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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, $\mu$ on a compact, connected, classical simple Lie group $G$ or its Lie algebra satisfy a dichotomy: either $\mu^{k}\in L^{2}$ or $\mu^{k}$ 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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L 2 -Singular Dichotomy for Orbital Measures on Complex Groups

$L^{2}$ -Singular Dichotomy for Orbital Measures on Complex Groups