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.
Highlights
In 1971 Calogero [Cal71] studied the quantum system describing N particles on the line pairwise interacting with the rational potentialUR(x1, . . . , xN ) =1≤i< j≤N (xi γ − x j )2+ ω2x2, N x2 = xi2. i =1 (1)Almost at the same time Sutherland [Sut71] considered the quantum system of N particles on a circle with trigonometric interactionUT (x1, . . . , xN ) γ a2 sin2 a(xi −, xj)
Under this equivalence integrals are mapped to integrals with the Hamiltonian of the rational system being mapped to the momentum of the trigonometric system. To derive this map Nekrasov used the Hamiltonian reduction, following the ideas of the work by Kazhdan, Kostant and Sternberg [KKS78]. This explains an earlier construction of Lassalle [Las91] of multivariable Hermite polynomials from Jack polynomials, which can be interpreted as a correspondence between the eigenfunctions of the two quantum systems with potentials (1) and (2)
Let DCT M and DCRM be the algebras of quantum integrals of the trigonometric CM system and the rational CM system with an harmonic term generated by the differential operators Lp and LHp, p ∈ ΛN, respectively
Summary
In 1971 Calogero [Cal71] studied the quantum system describing N particles on the line pairwise interacting with the rational potential. It came as a surprise when in 1997 Nekrasov [Nek97] discovered that the systems (both in the classical and the quantum case) are essentially equivalent He showed that there is a symplectomorphism from the phase space of the rational system onto the open (positive) part of the phase space of the trigonometric system. To derive this map Nekrasov used the Hamiltonian reduction, following the ideas of the work by Kazhdan, Kostant and Sternberg [KKS78] This explains an earlier construction of Lassalle [Las91] of multivariable Hermite polynomials from Jack polynomials, which can be interpreted as a correspondence between the eigenfunctions of the two quantum systems with potentials (1) and (2) (see Baker and Forrester’s paper [BF97] which, in particular, contains further unpublished results due to Lassalle).
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