Explicit Constructions of Automorphic L-Functions
The goal of this research monograph is to derive the analytic continuation and functional equation of the L-functions attached by R.P. Langlands to automorphic representations of reductive algebraic groups. The first part of the book (by Piatetski-Shapiro
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1254 Stephen Gelbart Ilya Piatetski-Shapiro Stephen Rallis
Explicit Constructions of Automorphic L-Functions
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann
1254 Stephen Gelbart Ilya Piatetski-Shapiro Stephen Rallis
Explicit Constructions of Automorphic L-Functions
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Authors
Stephen Gelbart Department of Theoretical Mathematics, The Weizmann Institute of Science Rehovot 76100, Israel lIya Piatetski-Shapiro Department of Mathematics, Tel-Aviv University Ramat Aviv 69978, Israel and Department of Mathematics Yale University New Haven, CT 06520, USA Stephen Rallis Department of Mathematics, Ohio State University Columbus, OH 43210, USA
Mathematics Subject Classification (1980): 11 F70, 11S37, 22E50
ISBN 3-540-17848-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17848-1 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Oruckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
These previously unpublished manuscripts describe certain L-functions attached to automorphic representations of the classical groups. Part A dates from 1983-84 and represents work of Piatetski-Shapiro and Rallis. The subject matter is a generalization of the method of Godement-Jacquet from GL(n) to a simple classical group G. Part B was written by Gelbart and Piatetski-Shapiro in the Fall of 1985, with an Appendix by all three authors.
This work concerns a generalization of Rankin-Selberg
convolution to G x GL(n), with G a classical reductive group of split rank n. Parts A and B appear with their own Introductions and Bibliography. For a discussion of how these results are related to the recent works of F. Shahidi, the reader is referred to the 'Postscript' in the Introduction to Part B, and also to S. Gelbart and F. Shahidi's new paper on "Analytic Properties of Automorphic L-functions". The expeditious preparation of the final form of this Lecture Note volume was done by Miriam Abraham of the Weizmann Institute, to whom we offer our thanks.
S. Gelbart
1. Piatetski-Shapiro S. Rallis
January 1987
Research supported in part by Grant No. 84-00139 from the United States-Israel Binational Science Foundation, Jerusalem, Israel.
CONTENTS Preface Part A:
L-FUNCTIONS FOR THE CLASSICAL GROUPS
§1.
A formal identity
§2.
Explicit constructions for t
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