On the Affine Analogue of Jack's and Macdonald's Polynomials

Introduction. Jack's and Macdonald's polynomials are an important class of symmetric functions associated to root systems. In this paper we deene and study an analogue of Jack's and Macdonald's polynomials for aane root systems. Our approach is based on representation theory of aane Lie algebras and quantum aane algebras, and follows the ideas of our recent papers EK1,EK2,EK3]. We start with a review of the theory of Jack (Jacobi) polynomials associated with the root system of a simple Lie algebra g. This theory was described in the papers of Heckman and Opdam HO,H1,O1,O2]. In these papers, Jack's polynomials are deened as a basis in the space of Weyl group invariant trigonometric polynomials which 1) diiers from the basis of orbitsums by a triangular matrix (with respect to the standard partial ordering on dominant integral weights) with ones on the diagonal, and 2) is an eigenbasis for a certain second order diierential operator (the Sutherland-Olshanetsky-Perelomov operator, Su,OP]). It turns out that these conditions determine Jack's polynomials uniquely. Orbitsums and characters for g turn out to be special cases of Jack's polynomials. These polynomials have a q-deformation, which is called Macdonald's polynomials; they have been introduced by I. Macdonald in his papers M1, M2] and have been intensively studied since that time. We generalize the deenition of Jack's polynomials to the case of aane root systems. We assign such a polynomial to every dominant integral weight of the aane root system. It is done in the same way as for the usual root systems: the only thing one has to do is replace the Sutherland operator by its aane analogue. This analogue is constructed in the same way as for usual root systems, and it turns out to be (after specialization of level) a parabolic diierential operator whose coeecients are elliptic functions. This operator was introduced in EK3] (for the root system A n?1) and is closely related to the Sutherland operator with elliptic coeecients considered in OP], but is more general. Analogously to the nite-dimensional case, orbitsums and characters (of integrable modules) for the aane Lie algebra ^ g are special cases of aane Jack's polynomials. For orbitsums and characters of aane Lie algebras, there is a beautiful theory of modular invariance described in K]. We generalize this theory to general aane Jack's polynomials. It turns out that the nite-dimensional space spanned by the Jack's polynomials of a given level is …