Geodesic
Calogero Operator and Lie Superalgebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 3, pp. 355-376 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {A. N. Sergeev},
     title = {Calogero {Operator} and {Lie} {Superalgebras}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2002_131_3_a0/}
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A. N. Sergeev. Calogero Operator and Lie Superalgebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 3, pp. 355-376. http://geodesic.mathdoc.fr/item/TMF_2002_131_3_a0/

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