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Super Jack-Laurent Polynomials

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Abstract

Let \(\mathcal {D}_{n,m}\) be the algebra of 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 multiplicity free and we prove that the image of the algebra \(\mathcal {D}_{n,m}\) in the algebra of endomorphisms of the generalised eigenspace is k[ε]⊗r where k[ε] is the algebra of dual numbers. The corresponding representation is the regular representation of the algebra k[ε]⊗r.

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Acknowledgements

This work has been funded by Russian Ministry of Education and Science (grant 1.492.2016/1.4) and partially by the Russian Academic Excellence Project ‘5-100’. I am very grateful to the anonymous referee for an excellent job, which help me to improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, Saratov State University, Astrakhanskaya 83, Saratov, 410012, Russia

    A. N. Sergeev

  2. National Research University Higher School of Economics, Moscow, Russia

    A. N. Sergeev

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  1. A. N. Sergeev
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Correspondence to A. N. Sergeev.

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Presented by: Michael Pevzner

Dedicated to my teacher A.A. Kirillov on the occasion of his 81st birthday.

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Sergeev, A.N. Super Jack-Laurent Polynomials. Algebr Represent Theor 21, 1177–1202 (2018). https://doi.org/10.1007/s10468-018-9778-4

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  • DOI: https://doi.org/10.1007/s10468-018-9778-4

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