeton University Press
and
University of Tokyo Press
Princaton, New Jersey
1979
Copyrightf} 1979 by Princeton University Press
All Rights Reserved
Published in Japan Exclusively
by University of Tokyo Press
in other parts of the world by
Princeton University Press
Printed in the United States of America
by Princeton University Press, Princeton, New Jersey
Library of Congress Cataloging in Publication Data will
be found on the last printed page of this book
FOREWORD
These notes represent the writer's attempt to organize and compre
hend the mathematics communicated to him by Barish-Chandra, both in
public lectures and private conversations, during the years 1971-1973. They
offer the reader an ab initio introduction to the theory of harmonic analysis
on reductive p-adic g;oups. Besides laying the foundations for a theory of
induced representations by presenting Jacquet' s theory, the Bruhat theory,
the theory of the constant term, and the Maass-Selberg relations, these
notes develop the theory of the Schwartz space on a p-adic group and the
theory of the Eisenstein integral in complete detail. They also give the con
struction of the algebras of wave packets as orthogonal components of Schwartz
space and prove Plancherel' s formula for induced series, Barish-Chandra's
commuting algebra theorem, and the sufficiency of the tempered spectrum
for rank one groups. Most notable among omissions from these notes is
Barish-Chandra's completeness theorem (i.e., that for arbitrary rank, the
tempered spectrum suffices) announced in (7f] and his theory of the characters
of admissible representations.
The reader will find a summary of a part of the contents of this
work given in Barish-Chandra's Williams College lectures ([7e]).
ACKNOWLEDGMENTS
The writer would like to express his great appreciation to Roger
Howe for his many perceptive comments regarding the manuscript. He
also wishes to thank Mark Krusemeyer, Paul Sally, and Nolan Wallach
for pointing out mistakes in the first draft of these notes. Evelyn Laurent
typed and several times corrected the manuscript. The writer owes her
a great debt for her patience and efficiency. The preparation of these notes
was supported by funds from the National Science Foundation.
September 1, 1978
ii
TABLE OF CONTENTS
Foreword
Acknowledgments.
Chapter O. On the Structure of Reductive p-adic Groups. 1
§0. 1. Some Definitions and Facts. 2
§0. 2. Cartan Subgroups and Split Tori in Reductive Groups. 4
§0. 3. Parabolic Subgroups of Reductive Groups. 5
§0. 4. On the Rational Points of p-adic Reductive Groups. 7
§0. 5. Lie Algebras, Roots, and Weyl Groups. 9
§0. 6. A0-good :Maximal Compact Subgroups of G. 12
Chapter 1. Generalities Concerning Totally Disconnected Groups and
Their Representations. 14
§1. 1. Functions and Distributions on Totally Disconnected
Spaces. 14
§1. 2. T. d. Groups and Relatively Invariant Measures on
Homogeneous Spaces. 17
§1. 2. 1. On the Haar Measure for a Parabolic Group. 22
§1. 2. 2. The Constant -y(G/P) for a Reductive p-adic Group. 2 3
§1. 3. Representations of Groups. 25
§1. 4. Smooth Representations of t. d. Groups. 28
§1. 5. Admissible Representations of t. d. Groups. 35
§1. 6. Intertwining Operators and Forms. 38
§1. 7. Smooth Induced Representations of t. d. Groups. 41
§1. 8. Some Results Concerning Distributions Defined on a t. d.
Group. 47
§1. 9. Invariant Distributions and Intertwining Forms for Induced
Representations. 52
§1.10. Automorphic Forms on a t. d. Group. 57
§1. ll. The Space of Finite Operators for an Admissible Module. 63
§1. 12. Double Representations of K and Automorphic forms. 69
§1.13. On the Characters of Admissible Representations. 74
Chapter 2. Jacquet' s Theory, Bruhat' s Theory, the Elementary
Theory of the Constant Term. 78
§2.1. A Decomposition Theorem for Certain Compact Open
Subgroups of G. 79
§2.2. J-supercuspidal Representations. 81
§2. 3. Jacquet' s Quotient Theorem. 86
§2.4. Dual Exponents, the Sub representation Theorem,
and ~· (G). 90
§2.5. Irreducibility and Intertwining Numbers for Certain
Induced Representations. 93
§2.6. The Constant Term. 100
§2. 7. Elementary Properties of the Constant Term. 106
§2. 8. The Constant Term and Supercusp Forms. ll2
iii
CONTENTS (cont'd)
Chapter 3. Exponents and the Maass-Selberg Relations. 116
§3. 1. Exponents. 116
§3. 2. Dual Exponents and Class Exponents. 120
§3. 3. On Exponents and Induced Representations. 125
§3. 4. Simple Classes and Negligibility. 130
§ 3. 5. The Maass-Selberg Relations. 139
Chapter 4. The Schwartz Spaces. 147
§4.1. Some Preliminaries. 148
§4.2. The Spherical Function - . 152
§4. 3. Inequalities. 158
§4. 4. The Schwartz Spaces and Square Integrable Forms. 174
§4. 5. Tempered Representations and the Weak Constant Term. 182
