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

李群的表示论是数学的核心部分。它涉及通过用矩阵形式表示对称性来理解系统。其中一个工具已被证明是非常有效的研究表示是理论的字符层的PI发起于20世纪80年代初。这个理论提供了李型有限单群的不可约特征标理论的几何化。建议继续研究约化代数群上的特征层，包括不连通的特征层。特别是它建议继续研究仿射Hecke代数不等参数，特别是建立一个几何解释其规范的基础。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半单p-adic群的特征标，这是作者与J.L. Kim.在这些主题的进展，预计有应用到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。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 tools which has proved to be highly effective for the study of representations is the theory of character sheaves which the PI initiated in the early 1980's. This theory provides a geometrization of the theory of irreducible characters of finite simple groups of Lie type. It is proposed to continue the project of studying character sheaves on reductive algebraic groups, including disconnected ones. In particular it is proposed to continue the study of affine Hecke algebras with unequal parameters and in particular to establish a geometric interpretation for their canonical basis. This should give new information on the representation theory of groups over p-adic fields. It is also proposed to continue the study of characters of semisimple p-adic groups continuing the author's earlier work with J.L. Kim. Progress in these topics 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.