JACK POLYNOMIALS FOR THE BCn ROOT SYSTEM AND GENERALIZED SPHERICAL FUNCTIONS

Functions on the double coset space K\G/K, where G is a group and K is a subgroup of G are called spherical functions. One can also study functions on G/K with values in a representation V of G which are equivariant with respect to the left action of K. This more general class of functions may be called vector valued spherical functions. The theory of such functions was developed by Harish-Chandra, Helgason and other authors [1],[2],[3]. In the case when G = K ×K, studying spherical functions is equivalent to studying functions on K, equivariant with respect to conjugation. When K is reductive over C, the Peter-Weyl theorem gives us a description of the space of conjugacy equivariant functions as the space spanned by vector valued characters. The article [4] deals with such kind of spherical functions in the case when K = SL(n,C). Namely, the Laplace operator on K restricted to the space of spherical functions can be written in terms of coordinates along the maximal torus. If V = SC, the resulting operator (the so called radial part of the Laplacian) coincides, up to an obvious conjugation, with the Sutherland differential operator, which is the Hamiltonian of the Calogero-Moser quantum mechanical system for the root system An−1 [8], [10], [9]. Differential operators on K corresponding to the higher Casimir operators can also be written in terms of coordinates along the maximal torus. These operators are quantum integrals of this Calogero-Moser system. Moreover, Weyl group invariant eigenfunctions of the Sutherland operator can be expressed as vector valued traces of some intertwining operators, which gives representation theoretic formulas for Jack polynomials for the root system An−1. The main result of this paper is that if we consider the case G = GL(m + n,C) (m ≥ n) and K = GL(m,C) × GL(n,C) and slightly modify the definition of spherical functions, then we get a similar theory for the root system BCn. Namely, in this case the Laplace operator on G, written in terms of coordinates of some torus inside G, gives us the Sutherland operator for the root system BCn, and the higher Casimir operators give us quantum integrals of the corresponding quantum mechanical system. Furthermore, restriction to the torus of some special matrix elements of irreducible representations Lλ of G, where λ ranges over a set isomorphic to the cone of dominant integral weights for Sp(n), yields W -invariant