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 $\mathfrak{gl}(n,m)$. 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 $k[\epsilon {]}^{\otimes r}$ where $k[\epsilon ]$ is the algebra of the dual numbers the corresponding representation is the regular representation of the algebra $k[\epsilon {]}^{\otimes r}$.

29 pages, Corrected typos, Added refferences

References in corpus (1)

• Deformed quantum Calogero-Moser problems and Lie superalgebras