A geometric proof of the $L^{2}$-singular dichotomy for orbital measures on Lie algebras and groups

Abstract

Let G be a compact, connected simple Lie group and $\mathfrak {g}$ its Lie algebra. It is known that if $\mu$ is any G-invariant measure supported on an adjoint orbit in $\mathfrak {g}$, then for each integer k, the k-fold convolution product of $\mu$ with itself is either singular or in $L^{2}$. This was originally proven by computations that depended on the Lie type of $\mathfrak {g}$, as well as properties of the measure. In this note, we observe that the validity of this dichotomy is a direct consequence of the Duistermaat–Heckman theorem from symplectic geometry and that, in fact, any convolution product of (even distinct) orbital measures is either singular or in $L^{2+\varepsilon }$ for some $\varepsilon >0$. An abstract transference result is given to show that the $L^{2}$-singular dichotomy holds for certain of the G-invariant measures supported on conjugacy classes in G.