Super Jack-Laurent Polynomials

TL;DR

This paper studies the spectral decomposition of quantum integrals related to deformed Calogero-Moser-Sutherland systems for Lie superalgebras, revealing a regular representation structure involving dual numbers.

Contribution

It characterizes the spectral decomposition of the algebra of quantum integrals acting on quasi-invariant Laurent polynomials for general parameter values.

Findings

Spectral decomposition is not simple for general parameters.

The image of the algebra in endomorphisms is isomorphic to a tensor product of dual number algebras.

The representation is the regular representation of the algebra of dual numbers.

Abstract

Let $\mathcal{D}_{n,m}$ be the algebra of the quantum integrals of the deformed Calogero-Moser-Sutherland problem corresponding to the root system of the Lie superalgebra $\frak{gl}(n,m)$. The algebra $\mathcal{D}_{n,m}$ acts naturally on the quasi-invariant Laurent polynomials and we investigate the corresponding spectral decomposition. Even for general value of the parameter $k$ the spectral decomposition is not simple and we prove that the image of the algebra $\mathcal{D}_{n,m}$ in the algebra of endomorphisms of the generalised eigen-space is $k[\varepsilon]^{\otimes r}$ where $k[\varepsilon]$ is the algebra of the dual numbers the corresponding representation is the regular representation of the algebra $k[\varepsilon]^{\otimes r}$.