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 μk εL2 or μk 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 GC is absolute continuous with respect to Haar measure.
|Number of pages||11|
|Journal||Bolletino dell Unione Matematica Italiana|
|Publication status||Published - 2010|
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