Representation-theoretic proof of the inner product and symmetry identities for MacDonald's polynomials
Pavel I. Etingof, Alexander A. Kirillov
Abstract
This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system An-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 Uq sln. 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 An-1. Macdonald's inner product identities (see [M2]) have been proved by combinatorial methods my Macdonald ([Macdonald, private communication]).
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