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

  Published:2016-06-27
 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
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Abstract

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 $k=k(X)$ 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 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 $A_{n}$.
Keywords: compact Lie algebra, orbital measure, absolutely continuous measure compact Lie algebra, orbital measure, absolutely continuous measure
MSC Classifications: 43A80, 17B45, 58C35 show english descriptions Analysis on other specific Lie groups [See also 22Exx]
Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]
Integration on manifolds; measures on manifolds [See also 28Cxx]
43A80 - Analysis on other specific Lie groups [See also 22Exx]
17B45 - Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]
58C35 - Integration on manifolds; measures on manifolds [See also 28Cxx]
 

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