ar X iv : h ep - t h / 93 12 10 3 v 1 1 3 D ec 1 99 3 MACDONALD ’ S POLYNOMIALS AND REPRESENTATIONS OF QUANTUM GROUPS

Recently I.Macdonald defined a family of systems of orthogonal symmetric polynomials depending of two parameters q, k which interpolate between Schur’s symmetric functions and certain spherical functions on SL(n) over the real and p-adic fields [M]. These polynomials are labeled by dominant integral weights of SL(n), and (as was shown by I.Macdonald) are uniquely defined by two conditions: 1) they are orthogonal with respect to a certain weight function, and 2) the matrix transforming them to Schur’s symmetric functions is strictly upper triangular with respect to the standard partial ordering on weights (“strictly” means that the diagonal entries of this matrix are equal to 1). Another definition of Macdonald’s polynomials is that they are (properly normalized) common eigenfunctions of a commutative set of n self-adjoint partial difference operators M1, ...,Mn (Macdonald’s operators) in the space of symmetric polynomials. In this paper we present a formula for Macdonald’s polynomials which arises from the representation theory of the quantum group Uq(sln). This formula expresses Macdonald’s polynomials via (weighted) traces of intertwining operators between certain modules over Uq(sln). The paper is organized as follows. In Section 1, we define Macdonald’s inner product, orthogonal polynomials, and commuting difference operators, and compute the eigenvalues of these operators. In Section 2, we review some facts about representations of quantum groups that will be needed in the following sections. In Section 3 we introduce weighted traces of intertwiners (vector-valued characters) and prove an analogue of the Weyl orthogonality theorem for them. In Section 4 we formulate the main result – the explicit formula for Macdonald’s polynomials for positive integer values of k – and give a complete proof of this formula. In Section 5, we generalize the result of Section 4 to the case of an arbitrary k. In Section 6, we construct Macdonald’s operators from the generators of the center of