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 or is singular to the Haar measure on G. The minimum exponent k for which 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 μk∈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.