The Smoothness of Orbital Measures on Exceptional Lie Groups and Algebras

Kathryn Hare¹ ; Paul Skoufranis²

¹Dept. of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
²Dept. of Mathematics, University of California, Los Angeles, CA 90095--1555, U.S.A.

Journal of Lie Theory, Tome 21 (2011) no. 4, pp. 987-1007

Résumé

\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.

Détail

Classification : 43A80, 22E30 58C3
Mots-clés : Compact Lie group, compact Lie algebra, orbital measure, orbit, conjugacy class