Moment Functions on Affine Groups

Abstract

Moments of probability measures on a hypergroup can be obtained from so-called (generalized) moment functions of a given order. The aim of this paper is to characterize generalized moment functions on a non-commutative affine group. We consider a locally compact group G and its compact subgroup K. First we recall the notion of the double coset space G /  / K of a locally compact group G and introduce a hypergroup structure on it. We present the connection between K-spherical functions on G and exponentials on the double coset hypergroup G /  / K. The definition of the generalized moment functions and their connection to the spherical functions is discussed. We study an important class of double coset hypergroups: specyfing K as a compact subgroup of the group of inverible linear transformations on a finitely dimensional linear space V we consider the affine group ${\mathrm {Aff\,}}K$. Using the fact that in the finitely dimensional case $({\mathrm {Aff\,}}K,K)$ is a Gelfand pair we give a description of exponentials on the double coset hypergroup ${\mathrm {Aff\,}}K//K$ in terms of K-spherical functions. Moreover, we give a general description of generalized moment functions on ${\mathrm {Aff\,}}K$ and specific examples for $K=SO(n)$, and on the so-called $ax+b$-group.