A Laplace-type representation of the generalized spherical functions associated to the root systems of type A

TL;DR

This paper develops a Laplace-type integral representation for generalized spherical functions related to root systems of type A, extending previous expressions and analyzing their support in Dunkl and trigonometric Dunkl settings.

Contribution

It introduces a new Laplace-type representation for these functions and characterizes the support of the associated Abel transform and Dunkl intertwining operator.

Findings

Derived a Laplace-type representation in Dunkl and trigonometric Dunkl settings.

Precisely described the support of the generalized Abel transform.

Established the support of the Dunkl intertwining operator.

Abstract

In this paper, we extend the iterative expression for the generalized spherical functions associated to the root systems of type $A$ previously obtained beyond regular elements. We also provide the corresponding expression in the flat case. From there, we derive a Laplace-type representation for the generalized spherical functions associated to the root systems of type $A$ in the Dunkl setting as well as in the trigonometric Dunkl setting. This representation leads us to describe precisely the support of the generalized Abel transform. Thanks to a recent result of Rejeb, this allows us to give the support for the Dunkl intertwining operator.