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Publications and source records attributed to Alexander A. Kirillov Jr..
These are the notes from a series of lectures the author gave at Harvard University in the Fall of 1994. The goal of these lectures is to give a self-contained exposition of recent result of Cherednik (\cite{C6}), who proved Macdonald's inner product conjectures (see \cite{M2}) for arbitrary root systems, using the algebraic structure he called ``double affine Hecke algebra''.
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]).