New Orthogonality Relations for Super-Jack Polynomials and an Associated Lassalle–Nekrasov Correspondence

Abstract

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