SINGULAR RIESZ MEASURES ON SYMMETRIC CONES

A fondamental theorem due to Gindikin [Russian Math. Surveys, 29 (1964), 1-89] says that the generalized power Delta(s)(-theta(-1)) defined on a symmetric cone is the Laplace transform of a positive measure R-s if and only if s is in a given subset Xi of R-r, where r is the rank of the cone. When s is in a well defined part of Xi, the measure R-s is absolutely continuous with respect to Lebesgue measure and has a known expression. For the other elements s of Xi, the measure R-s is concentrated on the boundary of the cone and it has never been explicitly determined. The aim of the present paper is to give an explicit description of the measure R-s for all s in Xi. The work is motivated by the importance of these measures in probability theory and in statistics since they represent a generalization of the class of measures generating the famous Wishart probability distributions.