Eigenvalues of Heckman-Polychronakos operators

TL;DR

This paper explicitly computes the eigenvalues of Heckman-Polychronakos operators, which are commuting differential-difference operators related to Dunkl operators, for various types of eigenfunctions, advancing understanding in integrable systems and symmetric functions.

Contribution

The paper provides explicit formulas for the eigenvalues of Heckman-Polychronakos operators for symmetric, skew-symmetric, and polynomial eigenfunctions, a novel contribution to the theory of these operators.

Findings

Eigenvalues for symmetric eigenfunctions are explicitly derived.

Eigenvalues for skew-symmetric eigenfunctions are explicitly derived.

Partial sums of eigenvalues for general polynomial eigenfunctions are computed.

Abstract

Heckman-Polychronakos operators form a prominent family of commuting differential-difference operators defined in terms of the Dunkl operators $\mathcal D_i$ as $\mathcal P_m= \sum_{i=1}^N (x_i \mathcal D_i)^m$. They have been known since 1990s in connection with trigonometric Calogero-Moser-Sutherland Hamiltonian and Jack symmetric polynomials. We explicitly compute the eigenvalues of these operators for symmetric and skew-symmetric eigenfunctions, as well as partial sums of eigenvalues for general polynomial eigenfunctions.