Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence

TL;DR

This paper explains the Lassalle-Nekrasov correspondence using rational Cherednik algebras, introduces $\\\\mathcal A$-Hermite polynomials as a new basis for quasi-invariants, and extends the correspondence to quasi-invariant settings.

Contribution

It provides a conceptual framework for the Lassalle-Nekrasov correspondence via rational Cherednik algebras and introduces new $\\mathcal A$-Hermite polynomials for quasi-invariant spaces.

Findings

Introduction of $\\mathcal A$-Hermite polynomials as a basis for quasi-invariants.

Establishment of a quasi-invariant version of the Lassalle-Nekrasov correspondence.

Extension of the correspondence to higher order analogues.

Abstract

Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero-Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations $\mathcal A$ of real hyperplanes with multiplicities admitting the rational Baker-Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call $\mathcal A$-Hermite polynomials. These polynomials form a linear basis in the space of $\mathcal A$-quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero-Moser operator with harmonic term. In the case of the Coxeter configuration of type $A_N$ this leads to a quasi-invariant version of the Lassalle-Nekrasov correspondence and its higher order analogues.