Eigenvalues of Heckman-Polychronakos operators

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.

20 pages, to appear in Algebraic Combinatorics