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Journal of Lie Theory 21 (2011), No. 1, 189--203
Copyright Heldermann Verlag 2011

Dirichlet Distribution and Orbital Measures

Fa´za Fourati
Department of Mathematics, Preparatory Institute of Engineering Studies, University of Tunis, 1089 Monfleury - Tunis, Tunisia


\def\C{{\Bbb C}} \def\F{{\Bbb F}} \def\R{{\Bbb R}} \def\HH{{\Bbb H}} The starting point of this paper is an observation by Okounkov concerning the projection of orbital measures for the action of the unitary group $U(n)$ on the space Herm$(n,\C)$ of $n\times n$ Hermitian matrices. The projection of such an orbital measure on the straight line generated by a rank one Hermitian matrix is a probability measure whose density is a spline function. More generally we consider the projection of orbital measures for the action of the group $U(n,\F)$ on the space Herm$(n,\F)$ for $\F=\R$, $\C$, $\HH$, and their relation with Dirichlet distributions.

Keywords: Dirichlet distribution, orbital measure, Markov-Krein correspondence, spline function, Jack polynomial.

MSC: 60B05, 65D07

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