β Scribed by Jay Jorgenson, Serge Lang (auth.)
No coin nor oath required. For personal study only.
Harish-Chandra's general Plancherel inversion theorem admits a much shorter presentation for spherical functions. The authors have taken into account contributions by Helgason, Gangolli, Rosenberg, and Anker from the mid-1960s to 1990. Anker's simplification of spherical inversion on the Harish-Chandra Schwartz space had not yet made it into a book exposition. Previous expositions have dealt with a general, wide class of Lie groups. This has made access to the subject difficult for outsiders, who may wish to connect some aspects with several if not all other parts of mathematics, and do so in specific cases of intrinsic interest. The essential features of Harish-Chandra theory are exhibited on SLn(R), but hundreds of pages of background can be replaced by short direct verifications. The material becomes accessible to graduate students with especially no background in Lie groups and representation theory. Spherical inversion is sufficient to deal with the heat kernel, which is at the center of the authors' current research. The book will serve as a self-contained background for parts of this research.
Front Matter....Pages i-xx
Iwasawa Decomposition and Positivity....Pages 1-32
Invariant Differential Operators and the Iwasawa Direct Image....Pages 33-73
Characters, Eigenfunctions, Spherical Kernel and W -Invariance....Pages 75-129
Convolutions, Spherical Functions and the Mellin Transform....Pages 131-175
Gelfand-Naimark Decomposition and the Harish-Chandra c -Function....Pages 177-218
Polar Decomposition....Pages 219-254
The Casimir Operator....Pages 255-275
The Harish-Chandra Series and Spherical Inversion....Pages 277-308
General Inversion Theorems....Pages 309-324
The Harish-Chandra Schwartz Space (HCS) and Ankerβs Proof of Inversion....Pages 325-371
Tube Domains and the L 1 (Even L p ) HCS Spaces....Pages 373-386
SL n (C)....Pages 387-410
Back Matter....Pages 411-426
Topological Groups, Lie Groups
π SIMILAR VOLUMES
<span>For the most part the authors are concerned with SLn(R) and with invariant differential operators, the invarinace being with respect to various subgroups. To a large extent, this book carries out the general results of Harish-Chandra.</span>
This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup ^D*G of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; th
<p><P>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 SL<SUB>2</SUB>(C). The authors begin with the realization of the heat kernel on SL<SUB>2</SUB>(C) through spherical
A memoir that defines and studies multi-variable Eisenstein series attached to heat kernels