Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

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