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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. [ABSTRACT FROM AUTHOR]

We study the orbits and polynomial invariants of certain affine action of the super Weyl groupoid of Lie superalgebra gl(n,m), depending on a parameter. We show that for generic values of the parameter all the orbits are finite and separated by certain explicitly given invariants. We also describe explicitly the special set of parameters, for which the algebra of invariants is not finitely generated and does not separate the orbits, some of which are infinite. [ABSTRACT FROM AUTHOR]

An integral formula for the solutions of Knizhnik–Zamolodchikov (KZ) equation with values in an arbitrary irreducible representation of the symmetric group SN is presented for integer values of the parameter. The corresponding integrals can be computed effectively as certain iterated residues determined by a given Young diagram and give polynomials with integer coefficients. The derivation is based on Schur–Weyl duality and the results of Matsuo on the original SU(n) KZ equation. The duality between the spaces of solutions with parameters m and −m is discussed in relation with the intersection pairing in the corresponding homology groups. [ABSTRACT FROM PUBLISHER]