Posn(R) and Eisenstein Series
Posn(R) and Eisenstein Series provides an introduction, requiring minimal prerequisites, to the analysis on symmetric spaces of positive definite real matrices as well as quotients of this space by the unimodular group of integral matrices. The approach i
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Jay Jorgenson · Serge Lang
Posn(R) and Eisenstein Series
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Authors Jay Jorgenson City College of New York 138th and Convent Avenue New York, NY 10031 USA e-mail: [email protected] Serge Lang Department of Mathematics Yale University 10 Hillhouse Avenue PO Box 208283 New Haven, CT 06520-8283 USA
Library of Congress Control Number: 2005925188 Mathematics Subject Classification (2000): 43A85, 14K25, 32A50 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-25787-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-25787-5 Springer Berlin Heidelberg New York DOI 10.1007/b136063 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: TEX output by the author Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
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Preface
We are engaged in developing a systematic theory of theta and zeta functions, to be applied simultaneously to geometric and number theoretic situations in a more extensive setting than has been done up to now. To carry out our program, we had to learn some classical material in several areas, and it wasn’t clear to us what would simultaneously provide enough generality to show the effectiveness of some new methods (involving the heat kernel, among other things), while at the same time keeping knowledge of some background (e.g. Lie theory) to a minimum. Thus we experimented with the quadratic model of G/K in the simplest case G = GLn (R). Ultimately, we gave up on the quadratic model, and reverted to the G/K framework used systematically by the Lie industry. However, the quadratic model still serves occasionally to verify some things explicitly and concretely for instance in elementary differential geometry. The quadratic forms people see the situation on K\G, with right G-action. We retabulated all the formulas with left G-action. Just this may be useful for readers since the shift from right to left is ongoing, but not yet universal. Some other people have found our notes useful. For instance, we include some reduction theory and Siegel’s formula (after Hlawka’s work
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