The absolute continuity of convolutions of orbital measures in symmetric spaces

TL;DR

This paper characterizes when convolution products of orbital measures on certain symmetric spaces are absolutely continuous, linking it to eigenspace dimensions and root system properties, with implications for measure regularity.

Contribution

It provides a complete characterization of absolute continuity for convolutions of orbital measures in classical irreducible Riemannian symmetric spaces of Cartan type III, using algebraic and combinatorial criteria.

Findings

Convolution of any rank continuous, K-bi-invariant measures is absolutely continuous in most symmetric spaces.

The characterization depends on eigenspace dimensions and root system properties.

Exceptions occur in spaces with root systems of type A_n or D_3, or when rank+1 conditions are not met.

Abstract

We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces $G/K$ of Cartan type $III$, where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, $KzK,$ in $G$. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements $z$. A consequence of our work is to show that the convolution product of any rank% $G/K,$ continuous, $K$-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type $A_{n}$ or $D_{3}$, when rank$G/K$ $+1$ is needed.