Coxeter群的Kazhdan-Lusztig多项式与胞腔理论对于研究Coxeter群及其对应的Hecke代数的表示、代数群的表示、李型有限群和李代数的表示起了非常重要的作用。其中Kazhdan-Lusztig多项式的首项系数对Hecke代数的表示理论有重要意义，本项目将确定对称群的某些Kazhdan-Lusztig多项式的首项系数。通过对Coxeter群的胞腔分解而得到的左胞腔表示将有助于研究其相应的Hecke代数的不可约表示。例如，A型Hecke代数的所有不可约表示可以由左胞腔表示给出。本项目将研究多参数下某些具体的Coxeter群的Kazhdan-Lusztig多项式和胞腔刻画，并验证Lusztig提出的某些猜想。对于拟分裂情形下的某些类型的仿射Weyl群，本项目将研究其胞腔刻画和Lusztig的关于胞腔的连通性猜想。本项目还将对某些Hecke代数的胞腔表示进行研究。

The Kazhdan-Lusztig polynomials and cell theory of a Coxeter group play a central role in the representation theory of the Coxeter group and its corresponding Hecke algebra, algebraic groups, finite groups of Lie type and Lie algebras. The leading coefficient of some Kazhdan-Lusztig polynomials are of great importance for the representation theory of Hecke algebra and we will determine the leading coefficient of some Kazhdan-Lusztig polynomials of the symmetric group. The decomposition for Coxeter groups of the left cell representations will be useful for studying the irreducible representations of the corresponding Hecke algebras. For example, in type A, the representations afforded by left cells give all the irreducible representations of the corresponding Hecke algebras. We will compute the Kazhdan-Lusztig polynomials of some Coxeter groups with unequal parameters, study the decomposition for the cells of the Coxeter groups, and verify the validity of some Lusztig's conjectures in these cases. For the affine Weyl groups of some types in the quasi-split case, we will study the decomposition for the cells of the affine Weyl groups, and verify the Lusztig's conjecture for the connectness of cells. We will also study cell representations of some Coxeter groups and their corresponding Hecke algebras.