Riesz measures and Wishart laws associated to quadratic maps

We introduce a natural definition of Riesz measures and Wishart laws associated to an $Ω$-positive (virtual) quadratic map, where $Ω\subset \real^n$ is a regular open convex cone. We give a general formula for moments of the Wishart laws. Moreover, if the quadratic map has an equivariance property under the action of a linear group acting on the cone $Ω$ transitively, then the associated Riesz measure and Wishart law are described explicitly by making use of theory of relatively invariant distributions on homogeneous cones.

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