Springer Monographs in Mathematics

Spherical Inversion on SLn(R)

Authors: Jorgenson, Jay, Lang, Serge

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About this book

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.

Reviews

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 SLn(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 SLn(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)


Table of contents (12 chapters)

  • Iwasawa Decomposition and Positivity

    Jorgenson, Jay (et al.)

    Pages 1-32

  • Invariant Differential Operators and the Iwasawa Direct Image

    Jorgenson, Jay (et al.)

    Pages 33-73

  • Characters, Eigenfunctions, Spherical Kernel and W-Invariance

    Jorgenson, Jay (et al.)

    Pages 75-129

  • Convolutions, Spherical Functions and the Mellin Transform

    Jorgenson, Jay (et al.)

    Pages 131-175

  • Gelfand-Naimark Decomposition and the Harish-Chandra c-Function

    Jorgenson, Jay (et al.)

    Pages 177-218

Buy this book

eBook $149.00
price for USA (gross)
  • ISBN 978-1-4684-9302-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $189.00
price for USA
  • ISBN 978-0-387-95115-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $189.00
price for USA
  • ISBN 978-1-4419-2883-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Spherical Inversion on SLn(R)
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2001
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag New York, Inc.
eBook ISBN
978-1-4684-9302-3
DOI
10.1007/978-1-4684-9302-3
Hardcover ISBN
978-0-387-95115-7
Softcover ISBN
978-1-4419-2883-2
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XX, 426
Number of Illustrations and Tables
1 b/w illustrations
Topics