Takeuchi M. Modern Spherical Functions

American Mathematical Society, 1994. — 282 p. — Functions (Translations of Mathematical Monographs 135) — ISBN10: 0821845802.This book presents an exposition of spherical functions on compact symmetric spaces, from the viewpoint of Cartan-Selberg. Representation theory, invariant differential operators, and invariant integral operators play an important role in the exposition.
The author treats compact symmetric pairs, spherical representations for compact symmetric pairs, the fundamental groups of compact symmetric spaces, and the radial part of an invariant differential operator. Also explored are the classical results for spheres and complex projective spaces and the relation between spherical functions and harmonic polynomials. This book is suitable as a graduate textbook.Spherical Functions
Spherical functions on compact groups
Differential operators
Invariant differential operators
Spherical functions on unimodular Lie groups
Compact Symmetric Pairs
Riemannian symmetric pairs
Maximal toral groups for compact symmetric pairs
Integral formulas for compact symmetric spaces
Spherical representations for compact symmetric pairs
Fundamental groups of compact symmetric spaces
Radial parts of invariant differential operators
Spherical Functions on Spheres and on Complex
Projective Spaces
Gegenbauer functions
Spherical functions on spheres
Spherical functions on complex projective spaces
Harmonic polynomials