Spherical Inversion on SLn(R)
By: by Jay Jorgenson, Serge Lang
Contributors: Lang, Serge
Series: Springer Monographs in Mathematics; Springer monographs in mathematics
Publisher: Springer New York
Published: 2001
Place of publication: New York, NY
Language: eng
Format: [electronic resource] /
Number of pages: 426
ISBN-13: 9781441928832; 9781468493023
Subjects: Mathematics; Topological groups
Description: 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.
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Table of contents: title: Iwasawa Decomposition and Positivity; title: Invariant Differential Operators and the Iwasawa Direct Image; title: Characters, Eigenfunctions, Spherical Kernel and W-Invariance; title: Convolutions, Spherical Functions and the Mellin Transform; title: Gelfand-Naimark Decomposition and the Harish-Chandra -Function; title: Polar Decomposition; title: The Casimir Operator; title: The Harish-Chandra Series for Eigenfunctions of Casimir; title: General Inversion; title: The Harish-Chandra Schwartz Space (HCS) and Anker's Proof of Inversion; title: Tube Domains and the L1 (Even Lp) HCS Spaces; title: SLn(C).