The Heat Kernel and Theta Inversion on SL2(C)
The present monograph develops the fundamental ideas and results surrounding heat kernels, spectral theory, and regularized traces associated to the full modular group acting on SL2(C). The authors begin with the realization of the heat kernel on SL2(C) t
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Jay Jorgenson Serge Lang
The Heat Kernel and Theta Inversion on SL2(C)
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Jay Jorgenson Department of Mathematics City College of New York New York, NY 10031 USA [email protected]
Serge Lang (deceased)
ISBN 978-0-387-38031-5 e-ISBN 978-0-387-38032-2 DOI 10.1007/978-0-387-38032-2 Library of Congress Control Number: 2006931199 Mathematics Subject Classification (2000): 11Fxx, 32Wxx c 2008 Springer Science+Business Media, LLC ° All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Preface
The draft of the present book was first completed in August 2005, one month prior to the passing of Serge Lang. As such, this book is the last completed in its entirety by Lang. Beginning the early 1990’s, Lang became fascinated with the prospect of using heat kernels and heat kernel analysis in analytic number theory. Specifically, we developed a program of study where one would define Selberg-type zeta functions associated to finite volume quotients of symmetric spaces, and we speculated that each such zeta function would admit a functional equation where lower level Selberg-type zeta functions would appear. In the case of the Riemann surface associated to PSL2 (Z), the Selberg zeta function has a functional equation which involves the Riemann zeta function. Lang and I began the work necessary to carry out our proposed analysis for quotients of SLn (C) by SLn (Z[i]). As with other mathematical works undertaken by Lang, he wanted to develop the foundations himself. The present book is the result of establishing, as only Lang could, the case n = 2. During the reviewing process, I have refrained from making any changes in order to preserve Lang’s style of exposition.
New York May 2008
Jay Jorgenson
v
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Part I Gaussians, Spherical Inversion, and the Heat Kernel 1
2
Spherical Inversion on SL2 (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Iwasawa Decomposition, Polar Decomposition, and Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Chara
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