Spherical Inversion on SLn(R) [Paperback]

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.Harish-Chandra?s general Plancherel inversion theorem admits a much shorter presentation for spherical functions. 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. In this book, the essential features of Harish-Chandra theory are exhibited on SLn(R), but hundreds of pages of background are replaced by short direct verifications. The material is accessible to graduate students with no background in Lie groups and representation theory.* Iwasawa Decomposition and Positivity * Invariant Differential Operators and the Iwasawa Direct Image * Characters, Eigenfunctions, Spherical Kernel and W-Invariance * Convolutions, Spherical Functions and the Mellin Transform * Gelfand-Naimark Decomposition and the Harish-Chandra --Function * Polar Decomposition * The Casimir Operator * The Harish-Chandra Series for Eigenfunctions of Casimir * General Inversion * The Harish-Chandra Schwartz Space (HCS) and Anker's Proof of Inversion * Tube Domains and the L^1 (Even L^p) HCS Spaces * SL_n(C)

From the reviews:

[This] book presents the essential features of the theory on SLn(R). This makes the book accessible to a wide class of readers, including nonexperts of Lie groups and representation theory and outsiders who would like to see connections of some aspects with other parts of mathematics. This feature is widely to be appreciated, together with the clearness of exposition and the way the book is structured. -Sergio Console, Zentralblatt

This book is devoted to Harish-Chandras Plancherel inversion formula in the special case of the group SlcF