PGL2 over the p-adics: its Representations, Spherical Functions, and Fourier Analysis
- PDF / 9,346,239 Bytes
- 212 Pages / 504 x 720 pts Page_size
- 11 Downloads / 136 Views
166
Allan J. Silberger Bowdoin College, Brunswick, Maine/USA
PGL2 over the p-adics: its Representations, Spherical Functions, and Fourier Analysis
Springer-Verlag Berlin · Heidelberg· NewYork 1970
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 1970. Library of Congress Catalog Card Number 70-139951.Printed in Germany. Title No. 1953
@
Offsetdruck: Julius Beltz, WeinheimlBergstr.
CONTENTS
Chapter I INTRODUCTION AND GENERALITIES 1.0
Preliminary remarks; the organization of this memoir ..••••.•. l
1.1
Notations, conventions, and some facts needed later •.•••••..• 3
1.2
Spherical functions on
1.3
Spherical functions of class u and the induced representation
G with respect to K..•...•.•.•.••.•. 5
= ind K+G
u .•...•.•.•...•••.•••.•.•.••...•.•• 7
1.4
The Plancherel's and inversion formulas for a type I group .. 12
1.5
The Plancherel's formula for Plancherel's formula for
S2 u
for all u implies the
L2(G) ••.•••••....•••••••••.•.•.•.• 15
1.6
The characters of
1.7
Exhaustion of algebraically irreducible representations of
G as locally constant class functions ••• 19
G and spherical functions •••...••••••••••••••.•.•.••..• 21
Chapter II THE IRREDUCIBLE UNITARY REPRESENTATIONS OF PGL(2,n) AND THEIR CHARACTERS 2.1
The principal series of irreducible unitary representations of PGL(2,n) ..•...•.••...............•.•••.•...•..•.••••••..• 34
2.2
The supplementary series of irreducible unitary representations of
G
39
2.3
The one-dimensional representations of
G.•••••••.••••••.••• 40
2.4
The special representations of
2.5
The discrete series of
2.6
The characters of the unitary representations of
G..•••••.•••••••.•.••••••••• 40
G•.••••••••••••••••••••••••••••••••• 41 G.•••••••• 45
iv
Chapter III A DESCRIPTION OF ALL IRREDUCIBLE REPRESENTATIONS OF K = PGL(2,Q) 3.1
Representation theory of the integer triangle subgroup T nx
3.2
of
K..............•.•.........................•......• 51
A description of the irreducible representations of contained in
K
ind 1 .•.•...•.•...•.....••..........•.••..••••• 55 BnKtK
3.2.1
Two miscellaneous results •.••.••••.••.•.....•.•..••.•...... 58
3.2.2
Some estimates on the dimensions of the spaces
3.3
The decomposition of the restrictions to
Su(v) •.•... 63
K of the
irreducible representations of the discrete series of G.....• 68 3.3.1
Classification of all the irreducible representations contained in the restrictions to
K of the discrete
series of G; proofs of irreducibility, equivalence, and continuity for the representations discrete series of
Tn
of the
G.....•.•••...
Data Loading...
