New orthogonality relations for super-Jack polynomials and an associated Lassalle--Nekrasov correspondence

Abstract

The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in $n+m$ variables, which reduce to the Jack polynomials when $n=0$ or $m=0$ and provide joint eigenfunctions of the quantum integrals of the trigonometric deformed Calogero-Moser-Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form $(p,q)\mapsto (L_pq)(0)$, with $L_p$ quantum integrals of the rational deformed Calogero-Moser-Sutherland system. In addition, we provide a new proof of the Lassalle-Nekrasov correspondence between trigonometric and rational harmonic deformed Calogero-Moser-Sutherland systems and infer orthogonality of super-Hermite polynomials, which provide joint eigenfunctions of the latter system.