Eigenvalues of Heckman–Polychronakos operators

Heckman–Polychronakos operators form a prominent family of commuting differential-difference operators defined in terms of the Dunkl operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝒟</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>𝒫</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>N</mml:mi> </mml:msubsup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>𝒟</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> </mml:math> . 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.