The Concept of a Spherical Function

Abstract

Let G be a locally compact group, second countable and unimodular; and let K ⊂ G be a compact subgroup. In this chapter we shall explore the concept of spherical functions for (G, K) in some generality. If δ is an equivalence class of irreducible representations of K, elementary spherical functions of type δ are essentially matrix elements of irreducible representations of G defined by vectors that transform according to δ. They may be characterized as eigenfunctions on G with respect to appropriate algebras of convolution operators. If G is a Lie group we can replace the convolution operators by suitable differential operators in the above characterization. So, in the context of a Lie group G there is a close analogy with classical spherical harmonics. It is now natural to raise the problem of harmonic analysis of functions on the homogeneous space G/K in terms of the elementary spherical functions, and to view this as an eigenfunction expansion problem, namely to “expand” a sufficiently arbitrary function on G/K as a “linear combination” of eigenfunctions with respect to suitable algebras of G-invariant differential operators on G/K.