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

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 $A_{n}$.