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_n-1 one can obtain Macdonald's polynomials - a new interesting class of symmetric functions recently defined by I. Macdonald {M1] - as weighted traces of intertwining operators between certain finite-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_n-1. Macdonald's inner product identities (see [M2]) have been proved by combinatorial methods my Macdonald ([Macdonald, private communication]). 
Pavel I. Etingof, Alexander A. Kirillov Jr.