Abstract view
Characterizing the Absolute Continuity of the Convolution of Orbital Measures in a Classical Lie Algebra


Published:20160627
Printed: Aug 2016
Sanjiv Kumar Gupta,
Dept. of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36 Al Khodh 123, Sultanate of Oman
Kathryn Hare,
Dept. of Pure Mathematics, University of Waterloo, Waterloo, Ont., Canada, N2L 3G1
Abstract
Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension
$d$. It is
a classical result that the convolution of any $d$ nontrivial,
$G$invariant,
orbital measures is absolutely continuous with respect to
Lebesgue measure on $\mathfrak{g}$ and the sum of any $d$ nontrivial
orbits
has nonempty interior. The number $d$ was later reduced to the
rank of the
Lie algebra (or rank $+1$ in the case of type $A_{n}$). More
recently, the
minimal integer $k=k(X)$ such that the $k$fold convolution of
the orbital
measure supported on the orbit generated by $X$ is an absolutely
continuous
measure was calculated for each $X\in \mathfrak{g}$.
In this paper $\mathfrak{g}$ is any of the classical, compact,
simple Lie
algebras. We characterize the tuples $(X_{1},\dots,X_{L})$, with
$X_{i}\in
\mathfrak{g},$ which have the property that the convolution of
the $L$orbital
measures supported on the orbits generated by the $X_{i}$ is
absolutely continuous and, equivalently, the sum of their orbits
has
nonempty interior. The characterization depends on the Lie type
of
$\mathfrak{g}$ and the structure of the annihilating roots of
the $X_{i}$.
Such a characterization was previously known only for type $A_{n}$.