A characterization of the Wishart exponential families by an invariance property

E is the space of real symmetric (d, d) matrices, andS and (bar S) are the subsets ofE of positive definite and semipositive-definite matrices. Let there be ap in $$Lambda = left{ {frac{1}{2},1,frac{3}{2}, ldots frac{{d - 1}}{2}} right} cup left] {frac{{d - 1}}{2}, + infty } right[$$ The Wishart natural exponential family with parameterp is a set of probability distributions on (bar S) defined by $$F_p = { exp [ - tfrac{1}{2}Tr(Gamma x)](detGamma )^p mu _p (dx);Gamma in S} $$ where μ_p is a suitable measure on (bar S) . LetGL(?^d) be the subset ofE of invertible matrices. Fora inGL(?^d), define the automorphismg _a ofE byg _a(x)=^t axa, where ^t a is the transpose ofa. The aim of this paper is to show that a natural exponential familyF onE is invariant byg _a for alla inGL(?^d) if and only if there existsp in Λ such that eitherF=F _p, orF is the image ofF _p byx??x. (Theorem).