Riesz measures and Wishart laws associated to quadratic maps

TL;DR

This paper introduces a new framework for defining Riesz measures and Wishart laws linked to quadratic maps on convex cones, providing explicit formulas and leveraging group symmetries for their characterization.

Contribution

It generalizes the theory of Riesz measures and Wishart laws to quadratic maps on convex cones, including explicit formulas and symmetry-based descriptions.

Findings

Derived a general formula for moments of Wishart laws.

Explicitly described Riesz measures and Wishart laws using group invariance.

Extended the theory to virtual quadratic maps on convex cones.

Abstract

We introduce a natural definition of Riesz measures and Wishart laws associated to an $\Omega$-positive (virtual) quadratic map, where $\Omega \subset \real^n$ 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 $\Omega$ transitively, then the associated Riesz measure and Wishart law are described explicitly by making use of theory of relatively invariant distributions on homogeneous cones.