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L 2 -Singular Dichotomy for Orbital Measures on Complex Groups

S. K. Gupta;  K. E. Hare — 2010

Bollettino dell'Unione Matematica Italiana

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 ∈ L 2 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.

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