Harmonic Analysis on Reductive p-adic Groups
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162
Harish-Chandra The Institute for Advanced Study School of Mathematics, Princeton, NJ/USA Notes by
G. van Dijk Harmonic Analysis on Reductive p-adic Groups
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. C by Springer-Verlag Berlin' Heidelberg 1970. Library of Congress CatalOg Card Number 79-138810 Printed in German.y. Title No. 3319
CONTENTS Part I.
Existence of characters for the discrete series. §1. §2. §3. §4.
Part II.
4 8 9 10
Square.integrable representations mod Z. Reductive .,..,..adic groups. Supercuspidal representations. A conjecture.
Existence of characters in the general case. §l.
§2. §3. Part III.
§2.
20
The space generated by a supercusp form. Some consequences.
26
The space ,)I.(G, T).
§l.
§2. §3. §4. Part V.
16
Supercusp forms and supercuspidal representations.
§l.
Part IV.
12 15
The Godement principle. A theorem of Bruhat and Tits. Proof of Theorem 4 (based on Conjecture I).
30 31 38 39
Conjecture III. The space ...4(G, T). Proof of Theorem 7. Proof of Theorem 8.
The behavior of the characters of the supercuspidal representations on the regular set. §l.
§2. §3. §4. Part VI,
The mapping
§L
§2. §3. §4. §5. §6. §7. §8.
43
Two fundamental theorems. manifolds and distributions. Invariant distributions on the regular set. Applications to the characters of the supercuspidal representations.
1" -Adtc
IIF{
(char"
48 51 57
= 0).
Introduction and elementary properties of the mapping The first step in the proof of Theorem 13. Some algebraic lemmas on nilpotent elements. A submersive map. Some more preparation. The second step in the proof of Theorem 13. Completion of the proof of Theorem 13. Lifting of Theorem 13 to the group.
114>{.
63 68 71 72 76 78 81 82
IV
1
Part VII. § 1.
§2. Part VIII.
The local summability of
ID I
Statem.ent of Theorem. 15. Proof of the m.ain lem.m.a.
-2'-£
(char 0 = 0).
Reduction to the Lie algebra.
86 90
The local summability of the characters of the sueercuseidal representations (char 0 = 0).
§1. The m.ain theorem. and its consequences. §2. Statem.ent'of the preparatory results for the proof of Theorem. 16. §3, Proof of the m.ain theorem.. §4. Proof of Lem.m.a 46. §5. Proof of Theorem. 18 (first step), §6. Proof of Theorem. 19. §7. Proof of Theorem. 18 (second step). §8. Proof of Theorem. 20.
92 95 98 103 106 108 114 116
Introduction The object of these lectures is to illustrate, what I like to call the Lefschetz principle, which, in the context of reductive groups, says that whatever is true for real groups is also true for
fl/ -adic
groups.
The
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