Characterizing the Absolute Continuity of the Convolution of Orbital Measures in aClassical Lie Algebra

Gupta, Sanjiv Kumar ; Hare, Kathryn

Canadian journal of mathematics, Tome 68 (2016) no. 4, pp. 841-875

Résumé

Let $\mathfrak{g}$ be a compact simple Lie algebra of dimension $d$ . It is a classical result that the convolution of any $d$ non-trivial, $G$ -invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on $\mathfrak{g}$ , and the sum of any $d$ non-trivial orbits has non-empty 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 $k\,=\,k\left( X \right)$ 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 $\left( {{X}_{1}},\,.\,.\,.\,,\,{{X}_{L}} \right)$ , 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 non-empty 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 ${{A}_{n}}$ .

Détail

DOI : 10.4153/CJM-2015-018-2

Mots-clés : 43A80, 17B45, 58C35, compact Lie algebra, orbital measure, absolutely continuous measure