Mathematical Sciences: Geometric methods in the representation theory of affine Hecke algebras, finite reductive groups and quantum groups

李群的表示论是数学的核心部分，它涉及通过矩阵形式来理解具有对称性的系统。表示论中最困难的领域之一是p-adic域上的群，它与数论有很强的联系。研究这些群的主要工具之一是仿射Hecke代数。G. Lusztig建议继续研究仿射Hecke代数与不等参数，特别是建立一个几何解释其规范的基础。并给出了相应的渐近Hecke代数的存在性.这应该给新的信息的代表性理论的群体在p-adic领域。本文还建议继续研究不连通约化群上的特征标层，并使理论达到与连通情形相同的完备性水平。这项研究是必要的，把伴随p-adic群（不一定是分裂群的内部形式）的幂幺表示的分类放在一个坚实的基础上。更一般的特征标层理论也将需要在一个有限域上定义的一个具有循环分支群的约化群的有理点群的不可约特征标的研究中。还建议继续研究小特征标中的幂幺元，特别是试图给出这类元的中心化子的分支群的统一描述。在上述主题上的进展有望应用于数学和理论物理的各个部分。群表示理论试图通过更易于计算的矩阵来研究对称性的思想。表示论最古老的应用之一是傅立叶级数理论，广泛应用于工程和应用科学。最近，表象论的思想被用于化学（晶体研究）和物理学（基本粒子理论）。G. Lusztig的研究关注的是应用方法的代数拓扑（研究的形状通过代数）和代数几何（几何研究方程），以获得新的成果，对团体表示这是无法获得的其他方法。

Representation theory of groups of Lie type is a central part of mathematics.It is concerned with understanding systems with symmetry by representing them in matrix form. One of the most difficult areas of representation theory is that of groups over p-adic fields, which has strong connections with number theory.One of the main tools in the study of these groups are the affine Hecke algebras. G. Lusztig proposes to continue the study of affine Hecke algebras with unequal parameters and in particular to establish a geometric interpretation for their canonical basis. Also it is proposed to establish the existence of the corresponding asymptotic Hecke algebras. This should give new information on the representation theory of groups over p-adic fields. It is also proposed to continue the study of character sheaves on disconnected reductive groups and bring the theory to the same level of completeness as that in the connected case. This study is necessary to put the classification of unipotent representations of adjoint p-adic groups (not necessarily inner forms of split groups) on a firm foundation. The more general theory of character sheaves will be also needed in the study of irreducible characters of the group of rational points of a reductive group with a cyclic group of components defined over a finite field.It is also proposed to continue the study of unipotent elements in small characteristic, in particular to try to give a uniform description of the group of components of centralizers of such elements. Progress on the topics above is expected to have applications to various parts of mathematics and theoretical physics.The theory of group representations attempts to study the idea of symmetry by means of matrices which are more amenable to computation. One of the oldest application of representation theory is the theory of Fourier series, widely used in engineering and applied science. More recently, ideas from representation theory have been used in chemistry (study of crystals) and physics (theory of elementary particles). G. Lusztig's research is concerned with applications of methods of algebraic topology (study of shapes by means of algebra) and algebraic geometry (geometric study of equations) to obtain new results on group representations which could not be obtained by other methods.