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This article is cited in 24 scientific papers (total in 24 papers)
Calogero Operator and Lie Superalgebras
A. N. Sergeev Balakovo Institute of Technique, Technology and Control
Abstract:
We construct a supersymmetric analogue of the Calogero operator $\mathcal S\mathcal L$ which depends on the parameter $k$. This analogue is related to the root system of the Lie superalgebra $\mathfrak {gl}(n|m)$. It becomes the standard Calogero operator for $m = 0$ and becomes the operator constructed by Veselov, Chalykh, and Feigin up to changing the variables and the parameter $k$ for $m = 1$. For $k = 1$ and 1/2, the operator $\mathcal S\mathcal L$ is the radial part of the second-order Laplace operator for the symmetric superspaces corresponding to the respective pairs $(\mathfrak {gl}\oplus \mathfrak {gl}, \mathfrak {gl})$, $(\mathfrak {gl},\mathfrak {osp})$.
We show that for any m and n, the supersymmetric analogues of the Jack polynomials constructed by Kerov, Okounkov, and Olshanskii are eigenfunctions of the operator $\mathcal S\mathcal L$. For $k = 1$ and 1/2, the supersymmetric analogues of the Jack polynomials coincide with the spherical functions on the above superspaces. We also study the algebraic analogue of the Berezin integral.
Received: 19.12.2001
Citation:
A. N. Sergeev, “Calogero Operator and Lie Superalgebras”, TMF, 131:3 (2002), 355–376; Theoret. and Math. Phys., 131:3 (2002), 747–764
Linking options:
https://www.mathnet.ru/eng/tmf334https://doi.org/10.4213/tmf334 https://www.mathnet.ru/eng/tmf/v131/i3/p355
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