
Journal of Lie Theory 21 (2011), No. 4, 9871007 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 900951555, U.S.A. pskoufra@math.ucla.edu [Abstractpdf] \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 nonempty 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 nontrivial conjugacy classes or adjoint orbits of minimal nonzero 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 [ Fulltextpdf (342 KB)] for subscribers only. 