BC-infinity Calogero-Moser operator and super Jacobi polynomials

TL;DR

This paper introduces an infinite-dimensional Calogero-Moser operator of BC-type and explores its associated super Jacobi polynomials, establishing their algebraic structure, integrability, and connections to finite-dimensional systems.

Contribution

It develops the theory of an infinite-dimensional BC-type Calogero-Moser operator and links it to super Jacobi polynomials, extending finite-dimensional results and integrability.

Findings

Introduction of an infinite-dimensional BC-type Calogero-Moser operator.

Derivation of algebraic properties and formulas for super Jacobi polynomials.

Establishment of integrability of related quantum systems.

Abstract

An infinite-dimensional version of Calogero-Moser operator of $BC$-type and the corresponding Jacobi symmetric functions are introduced and studied, including the analogues of Pieri formula and Okounkov's binomial formula. We use this to describe all the ideals linearly generated by the Jacobi symmetric functions and show that the deformed $BC(m,n)$ Calogero-Moser operators, introduced in our earlier work, appear here in a natural way as the restrictions of the $BC_{\infty}$ operator to the corresponding finite-dimensional subvarieties. As a corollary we have the integrability of these quantum systems and all the main formulas for the related super Jacobi polynomials.