  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.
