A geometric proof of the $$L^{2}$$L2-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.