L²- Singular dichotomy for orbital measures on complex groups

It is known that all continuous orbital measures, μ,on a compact, connected, classical simple Lie group G or its Lie algebra satisfy a dichotomy: either μ^k εL² or μ^k is purely singular to Haar measure. In this note we prove that the same dichotomy holds for the dual situation, continuous orbital measures on the complex group G ^C. We also determine the sharp exponent k such that any k-fold convolution product of continuous G-bi-invariant measures on G^C is absolute continuous with respect to Haar measure.