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

李型群的表示理论是数学的一个核心部分。它关注的是通过用矩阵形式表示对称系统来理解它们。其中一个被证明对表征研究非常有效的工具是PI在20世纪80年代早期发起的字符束理论。该理论提供了李型有限单群不可约性理论的一个几何化。建议继续研究约化代数群（包括不连通代数群）上的特征束。特别提出继续研究具有不等参数的仿射Hecke代数，特别是建立其正则基的几何解释。这将为p进域上群的表示理论提供新的信息。本文还建议继续研究半单p元群的性质，以延续作者与J.L. Kim的早期工作。这些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的概念。傅立叶级数是表征理论最古老的应用之一，广泛应用于工程和应用科学。最近，表征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的研究涉及代数拓扑（通过代数研究形状）和代数几何（方程的几何研究）方法的应用，以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 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.