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arXiv · 1905.09493

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

Margit Rösler
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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.

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Margit Rösler. 2019-05-23. Riesz distributions and Laplace transform in the Dunkl setting of type A. https://arxiv.org/abs/1905.09493

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