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arXiv · 2008.06010

Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence

M. V. Feigin·M. A. Hallnäs·A. P. Veselov
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Abstract

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.

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M. V. Feigin, M. A. Hallnäs, A. P. Veselov. 2020-08-13. Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence. https://doi.org/10.1007/s00220-021-04036-8

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