The Absolute continuity of convolutions of orbital measures in SO(2n+1)

TL;DR

This paper characterizes when the convolution of orbital measures in the Lie group SO(2n+1) has a non-empty interior, based on eigenspace dimensions, extending previous results from type A groups.

Contribution

It provides the first complete characterization for type B groups, specifically SO(2n+1), of tuples leading to absolutely continuous convolutions of orbital measures.

Findings

Characterization depends on largest eigenspaces of elements.

Convolution product has non-empty interior under specific eigenspace conditions.

Extends known results from type A to type B Lie groups.

Abstract

Let $G$ be a compact Lie group of Lie type $B_{n},$ such as $SO(2n+1)$. We characterize the tuples\ $(x_{1},...,x_{L})$ of the elements $x_{j}\in G$ which have the property that the product of their conjugacy classes has non-empty interior. Equivalently, the convolution product of the orbital measures supported on their conjugacy classes is absolutely continuous with respect to Haar measure. The characterization depends on the dimensions of the largest eigenspaces of each $x_{j}$. Such a characterization was previously only known for the compact Lie groups of type $A_{n}$.