Spherical Inversion on Sln(r)

ISBN-13: 9780387951157 / Angielski / Twarda / 2001 / 426 str.

Jay Jorgenson; Serge Lang

Opis

Harish-Chandras 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.

Recenzje

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-Chandra's Plancherel inversion formula in the special case of the group SL_n(R) and for spherical functions. ... the book is easily accessible and essentially self contained." (A. Cap, Monatshefte für Mathematik, Vol. 140 (2), 2003)

"Roughly, this book offers a 'functorial exposition' of the theory of spherical functions developed in the late 1950s by Harish-Chandra, who never used the word 'functor'. More seriously, the authors make a considerable effort to communicate the theory to 'an outsider'. .... However, even an expert will notice several new and pleasing results like the smooth version of the Chevally restriction theorem in Chapter 1." (Tomasz Przebinda, Mathematical Reviews, Issue 2002 j)

"This excellent book is an original presentation of Harish-Chandra's general results ... . Unlike previous expositions which dealt with general Lie groups, the present book presents the essential features of the theory on SL_n(R). This makes the book accessible to a wide class of readers, including nonexperts ... . This feature is widely to be appreciated, together with the clearness of exposition and the way the book is structured. Very nice is, for instance, the ... table of the decompositions of Lie groups." (Sergio Console, Zentralblatt MATH, Vol. 973, 2001)

Spis treści

* 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)