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 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 A -Hermite polynomials. These polynomials form a linear basis in the space of 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.