Orthogonality of super-Jack polynomials and a Hilbert space interpretation of deformed Calogero-Moser-Sutherland operators

TL;DR

This paper establishes orthogonality and explicit norms for super-Jack polynomials under a specific Hermitian product, extending classical results and providing a Hilbert space framework for deformed Calogero-Moser-Sutherland operators.

Contribution

It introduces a new positive semi-definite Hermitian product for super-Jack polynomials, generalizing Macdonald's inner product, and offers a Hilbert space interpretation of deformed Calogero-Moser-Sutherland operators.

Findings

Proved orthogonality of super-Jack polynomials under the new inner product.

Explicitly computed quadratic norms for super-Jack polynomials.

Identified the kernel of the Hermitian form and connected to deformed Calogero-Moser-Sutherland operators.

Abstract

We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials $SP_\lambda((z_1,\ldots,z_n),(w_1,\ldots,w_m);\theta)$ with respect to a natural positive semi-definite, but degenerate, Hermitian product $\langle\cdot,\cdot\rangle_{n,m}^\prime$. In case $m=0$ (or $n=0$), our product reduces to Macdonald's well-known inner product $\langle\cdot,\cdot\rangle_n^\prime$, and we recover his corresponding orthogonality results for the Jack polynomials $P_\lambda((z_1,\ldots,z_n);\theta)$. From our main results, we readily infer that the kernel of $\langle\cdot,\cdot\rangle_{n,m}^\prime$ is spanned by the super-Jack polynomials indexed by a partition $\lambda$ not containing the $m\times n$ rectangle $(m^n)$. As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero-Moser-Sutherland operators of type $A(n-1,m-1)$.