Harmonic Analysis on Real Reductive Groups
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576 V. S. Varadarajan
Harmonic Analysis on Real Reductive Groups
Springer-Verlag Berlin. Heidelberg-New York 1977
Author
V. S. Varadarajan Department of Mathematics University of California at Los Angeles Los Angeles, C A 9 0 0 2 4 / U S A
Library of Congress Cataloging ila Publication Data
Varadarajan,
V S Harmonic analysis on real reductive groups.
(Lecture notes in mathematics ; 576) Includes bibliographical references. 1. Lie groups. 2. Lie algebras. 3. Harmonic analysis. I. Title. II. Title : Real reductive oups. III. series: Lecture notes in mathematics erlin) ; 576. 0A3.L28 no. 576 [QA387] 510'.8s [53-2'.55] 77-22]-6
~B
AMS Subject Classifications (1970): 22 E30, 2,.r ) E45 ISBN 3-540-08135-6 ISBN 0-387-08135-6
Springer-Verlag Berlin- Heidelberg. New York Springer-Verlag New York • Heidelberg • Berlin
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 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE
The contents of these notes are essentially the same as those of a Seminar on semisimple groups that I conducted during 1969-1975 at the University of California at Los Angeles.
I am very grateful to Professors Gangolli and Eckmann
for suggesting that this material appear in the Springer Lecture Notes Series and encouraging me to prepare them for publication. My aim here has been to give a more or less self-contained exposition of Harish-Chandra's work on harmonic analysis on real reductive groups, leading to the complete determination of the discrete series.
I have kept quite close to
his view of the subject although the informed reader may perceive departures in detail here and there. These notes are in two parts.
Part one deals with the problems of invariant
analysis on a real reductive Lie algebra.
It contains a full treatment of regular
orbital integrals and their Fourier transforms; the theorem that invariant eigendistributions
it presents a detailed proof of
are locally integrable functions;
and concludes with the proof of the theorem that an analytic invariant differential operator that kills all invariant distributions.
C~
functions, kills all invariant
Part two treats the theory on the group, with descent to Lie al-
gebra playing a key role in many proofs.
Here I have proved that invariant
eigendistributions on real reductive groups are locally integrable functions~ given the explicit construction of the characters of the discrete series~ and treated all the aspects of Schwartz space and tempered distributions that are needed to reach the goals I set
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