  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.
