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

Journal article, 2024

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 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) ↦ (Lpq) (0) , with Lp quantum 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.