Quasi-Invariant Hermite Polynomials And Lassalle–Nekrasov Correspondence.pdf

The article explores the quasi-invariant Hermite polynomials and the Lassalle–Nekrasov correspondence within the context of digital objects (DOI) and mathematical physics. It builds upon previous works by Calogero, Sutherland, Moser, Adler, Perelomov, Nekrasov, and Lassalle among others.

Key Points:

1. Quantum Systems and Their Equivalence: The authors focus on two quantum systems: one with a rational potential (Calogero system) and another with a trigonometric interaction (trigonometric Calogero system). Nekrasov discovered their surprising equivalence, known as the Lassalle–Nekrasov correspondence, which maps integrals from the rational to the trigonometric system.

2. Rational Cherednik Algebra and Quasi-Invariant Polynomials: They introduce the rational Cherednik algebra and its quasi-invariant polynomials QA for a general configuration A of hyperplanes in Euclidean space RN with multiplicities mα. These polynomials are invariant up to order 2mα under orthogonal reflections about the hyperplanes.

3. A-Hermite Polynomials: The authors define A-Hermite polynomials Hqi(x) as generating functions related to the rational Baker–Akhiezer (BA) function φ(x, λ). These polynomials form a linear basis in QA and are non-homogeneous with the highest degree term qi(x), where qi are quasi-invariants.

4. Hermitisation Map: A Hermitisation map χH: QA → QA is introduced, which sends q to Hq. This map has explicit formulas involving the rational Calogero–Moser (CM) Hamiltonian L and its quantum integral with highest order term q(∂). It generalises properties of classical Hermite polynomials.

5. Generalised CM Hamiltonian and Eigenfunctions: The A-Hermite polynomials are related to eigenfunctions Ψq(x) of the generalised CM Hamiltonian (3) with a harmonic term, where Am(x) = α∈A(x, α)mα.

6. Quasi-Invariant Lassalle–Nekrasov Correspondence: In Section 5, they extend the Lassalle–Nekrasov correspondence to quasi-invariants for the Coxeter configuration A of type AN−1 with all roots of multiplicity m. This demonstrates how the correspondence intertwines the action on QA of quantum i