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Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
journal contributionposted on 08.02.2021, 11:19 by Misha Feigin, Martin Hallnas, Alexander Veselov
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 quasiinvariant 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-quasiinvariants, which is an eigenbasis for the corresponding generalised rational Calogero–Moser operator with harmonic term. In the case of the Coxeter configuration of type AN this leads to a quasi-invariant version of the Lassalle– Nekrasov correspondence and its higher order analogues.
Swedish Research Council (Reg. nr. 2018-04291)
Russian Science Foundation grant no. 20-11-20214
- Mathematical Sciences