Calogero operator and Lie superalgebras. (English. Russian original) Zbl 1039.81028

Sergeev, A. N.

Theor. Math. Phys. 131, No. 3, 747-764 (2002); translation from Teor. Mat. Fiz. 131, No. 3, 355-376 (2002).

Summary: We construct a supersymmetric analogue of the Calogero operator \(\mathcal{SL}\), 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 A. P. Veselov, M. V. Feigin and O. A. Chalykh [Russ. Math. Surv. 51, No. 3, 573–574 (1996); translation from Usp. Mat. Nauk 51, No. 3, 185–186 (1996; Zbl 0874.35098)], and up to changing the variables and the parameter \(k\) for \(m = 1\). For \(k = 1\) and 1/2, the operator \(\mathcal{SL}\) is the radial part of the second-order Laplace operator for the symmetric superspaces corresponding to the respective pairs \((\mathfrak{gl}\oplus\mathfrak{gl},\mathfrak{gl})\) and \((\mathfrak{gl},\mathfrak{osp})\). We show that for any \(m\) and \(n\), the supersymmetric analogues of the Jack polynomials constructed by S. Kerov, A. Okounkov and G. Olshanskii [Int. Math. Res. Not. 1998, No. 4, 173–199 (1998; Zbl 0960.05107)] are eigenfunctions of the operator \(\mathcal{SL}\). 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. This paper is a detailed version of [the author, J. Nonlinear Math. Phys. 8, 59–64 (2001; Zbl 0972.35114)].

MSC:

81Q60	Supersymmetry and quantum mechanics
81R12	Groups and algebras in quantum theory and relations with integrable systems
17B80	Applications of Lie algebras and superalgebras to integrable systems

Keywords:

Calogero operator; Jack polynomial; Lie superalgebra; symmetric superspace; spherical function