The absolute continuity of convolutions of orbital measures in symmetric spaces

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.