Superanalogs of the Calogero operators and Jack polynomials

Alexander Sergeev

doi:10.2991/jnmp.2001.8.1.7

paper

Superanalogs of the Calogero operators and Jack polynomials

arXiv:math/0106222 · doi:10.2991/jnmp.2001.8.1.7

Abstract

A depending on a complex parameter $k$ superanalog $S L$ of Calogero operator is constructed; it is related with the root system of the Lie superalgebra $gl (n ∣ m)$ . For $m = 0$ we obtain the usual Calogero operator; for $m = 1$ we obtain, up to a change of indeterminates and parameter $k$ the operator constructed by Veselov, Chalykh and Feigin [2,3]. For $k = 1, \frac{1}{2}$ the operator $S L$ is the radial part of the 2nd order Laplace operator for the symmetric superspaces corresponding to pairs $(G L (V) \times G L (V), G L (V))$ and $(G L (V), O S p (V))$ , respectively. We will show that for the generic $m$ and $n$ the superanalogs of the Jack polynomials constructed by Kerov, Okunkov and Olshanskii [5] are eigenfunctions of $S L$ ; for $k = 1, \frac{1}{2}$ they coinside with the spherical functions corresponding to the above mentioned symmetric superspaces. We also study the inner product induced by Berezin's integral on these superspaces.

Superanalogs of the Calogero operators and Jack polynomials

Abstract

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