### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 033, 27 pages      arXiv:1511.06721      https://doi.org/10.3842/SIGMA.2016.033
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials

Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA

Received November 26, 2015, in final form March 23, 2016; Published online March 27, 2016

Abstract
For each irreducible module of the symmetric group on $N$ objects 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, self-adjoint with respect to certain Hermitian forms. These polynomials were studied by the author and J.-G. Luque using a Yang-Baxter graph technique. This paper constructs a matrix-valued measure on the $N$-torus for which the polynomials are mutually orthogonal. The construction uses Fourier analysis techniques. Recursion relations for the Fourier-Stieltjes coefficients of the measure are established, and used to identify parameter values for which the construction fails. It is shown that the absolutely continuous part of the measure satisfies a first-order system of differential equations.

Key words: nonsymmetric Jack polynomials; Fourier-Stieltjes coefficients; matrix-valued measure; symmetric group modules.

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