# Riesz distributions and Laplace transform in the Dunkl setting of type A

## TL;DR

This paper develops a rigorous Laplace transform framework for Riesz distributions in the Dunkl setting of type A, characterizing positivity through a generalized Wallach set, extending classical results to a new algebraic context.

## Contribution

It provides a rigorous formulation of the Laplace transform for Riesz distributions in Dunkl theory and characterizes positivity via a generalized Wallach set, extending Gindikin's classical result.

## Key findings

- Established a formal Laplace transform for Dunkl kernels in type A.
- Characterized positivity of Riesz distributions using a generalized Wallach set.
- Extended classical symmetric cone results to the Dunkl setting.

## Abstract

We study Riesz distributions in the framework of rational Dunkl theory associated with root systems of type A. As an important tool, we employ a Laplace transform involving the associated Dunkl kernel, which essentially goes back to Macdonald, but was so far only established at a formal level. We give a rigorous treatment of this transform based on suitable estimates of the type A Dunkl kernel. Our main result is a precise analogue in the Dunkl setting of a well-known result by Gindikin, stating that a Riesz distribution on a symmetric cone is a positive measure if and only if its exponent is contained in the Wallach set. For Riesz distributions in the Dunkl setting, we obtain an analogous characterization in terms of a generalized Wallach set which depends on the multiplicity parameter on the root system.