■型仿射Weyl群的Kazhdan-Lusztig多项式的首项系数

Leading Coefficients of the Kazhdan-Lusztig Polynomials for an Affine Weyl Group of Type ■

Kazhdan-Lusztig(以下简称“K-L”)多项式是K-L理论中一个核心的研究对象,当考虑Weyl群或者仿射Weyl群时,它们的K-L多项式的首项系数对理解全部的K-L多项式起着关键的作用,同时在表示理论及李理论中也有深刻的意义.然而,关于这些K-L多项式的首项系数研究结果并不多.为了研究仿射Weyl群W-图的非局部有限性,Lusztig引入了与K-L多项式的首项系数μ(y,w)相关的一些半线性方程,这些半线性方程对于求解首项系数起着重要作用.在此主要借助半线性方程这一有力工具,来计算A3型仿射Weyl群的K-L多项式的部分首项系数.

The Kazhdan-Lusztig(abbreviated as“K-L”)polynomial is a core research object in K-L theory.When Weyl groups or affine Weyl groups are considered,the leading coefficients of their K-L polynomials play a key role in understanding all K-L polynomials.At the same time,they are also profound in representation theory and Lie theory.However,for the leading coefficients of the K-L polynomials,the results of the study are sporadic.In order to study the nonlocal finiteness of W-graphs in affine Weyl groups,some semi-linear equations related to the leading coefficientsμ(y,w)of the K-L polynomials are introduced by Lusztig.They play an important role in solving the leading coefficients.In this paper,for the K-L polynomials of A3 affine Weyl groups,the powerful tool of semi-linear equation is used to calculate some of the leading coefficients.