Olshanski Spherical Functions for Infinite Dimensional Motion Groups of Fixed Rank

M. R�sler ; M. Voit

Journal of Lie theory, Tome 23 (2013) no. 4, pp. 899-92.

Résumé

\def\C{{\Bbb C}} \def\F{{\Bbb F}} \def\H{{\Bbb H}} \def\R{{\Bbb R}} Consider the Gelfand pairs $(G_p, K_p):=(M_{p,q} \rtimes U_p, U_p)$ associated with motion groups over the fields $\F = \R,\C,\H$ with $p\geq q$ and fixed $q$ as well as the inductive limit for $p\to\infty$, 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.\par We also extend the results to the pairs $(M_{p,q} \rtimes (U_p\times U_q), (U_p\times U_q))$ 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.

Détail

Classification : 43A90, 22E66, 33C80, 43A85
Mots-clés : Spherical functions, Olshanski spherical pairs, Bessel functions on matrix cones, Dunkl theory, positive definite functions, multivariate beta distributions