Manifolds with Cusps of Rank One Spectral Theory and L2-Index Theore
The manifolds investigated in this monograph are generalizations of (XX)-rank one locally symmetric spaces. In the first part of the book the author develops spectral theory for the differential Laplacian operator associated to the so-called generalized D
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1244 Werner Muller
Manifolds with Cusps of Rank One Spectral Theory and L2-lndex Theorem
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches Institut der Universitat und Max-Planck-Institut fur Mathematik, Bonn - vol. 9 F. Hirzebruch Adviser:
1244 Werner Muller
Manifolds with Cusps of Rank One Spectral Theory and L2-lndex Theorem
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author Werner Muller Akademie der Wissenschaften der DDR Karl-WeierstraB-lnstitut fur Mathematik MohrenstraBe 39, DDR - 1086 Berlin, German Democratic Republik
Mathematics Subject Classification (1980): 58G 10, 58G 11, 58G25
ISBN 3-540-17696-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17696-9 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: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
INTRODUCTION
Let G be a connected real semisimple Lie group of noncompact type, K a maximal compact subgroup of G and G!K the associated globally symmetric space. Consider a discrete torsion-free subgroup r of G with finite covolume and let r\G!K be the corresponding locally symmetric space. Let V,W be finite-dimensional unitary K-modules and denote by E, F the induced homogeneous vector bundles over G!K. E and F can be pushed down to locally homogeneous vector bundles E r\E' and F rV aver r\G!K. Let V: c""( G!K,E)
be an invariant elliptic differential operator. Then elliptic differential operator V:
COO(r\G!K,E)
--+
V induces an
COO(r\G!K,n
It is proved in [61] that V has a well-defined L2-index which, as in the compact case, depends only on ch V - ch W. Using Selberg's trace formula, Barbasch and Moscovici [15] derived an explicit formula for the L2-index of V i f the locally symmetric space r\G!K has strictly negative curvature or, equivalently, if the real rank of G equals one. It seems to be very interesting to have an explicit formula for the L2-index in the general case. In this book we shall investigate the case of a locally symmetric space of (Q-rank one. Actually, we shall work with a larger class of manifolds. Each of these manifolds is locally symmetric near infinity with ends generalizing the case of a cusp of a (Q-rank one locally symmetric space. This is motivated by our approach to the proof of a conjecture of Hirzebruch (c.f. [63,§6]). In [63] we investigated
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