Continuous Measures on Homogenous Spaces

TL;DR

This paper extends Wiener's characterization of continuous measures to compact homogeneous spaces, providing necessary and sufficient conditions on probability measures on Lie groups and nilmanifolds using heat kernel properties.

Contribution

It introduces a generalization of Wiener's theorem to compact homogeneous spaces, utilizing heat kernel techniques for characterizing continuous measures.

Findings

Necessary and sufficient conditions for measures to be continuous on Lie groups.

Application of heat kernel properties to measure characterization.

Extension of classical results to broader classes of manifolds.

Abstract

In this paper we generalize Wiener's characterization of continuous measures to compact homogenous manifolds. In particular, we give necessary and sufficient conditions on probability measures on compact semisimple Lie groups and nilmanifolds to be continuous. The methods use only simple properties of heat kernels.