The Laplace transform in the Dunkl setting

In the analysis on symmetric cones and the classical theory of hypergeometric functions of matrix argument, the Laplace transform plays an essential role. In an unpublished manuscript dating back to the 1980ies, I.G. Macdonald proposed a generalization of this theory, where the spherical polynomials of the underlying symmetric cone - such as the cone of positive definite matrices - are replaced by Jack polynomials with arbitrary index. He also introduced a Laplace transform in this context, but many of the statements in his manuscript remained conjectural. In the late 1990ies, Baker and Forrester took up these matters in their study of Calogero-Moser models, still at a rather formal level, and they noticed that they were closely related to Dunkl theory.