Vector valued Riesz distributions on Euclidian Jordan algebras

Abstract

Let V be a Euclidean Jordan algebra, Гthe associated symmetric cone and G be the identity component of the linear automorphism group of Г.In this paper we associate to a certain class of spherical representations (ρ, ɛ) of G certain ɛ-valued Riesz distributions generalizing the classical scalar valued Riesz distributions on V. Our construction is motivated by the analytic theory of unitary highest weight representations where it permits to study certain holomorphic families of operator valued Riesz distributions whose positive definiteness corresponds to the unitarity of a representation of the automorphism group of the associated tube domain Г +iV.