Olshanski spherical functions for infinite dimensional motion groups of fixed rank

Abstract

Consider the Gelfand pairs $(G_p,K_p):=(M_{p,q}⋊U_p,U_p)$ associated with motion groups over the fields $\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$ with $p\ge q$ and fixed $q$ as well as the inductive limit $p\to \infty$,the Olshanski spherical pair $(G_{\infty },K_{\infty })$. We classify all Olshanski spherical functions of $(G_{\infty },K_{\infty })$ as functions on the cone ${\Pi }_q$ of positive semidefinite $q\times q$-matrices and show that they appear as (locally) uniform limits of spherical functions of $(G_p,K_p)$ as $p\to \infty$. The latter are given by Bessel functions on ${\Pi }_q$. Moreover, we determine all positive definite Olshanski spherical functions and discuss related positive integral representations for matrix Bessel functions. We also extend the results to the pairs $(M_{p,q}⋊(U_p\times U_q),(U_p\times U_q))$ which are related to the Cartan motion groups of non-compact Grassmannians. Here Dunkl-Bessel functions of type B (for finite $p$) and of type A (for $p\to \infty$) appear as spherical functions.

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