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A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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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, self-adjoint with respect to two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued 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 $(N-1)!$ connected components by the hyperplanes $x_{i}=x_{j}$, $i$, 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.
Keywords: nonsymmetric Jack polynomials; matrix-valued weight function; symmetric group modules. 
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     author = {Charles F. Dunkl},
     title = {A {Linear} {System} of {Differential} {Equations} {Related} to {Vector-Valued} {Jack} {Polynomials} on the {Torus}},
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Charles F. Dunkl. A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a39/

[1] Beerends R. J., Opdam E. M., “Certain hypergeometric series related to the root system $BC$”, Trans. Amer. Math. Soc., 339 (1993), 581–609 | DOI | MR | Zbl

[2] Dunkl C. F., “Symmetric and antisymmetric vector-valued {J}ack polynomials”, Sém. Lothar. Combin., 64 (2010), B64a, 31 pp., arXiv: 1001.4485 | MR | Zbl

[3] Dunkl C. F., “Orthogonality measure on the torus for vector-valued {J}ack polynomials”, SIGMA, 12 (2016), 033, 27 pp., arXiv: 1511.06721 | DOI | MR | Zbl

[4] Dunkl C. F., “Vector-valued Jack polynomials and wavefunctions on the torus”, J. Phys. A: Math. Theor., 50 (2017), 245201, 21 pp., arXiv: 1702.02109 | DOI

[5] Dunkl C. F., Luque J. G., “Vector-valued Jack polynomials from scratch”, SIGMA, 7 (2011), 026, 48 pp., arXiv: 1009.2366 | DOI | MR | Zbl

[6] Felder G., Veselov A. P., “Polynomial solutions of the Knizhnik–Zamolodchikov equations and Schur–Weyl duality”, Int. Math. Res. Not., 2007 (2007), rnm046, 21 pp., arXiv: math.RT/0610383 | DOI | MR | Zbl

[7] Griffeth S., “Orthogonal functions generalizing Jack polynomials”, Trans. Amer. Math. Soc., 362 (2010), 6131–6157, arXiv: 0707.0251 | DOI | MR | Zbl

[8] James G., Kerber A., The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981 | MR | Zbl

[9] Opdam E. M., “Harmonic analysis for certain representations of graded Hecke algebras”, Acta Math., 175 (1995), 75–121 | DOI | MR | Zbl

[10] Stembridge J. R., “On the eigenvalues of representations of reflection groups and wreath products”, Pacific J. Math., 140 (1989), 353–396 | DOI | MR | Zbl

[11] Stoer J., Bulirsch R., Introduction to numerical analysis, Springer-Verlag, New York–Heidelberg, 1980 | DOI | MR | Zbl