Geodesic
$L^{2}$-Singular Dichotomy for Orbital Measures on Complex Groups
Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 3, pp. 409-419.

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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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Gupta, S. K.; Hare, K. E. $L^{2}$-Singular Dichotomy for Orbital Measures on Complex Groups. Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 3, pp. 409-419. http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_3_a0/

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