Matrix Spherical Functions on Finite Groups

In this paper we focus our attention on matrix or operator-valued spherical functions associated to finite groups (G, K), where K is a subgroup of G. We introduce the notion of matrix-valued spherical functions on G associated to any K-type \(\delta \in \hat{K}\) by means of solutions of certain associated integral equations. The main properties of spherical functions are established from their characterization as eigenfunctions of right convolution multiplication by functions in \(A[G]^K\), the algebra of K-central functions in the group algebra A[G]. The irreducible representations of \(A[G]^K\) are closely related to the irreducible spherical functions on G. This allows us to study and compute spherical functions via the representations of this algebra.