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.
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.
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)
