Riesz exponential families on symmetric cones

Let E be a simple Euclidean Jordan algebra of rank r and let Omega be its symmetric cone. Given a Jordan frame on E, the generalized power defined on -Omega is the Laplace transform of some positive measure R-s on E if and only if s is in a given subset Xi of R'. The aim of this paper is to study the natural exponential families (NEFs) F(R-s) associated to the measures R-s. We give a condition on s so that R-s generates a NEF, we calculate the variance function of F(R-s) and we show that a NEF F on E is invariant by the triangular group if and only if there exists s in Xi such that either F = F(R-s) or F is the image of F(R-s) under the map x --> - x.