Super Jack-Laurent Polynomials

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 $k[\varepsilon]^{\otimes r}$.