
SIGMA 13 (2017), 040, 41 pages arXiv:1612.01486
https://doi.org/10.3842/SIGMA.2017.040
A Linear System of Differential Equations Related to VectorValued Jack Polynomials on the Torus
Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 229044137, USA
Received December 11, 2016, in final form June 02, 2017; Published online June 08, 2017
Abstract
For each irreducible module of the symmetric group $\mathcal{S}_{N}$ there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, selfadjoint with respect to two Hermitian forms, one called the contravariant form and the other is with respect to a matrixvalued measure on the $N$torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper [SIGMA 12 (2016), 033, 27 pages] proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail. The $N$torus is divided into $(N1)!$ connected components by the hyperplanes $x_{i}=x_{j}$, $i$<$j$, which are the singularities of the system. The main result is that the orthogonality measure has no singular part with respect to Haar measure, and thus is given by a matrix function times Haar measure. This function is analytic on each of the connected components.
Key words:
nonsymmetric Jack polynomials; matrixvalued weight function; symmetric group modules.
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