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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
DOI:
10.4213/tmf334