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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Lectures on affine Hecke algebras and Macdonald’s conjectures
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by Alexander A. Kirillov Jr. PDF
Bull. Amer. Math. Soc. 34 (1997), 251-292 Request permission

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
  • 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 1997 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 34 (1997), 251-292
  • MSC (1991): Primary 05E35; Secondary 33D80
  • DOI: https://doi.org/10.1090/S0273-0979-97-00727-1
  • MathSciNet review: 1441642