The Spherical Transform of any k-Type in a Locally Compact Group

Abstract

Given a locally compact group G and a compact subgroup K, we develop and study a spherical transform on the convolution algebra Cc;(G) of all continuous functions f with compact support on G such that f = f = f . Here denotes the character of a unitary irreducible representation of K times its dimension. We obtain an inversion formula for the spherical transform by using the Fourier inversion formula in G. The case of the group G = SU(2; 1) and the compact subgroup K = U(2) is discussed in detail. We give explicit expressions for the spherical transform and the corresponding inversion formula in terms of the matrix hypergeometric function 2H1.