A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus

TL;DR

This paper analyzes a system of linear differential equations related to vector-valued Jack polynomials on the torus, showing the measure's absolute continuity and analyticity within connected components, advancing understanding of their spectral properties.

Contribution

It provides a detailed analysis of the differential system associated with vector-valued Jack polynomials, proving the measure's absolute continuity and analyticity on each component of the torus.

Findings

The orthogonality measure has no singular part and is absolutely continuous.

The measure is given by a matrix function times Haar measure.

The matrix function is analytic on each connected component.

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, 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, arXiv:1511.06721] 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 $x_{i}=x_{j}$, $i<j$, 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.