Characterizing the absolute continuity of the convolution of orbital measures in a classical Lie algebra

TL;DR

This paper characterizes when convolutions of orbital measures in classical Lie algebras are absolutely continuous, extending previous results from type A to all classical types based on orbit structures.

Contribution

It provides a complete characterization for classical Lie algebras, generalizing known results from type A to all classical types based on annihilating roots.

Findings

Characterization depends on Lie type and annihilating roots.

Convolution absolute continuity linked to sum of orbits having interior.

Extends previous type A results to all classical Lie algebras.

Abstract

Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension $d$. It is a classical result that the convolution of any $d$ non-trivial, $G$ -invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on $\mathfrak{g}$ and the sum of any $d$ non-trivial orbits has non-empty interior. The number $d$ was later reduced to the rank of the Lie algebra (or rank $+1$ in the case of type $A_{n}$). More recently, the minimal integer $k=k(X)$ such that the $k$-fold convolution of the orbital measure supported on the orbit generated by $X$ is an absolutely continuous measure was calculated for each $X\in \mathfrak{g}$. In this paper $\mathfrak{g}$ is any of the classical, compact, simple Lie algebras. We characterize the tuples $(X_{1},...,X_{L})$, with $X_{i}\in \mathfrak{g},$ which have the property that the convolution of the $L$ -orbital measures supported on the orbits generated by the $X_{i}$ is absolutely continuous and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of $ \mathfrak{g}$ and the structure of the annihilating roots of the $X_{i}$. Such a characterization was previously known only for type $A_{n}$.