Olshanski spherical functions for infinite dimensional motion groups of fixed rank

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 $p\to \mathrm{\infty }$,the Olshanski spherical pair $(G_{\mathrm{\infty }},K_{\mathrm{\infty }})$. We classify all Olshanski spherical functions of $(G_{\mathrm{\infty }},K_{\mathrm{\infty }})$ as functions on the cone ${\mathrm{\Pi }}_q$ 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 $p\to \mathrm{\infty }$. The latter are given by Bessel functions on ${\mathrm{\Pi }}_q$. 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 $p\to \mathrm{\infty }$) appear as spherical functions.