Laplace-type representation for some generalized spherical functions of type BC

TL;DR

This paper provides a Laplace-type integral representation for generalized spherical functions of type BC, linking them to type A functions and exploring their support and Dunkl setting representations.

Contribution

It introduces a Laplace-type formula for generalized spherical functions of type BC and describes the support of the associated intertwining operator in Dunkl settings.

Findings

Laplace-type representation for BC spherical functions

Support description of the generalized Abel transform

Support analysis of the Dunkl intertwining operator

Abstract

In [M. R\"osler and M. Voit. Integral Representation and Uniform Limits for Some Heckman-Opdam Hypergeometric Functions of type BC, Transactions of the American Mathematical Society, Vol. 368, No. 8, 6005-6032, 2016.], R\"osler and Voit give a formula for generalized spherical functions of type $BC$ in terms of the spherical functions of type A. We use this formula to describe precisely the support of the associated generalized Abel transform. Furthermore, we derive a similar formula for the generalized spherical functions in the rational Dunkl setting. The support of the intertwining operator V is also deduced. We also show, as a consequence, that a Laplace-type expression exists for the generalized spherical functions both in the trigonometric Dunkl setting and in the rational Dunkl setting.