§4. 6. The General Maass-Selberg Relations. 194
§4. 7. The Steinberg Character of G and an Application. 196
§4. 8. Howe's Theorem and Consequences. 211
Chapter 5. The Eisenstein Integral and Applications. 221
§5.1. On the Matrix Coefficients of Admissible Representations
and Their Constant Terms. 222
§5. 2. The Eisenstein Integral and Its Functional Equations:
The Formal Theory. 230
§5. 2. l. Preliminaries: The Context. 2 30
§5. 2. 2. Definition and Elerrentary Properties of the Eisenstein
Integral. 240
§5. 2. 3. Eisenstein Integrals as Matrices for Induced Repre-
sentations. 247
§5. 2. 4. Functional Equations for the Eisenstein Integral. 249
§5. 3. The Eisenstein Integral and Its Functional Equations:
The Analytic Theory. 263
§5. 3. l. The Complex Structure on ~<C(M). 263
§5. 3. 2. On the Simplicity of Exponents I. 268
§5. 3. 3. On the Simplicity of Exponents II. 272
§5. 3. 4. An Integral Formula for E P, 1(P : If; : v ). 2 78
§5. 3. 5. Analytic Functional E1uations for the Eisenstein Integral. 281
§5. 4. Applications and Further Development of the Analytic
Theory. 287
§5. 4.1. Some Ideas of Cas selrran. 287
§5. 4. 2. Some Results for the Case p-rank Equal One. 292
§5. 4. 3. The Product Formula and Consequences. 299
§5. 4. 4. The Composition Series Theorem. 307
§5. 4. 5. On the Existence and Finiteness of the Set of Special
Representations Associated to a Complex Orbit. 311
§5. 5. Wave Packets, the Plancherel Measure, and Inter
twining Operators. 318
iv
CONTENTS (cont' d)
§5. 5. 1. The Algebras of Wave Packets. 318
§5. 5. 2. The Plancherel Formula. 342
§5. 5. 3. The Commuting Algebra Theorem. 344
§5. 5. 4. An Almost Explicit Plancherel' s Formula for G of
Semi-Simple Rank One. 351
Refe:rences. 362
Selected Terminology. 365
Selected Notations. 370
Chapter 0. On the Structure of Reductive p-adic Groups.
The theory to be developed in the five later chapters of these notes
depends upon the structure theory for reductive groups with points in a p-adic
field. Fortunately, this theory has been worked out in detail (cf. , [Za ], [Zc] for
the essentially algebraic aspects and [4c] for the directly related topological
part). In this chapter we very briefly review only those facts from the structure
theory which we shall need later. The reader may consult the references both
for proofs and more details.
If G is any group and H a subgroup, we write ZG(H)[NG(H)] for
g -1
the centralizer [normalizer] of H in G. Given x, g E G, we write x = gxg
g -1
and H = gHg .
We write [X] for the cardinality of a finite set X.
x
For any ring R we write R for its group of units. If n is a
positive integer and R is a commutative ring, we write GLn (R) for the group
of all n X n matrices (c ij ) l:'.:_i, j:'.:_n with entries c L.J. E R and determinant in
Rx. We write det(x) or det(c .. ) to denote the determinant of a matrix x or
lJ
(c .. ).
lJ
We write 22: for the ring of ordinary (rational) integers, «]!, JR, and
CC for the fields, respectively, of rational, real, and complex numbers.
2
§0.1. Some Definitions and Facts.
Let Q be a field and Q an algebraic closure of Q. For any positive
integer m the space nm of all m-vectors with components in n carries the
Zariski and Q topologies: A subset S of nm is called Zariski closed (Q-closed]
if S is the zero set of some finite set of polynomials in n[x1, ... ,xm)(Q(x1, •.. ,xm)),
The complement of a Zariski closed (Q-closed) subset of nm is termed Zariski
open (Q-open). Obviously, the Zariski topology on nm is finer than the Q
topology. A subset s1 of a Zariski closed (Q-closed) set S2 C nm is called
-m
Zari ski dense (Q-dense] in s2 if every Zariski closed (Q-closed) subset of Q
which contains sl also contains s2.
Under an obvious identification, we may, for any positive integer n,
2
I
regard GL (n) as the Q-open subset det(x .. ) 0 of nn or, more conveniently
n LJ 2
=
for what follows, the Q-closed subset y det(x .. ) 1 of nn +l A linear algebraic
lJ
is a subgroup G C GL (n) which is Zariski closed as a subset
- n
is an 1. a. g., the group law (x, y) ,_.. x -ly for G is given by a
2 2 2 -
-n -n -n
polynomial mapping n x n - n . An 1. a. g. G is said to be defined over
n or called an n-~ if _9 is Q-closed in n-n 2 +l and if the group law is given
n.
by polynomials with coefficients in
We will denote linear algebraic groups by Roman letters with wavy
underlines. If 9 is an l.a.g. and R is a subring of Q, we write _9(R) for
gn GLn(R). If 9 is an Q-group, we sometimes write G = 9(Q).
A morphism of _!. ~· ,g_. ' s is a group homomorphism which is at the
same time a polynomial mapping. An Q-morphism is a morphism of Q-groups
in which the polynomials defining the mapping have coefficients in n.