Automorphic Forms on Semisimple Lie Groups
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62 Harish-Chandra Institute for Advanced Study, Princeton, New Jersey
Automorphic Forms on Semisimple Lie Groups 1968
Notes by J. G. M. Mars Mathematisches Institut der Universitat Utrecht
Springer-Verlag Berlin· Heidelberg· New York
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
62 Harish-Chandra Institute for Advanced Study, Princeton, New Jersey
Automorphic Forms on Semisimple Lie Groups 1968
Notes by J. G. M. Mars Mathematisches Institut der Universitat Utrecht
Springer-Verlag Berlin· Heidelberg· New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer-Verlag Berlin' Heidelberg 1968 Library of Congress Catalog Card Number 68-55417. Printed in Germany. Title No. 3668
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Preface These lectures are largely based on an important but, unfortunately, as yet unpublished, manuscript of R. P. Lang1ands, with the title "On the functional equations satisfied by Eisenstein Series." However they do not cover the last and the most difficult part (Chapter VII) of this manuscript. Lang1ands himself has given a brief account of his work in a short paper ("Eisenstein Series", pp. 235252, Algebraic Groups and Discontinuous Subgroups, 1966, Amer. Math. Soc.). In the last ten years, the analytic theory of automorphic forms has been pushed forward mainly through the contributions of Selberg, Gelfand and PiatetskyShapiro, and Lang1ands. In my opinion, here Selberg's ideas were decisive. But since only a very sketchy account of Selberg's work is available (see "Discontinuous groups and harmonic analysis", pp. 177189, Proceedings of the International Congress of Mathematicians, 1962), they have not attracted the notice which they undoubtedly deserve. After some of the arithmetic aspects of the theory of discontinuous groups began to be understood, it became clear that one needed analytical tools, in order to apply this newly acquired knowledge to the theory of automorphic forms. It is here that Lang1ands succeeded in adapting and extending Selberg's methods so as to fit them to the more general situation. Let me now give an outline of the main steps which lead to the analytic continuation and the functional equations of the Eisenstein series. 1) Definition of the space of cusp forms and proof of the theorem of Gelfand and a ECco(G), the operator °i\(a) is compact c (Theorem 2, §2, Chap. I). We also verify that dim J4(G/r,u,x) < co (Theorem 1, §2,
PiatetskyShapiro which says that for any
Chap. I) and prove a simple but important result of Lang1ands (Theorem 4, §5, Chap. I). 2) Definition and elementary properties of Eisenstein series corresponding to a cuspida1 subgroup q
(G/r,u)
P
of rank
q.
In particular we show that this series lies in
(see Chap. II, §§4,8). Also we derive a certain scalar product formula
( Lemma 40). 3) Given terms
fp
fE
u4q (G/r,u),
we obtain an estimate for
corresponding to the cuspida1 subgroups
P
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