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

李型群的表示理论是数学的核心部分，它通过用矩阵的形式表示系统来理解具有对称性的系统。P-ady域上的群是表示论中最困难的领域之一，它与数论有着很强的联系，仿射Hecke代数是研究这类群的主要工具之一。G.Lusztig建议继续研究具有不等参数的仿射Hecke代数，特别是建立其标准基的几何解释。并证明了相应的渐近Hecke代数的存在性。这将给p-ady域上的群的表示理论提供新的信息。并建议继续研究不连通约化群上的特征标层，使之达到与连通情形相同的完备性水平。这项研究是必要的，以使伴随p-ady群(不一定是分裂群的内部形式)的幂等表示的分类有一个坚实的基础。在研究定义在有限域上的具有循环分支群的约化群的有理点群的不可约特征时，还需要更一般的特征标理论.还建议继续研究小特征的幂等元，特别是试图给出这类元的中心化子的分支群的统一描述.上述课题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里叶级数理论，广泛应用于工程和应用科学。最近，表象理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的研究涉及应用代数拓扑学(通过代数研究形状)和代数几何(研究方程的几何)的方法来获得其他方法无法获得的关于群表示的新结果。

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.