Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence

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.