与a-函数相关的若干问题

a-函数是Kazhdan-Lusztig理论中的一个核心概念。关于a-函数有如下两个公开猜想：任意Coxeter群的a-函数的有界性猜想以及关于不等参数Hecke代数的猜想P1-P15。围绕a-函数，本项目拟：.1) 对双曲Coxeter群证明a-函数的有界性;.2) 对仿射李代数的最高权单模定义“Gelfand-Kirillov余维数”的概念，并证明它由仿射Weyl群的a-函数给出;.3) 将半单李代数的BGG模范畴中倾斜模的投射维数的概念推广至仿射李代数情形，并证明它由仿射Weyl群的a-函数给出;.4) 对秩3双曲Coxeter群考察猜想P1-P15;.5) 给出关于BCD型Weyl群的a-函数的一个具体的组合算法;.6) 探寻由p-典范基定义的双边胞腔、a-函数的基本性质。.这些研究能增进对a-函数的理解，并有助于进一步研究有界性猜想和猜想P1-P15。

Lusztig’s a-function is one of the central concepts in the Kazhdan-Lusztig theory. It is related to the following two open conjectures: the boundedness conjecture of the a-function of an arbitrary Coxeter group and conjectures P1-P15 for a Hecke algebra with unequal parameters. Around the a-function, in this project we plan to: .1) prove the boundedness of the a-function of a hyperbolic Coxeter group；.2) define the “Gelfand-Kirillov codimension” of a simple highest weight module of an affine Lie algebra, and prove that it is given by the a-function of the affine Weyl group;.3) generalize the projective dimension of the tilting module in the BGG category of a semisimple Lie algebra to the affine Lie algebra case, and prove that it is given by the a-function of the affine Weyl group;.4) investigate conjectures P1-P15 for hyperbolic Coxeter groups of rank 3;.5) give an explicit combinatorial algorithm for the a-function of a Weyl group of type BCD;.6) explore the basic properties of the cells and a-functions defined using the p-canonical basis..Study on these topics will enhance our understanding of the a-function，and will be helpful for us to further study the boundedness conjecture and conjectures P1-P15.