Representation-theoretic Proof of the Inner Product and Symmetry Identities for Macdonald's Polynomials

This paper is a continuation of our papers EK1, EK2]. In EK2] we showed that for the root system A n1 one can obtain Macdonald's polynomials { a new interesting class of symmetric functions recently deened by I. Macdonald M1] { as weighted traces of intertwining operators between certain nite-dimensional representations of U q sl n. The main goal of the present paper is to use this construction to give a representation-theoretic proof of Macdonald's inner product and symmetry identities for the root system A n1. Macdonald's inner product identities (see M2]) have been proved by combinatorial methods by Macdonald (unpublished) for the root system A n1 and by Cherednik in the general case; symmetry identities for the root system A n1 have also been proved by Macdonald ((Macdonald, private communication]). The paper is organized as follows. In Section 1 we brieey list the basic deenitions. In Section 2 we deene Macdonald's polynomials P and recall the construction of P and the inner product between them for the root system A n1 in terms of intertwining operators. By deenition, hP ; P i = 0 if 6 = , and we show that hP ; P i can be expressed as a certain matrix element of product of two intertwining operators. In Section 3 we use the Shapovalov determinant formula to analyze the poles of matrix coeecients of an intertwining operator, and this allows us to express the product of two intertwining operators in terms of a single intertwiner. Applying this to the formula for hP ; P i obtained in Section 2, we prove the Macdonald's inner product identity, and the right-hand side is obtained as a product of linear factors in Shapovalov determinant formula. In Section 4 we prove the symmetry identity, which relates the values of P (q 2(+k)) and P (q 2(+k)); the proof is based on the construction of Section 2 and the technique of representing identities in the category of representations of a quantum group by ribbon graphs ((RT1, RT2]). In Section 5 we use the symmetry identities and the fact that Macdonald polynomials are eigenfunctions of certain diierence operators (Macdonald operators) to derive recurrence relation for Macdonald polynomials.