Journal of Lie Theory 21 (2011), No. 4, 987--1007
Copyright Heldermann Verlag 2011
The Smoothness of Orbital Measures on Exceptional Lie Groups and Algebras
Kathryn Hare
Dept. of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
kehare@uwaterloo.ca
Paul Skoufranis
Dept. of Mathematics, University of California, Los Angeles, CA 90095--1555, U.S.A.
pskoufra@math.ucla.edu
[Abstract-pdf]
\def\g{{\frak g}} Suppose that $G$ is a compact, connected, simple, exceptional Lie group with Lie algebra $\g$. We determine the sharp minimal exponent $k_{0}$, which depends on $G$ or $\g$, such that the convolution of any $k_{0}$ continuous, $G$-invariant measures is absolutely continuous with respect to Haar measure. The exponent $k_{0}$ is also the minimal integer such that any $k_{0}$-fold product of conjugacy classes in $G$ or $k_{0}$-fold sum of adjoint orbits in $\g$ has non-empty interior. Unlike in the classical case, the answer can be less than the rank of $G$ or $\g$.\par We also establish a dichotomy for orbital measures $\mu$, supported on non-trivial conjugacy classes or adjoint orbits of minimal non-zero dimension: for each $k$, either $\mu^{k}\in L^{2}$ or $\mu^{k}$ is singular with respect to Haar measure.
Keywords: Compact Lie group, compact Lie algebra, orbital measure, orbit, conjugacy class.
MSC: 43A80; 22E30 58C3
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