Consider the Gelfand pairs (G p , K p ) := (M p,q U p , U p ) associated with motion groups over the fields F = R, C, H with p q and fixed q as well as the inductive limit for p , the Olshanski spherical pair (G , K ) .We classify all Olshanski spherical functions of (G , K ) as functions on the cone q of positive semidefinite q q -matrices and show that they appear as (locally) uniform limits of spherical functions of (G p , K p ) as p .The latter are given by Bessel functions on 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 U q ), (U p 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 ) appear as spherical functions.
Rösler et al. (Tue,) studied this question.