Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence

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 ex...

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Main Authors: Misha Feigin, Martin Hallnas, Alexander Veselov
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Published: 2021
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Online Access:https://hdl.handle.net/2134/13712527.v1
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spelling rr-article-137125272021-03-15T00:00:00Z Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence Misha Feigin (10096675) Martin Hallnas (7159559) Alexander Veselov (1259028) Foundations of quantum mechanics Mathematical Physics Pure Mathematics Quantum Mechanics 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. 2021-03-15T00:00:00Z Text Journal contribution 2134/13712527.v1 https://figshare.com/articles/journal_contribution/Quasi-invariant_Hermite_polynomials_and_Lassalle-Nekrasov_correspondence/13712527 CC BY 4.0
institution Loughborough University
collection Figshare
topic Foundations of quantum mechanics
Mathematical Physics
Pure Mathematics
Quantum Mechanics
spellingShingle Foundations of quantum mechanics
Mathematical Physics
Pure Mathematics
Quantum Mechanics
Misha Feigin
Martin Hallnas
Alexander Veselov
Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
description 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.
format Default
Article
author Misha Feigin
Martin Hallnas
Alexander Veselov
author_facet Misha Feigin
Martin Hallnas
Alexander Veselov
author_sort Misha Feigin (10096675)
title Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
title_short Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
title_full Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
title_fullStr Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
title_full_unstemmed Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
title_sort quasi-invariant hermite polynomials and lassalle-nekrasov correspondence
publishDate 2021
url https://hdl.handle.net/2134/13712527.v1
_version_ 1801898377492824064