Distributions vectorielles homog\`enes sur une alg\`ebre de Jordan

TL;DR

This paper characterizes and constructs homogeneous vector-valued distributions on Euclidean Jordan algebras, linking their existence to spherical representations and providing explicit constructions.

Contribution

It establishes the equivalence between the existence of such distributions and spherical representations, and computes their dimension and explicit forms.

Findings

Distributions exist iff the representation is spherical.

Dimension of distribution space is r+1, where r is the rank of the algebra.

Provides explicit constructions of these distributions.

Abstract

We study distributions on a Euclidean Jordan algebra V with values in a finite dimensional representation space for the identity component G of the structure group of V and homogeneous equivariance condition. We show that such distributions exist if and only if the representation is spherical, and that then the dimension of the space of these distributions is r+1 (where r is the rank of V). We give also construction of these distributions and of those that are invariant under the semi-simple part of G.