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

数学科学：仿射 Hecke 代数、有限约简群和特征轮表示论中的几何方法

• 批准号: 1303060
• 负责人: George Lusztig
• 金额: $ 22.5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303060&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; methods; representation

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.

李型群的表示理论是数学的核心部分。它涉及通过用矩阵形式表示具有对称性的系统来理解它们。特征标理论是20世纪80年代初S等人创立的研究表示的有效工具之一，它为李型有限单群的不可约特征标理论提供了一个几何化的方法。建议继续研究约化代数群，包括不连通代数群上的特征标层。特别是，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。这将给p-ady域上的群的表示理论提供新的信息。本文还建议继续研究半单p-进群的特征标，继续作者早先与J.L.Kim的工作。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。