##
**Harmonic analysis and special functions on symmetric spaces.**
*(English)*
Zbl 0836.43001

Perspectives in Mathematics. 16. Orlando, FL: Academic Press, Inc. xii, 225 p. (1994).

This book consists of three parts. The two first (main) parts (Hypergeometric and Spherical Functions, by G. Heckman, and Harmonic Analysis on Semisimple Symmetric Spaces, by H. Schlichtkrull) are based on lectures given for the European School of Group Theory (Lumini, 1991 and Twente, 1992). A shorter third part (G. Heckman) is added in order to explain the connection between the two topics.

In Part I firstly the hypergeometric function theory associated with a root system is exposed and secondly for a semisimple Lie group \(G\) elementary spherical functions corresponding to an arbitrary one- dimensional \(K\)-type are considered from the point of view of the hypergeometric theory (\(K\) is a maximal compact subgroup of \(G\)).

Part II is a survey of harmonic analysis on semisimple symmetric spaces \(G/H\) leading to the most continuous part of the Plancherel formula (recent works of E. van den Ban and the author).

In part III the spherical \(K\)-part of \(L^2 (G/H)\) is considered and it is shown that the spectral decomposition of it can be described in terms of hypergeometric functions associated with a root system. Furthermore, for a commutative algebra of differential operators associated with a root system, some spectral problems are discussed. In conclusion some open problems are formulated.

Reviewer’s note to part II. The notion of the Fourier transform which is based on \(H\)-invariants was first introduced and used in works of the reviewer in the later sixties and the subsequent years.

In Part I firstly the hypergeometric function theory associated with a root system is exposed and secondly for a semisimple Lie group \(G\) elementary spherical functions corresponding to an arbitrary one- dimensional \(K\)-type are considered from the point of view of the hypergeometric theory (\(K\) is a maximal compact subgroup of \(G\)).

Part II is a survey of harmonic analysis on semisimple symmetric spaces \(G/H\) leading to the most continuous part of the Plancherel formula (recent works of E. van den Ban and the author).

In part III the spherical \(K\)-part of \(L^2 (G/H)\) is considered and it is shown that the spectral decomposition of it can be described in terms of hypergeometric functions associated with a root system. Furthermore, for a commutative algebra of differential operators associated with a root system, some spectral problems are discussed. In conclusion some open problems are formulated.

Reviewer’s note to part II. The notion of the Fourier transform which is based on \(H\)-invariants was first introduced and used in works of the reviewer in the later sixties and the subsequent years.

Reviewer: V.F.Molchanov (Glasgow)

### MSC:

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

43A85 | Harmonic analysis on homogeneous spaces |

22E46 | Semisimple Lie groups and their representations |

33C70 | Other hypergeometric functions and integrals in several variables |