The aim of this book is to give an introduction to Harish-Chandra’s inversion for spherical functions, concentrating SLn(R) and, where possible, using a list of axioms that is readily verified in this particular case. This makes the book accessible to graduate students and it enables one to read its parts independently. All required material is developed from the beginning, while keeping the book to a reasonable length. This approach is to be contrasted with other modern accounts that are completely general (treating all semisimple groups at once), at the expense of being impenetrable to outsiders.

The book starts with a description of various decompositions (Iwasawa, Bruhat, etc.), computing corresponding Jacobian determinants, and with the study of invariant differential operators. The Casimir operator is discussed as a useful tool. The book culminates with Rosenberg’s proof of Harish-Chandra’s inversion on Paley-Wiener spaces and with Anker’s proof of the inversion on Schwartz spaces. The final chapter shows how the theory simplifies on SLn(C).

Reviewer:

pš