The absolute continuity of convolution products of orbital measures in exceptional symmetric spaces

TL;DR

This paper determines the minimal number of convolutions of orbital measures in exceptional symmetric spaces needed to ensure absolute continuity and non-empty interior of product sets, revealing that this number is typically 2 or 3.

Contribution

It establishes the minimal convolution count for absolute continuity in exceptional symmetric spaces, a novel result in harmonic analysis on these spaces.

Findings

Minimal convolution number L(G) is 2 or 3.

Products of L(G) double cosets have non-empty interior.

L(G) is often less than the rank of G.

Abstract

Let $G$ be a non-compact group, $K$ the compact subgroup fixed by a Cartan involution and assume $G/K$ is an exceptional, symmetric space, one of Cartan type $E,F $ or $G$. We find the minimal integer, $L(G),$ such that any convolution product of $L(G)$ continuous, $K$-bi-invariant measures on $G$ is absolutely continuous with respect to Haar measure. Further, any product of $L(G)$ double cosets has non-empty interior. The number $L(G)$ is either $2$ or $3$% , depending on the Cartan type, and in most cases is strictly less than the rank of $G$.