Lectures on affine Hecke algebras and Macdonald’s conjectures
Lectures on affine Hecke algebras and Macdonald’s conjectures
复制标题
DOI：
10.1090/s0273-0979-97-00727-1
复制
发表时间：
1997-07
影响因子：
1.3
通讯作者：
A. Kirillov
中科院分区：
文献类型：
--
作者：
A. Kirillov
本文综述了切雷德尼克关于麦克唐纳多项式及相关特殊函数的表示论方法的结果。Macdonald多项式是一个显着的双参数多项式族，它可以与任何根系相关联。作为特殊情况，它们包括Schur函数，q-Jacobi多项式，以及真实的和p-adic对称空间上的某些球面函数。它们具有许多优雅的组合性质，然而，这些性质极难证明。在本文中，我们表明，一个自然的设置为研究这些多项式的Hecke代数的表示理论，并显示如何可以用来证明一些组合的麦克唐纳多项式的身份。
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.