L²-singular dichotomy for orbital measures of classical simple Lie algebras

We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μ^k is singular to Lebesgue measure or μ^k ∈ L². The minimum k for which μ^k ∈ L² is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure.