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Journal of Lie Theory 23 (2013), No. 4, 899--920
Copyright Heldermann Verlag 2013

Olshanski Spherical Functions for Infinite Dimensional Motion Groups of Fixed Rank

Margit Rösler
Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

Michael Voit
Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44221 Dortmund, Germany


\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.

Keywords: Spherical functions, Olshanski spherical pairs, Bessel functions on matrix cones, Dunkl theory, positive definite functions, multivariate beta distributions.

MSC: 43A90, 22E66, 33C80, 43A85

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