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Lectures on affine Hecke algebras and Macdonald's conjectures

Author(s): Alexander A. Kirillov Jr.
Journal: Bull. Amer. Math. Soc. 34 (1997), 251-292.
MSC (1991): Primary 05E35; Secondary 33D80
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Abstract: This paper gives a review of Cherednik's results on the
representation-theoretic approach to Macdonald polynomials and related special functions. Macdonald polynomials are a remarkable 2-parameter family of polynomials which can be associated to every root system. As special cases, they include the Schur functions, the $q$-Jacobi polynomials, and certain spherical functions on real and $p$-adic symmetric spaces. They have a number of elegant combinatorial properties, which, however, are extremely difficult to prove. In this paper we show that a natural setup for studying these polynomials is provided by the representation theory of Hecke algebras and show how this can be used to prove some of the combinatorial identities for Macdonald polynomials.

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Additional Information:

Alexander A. Kirillov Jr.
Affiliation: Department of Mathematics, MIT, Cambridge, Massachusetts 02139
Email: kirillov@math.mit.edu

DOI: 10.1090/S0273-0979-97-00727-1
PII: S 0273-0979(97)00727-1
Received by editor(s): January 19, 1995,
Received by editor(s) in revised form: April 3, 1997
Additional Notes: The author was supported by Alfred P. Sloan dissertation fellowship.
Copyright of article: Copyright 1997, American Mathematical Society

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