L²-singular dichotomy for orbital measures of classical compact Lie groups

We prove that for any classical, compact, simple, connected Lie group G, the G-invariant orbital measures supported on non-trivial conjugacy classes satisfy a surprising L²-singular dichotomy: Either μ_h^k ∈ L² (G) or μ_h^k is singular to the Haar measure on G. The minimum exponent k for which μ_h^k ∈ L² is specified; it depends on Lie properties of the element h ∈ G. As a corollary, we complete the solution to a classical problem - to determine the minimum exponent k such that μ^k ∈ L¹ (G) for all central, continuous measures μ on G. Our approach to the singularity problem is geometric and involves studying the size of tangent spaces to the products of the conjugacy classes.