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.