Abstract
Let G be a locally compact group. Let mu be a complex, bounded, hermitian and idempotent measure on G. We introduce the concePt of mu-biinvariant function and mu-spherical function valued in a complex finite dimensional Hilbert space. We establish a theory for the mu-spherical functions and characterize them by the theorem 3.10. When G is unimodular, let K be a compact subgroup of G and let tau be a continuous, unitary and irreducible representation of K. Let mu(tau) = chi(tau)dk, where chi(tau) and dk is the normalized character of tau such that chi(tau) = chi(tau) * chi(tau) and dk is the normalized Haar measure of the group K. Then the K-spherical functionS Of type tau are the mu(tau)-spherical functions. The theorem 3.10 may be considered as a generalization of a Godement's theorem in the classical theory of K-spherical functions.