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.

谎言类型群体的表示理论是数学的核心部分。它与以矩阵形式表示对称系统的理解系统有关。事实证明，一种对表示表示的工具之一是PI在1980年代初期发起的性格滑轮理论。该理论提供了有限简单类型群体的不可约特征理论的几何化。有人建议继续研究还包括脱离连接的还原代数群体上的角色束带。特别是建议继续研究具有不平等参数的仿射Hecke代数，尤其是为其规范基础建立几何解释。这应该提供有关P-ADIC领域组的表示理论的新信息。还建议继续研究半imple p-adic群体的角色，继续作者与J.L. Kim的早期工作。这些主题的进展有望在数学和理论物理学的各个部分中应用。小组表示理论试图通过更适合计算的矩阵来研究对称性的概念。表示理论的最古老的应用之一是傅立叶系列的理论，该理论广泛用于工程和应用科学。最近，代表理论的思想已用于化学（晶体研究）和物理学（基本颗粒理论）。 G. Lusztig的研究与代数拓扑方法（通过代数的形状研究）和代数几何形状（方程式几何研究）有关，以获得无法通过其他方法获得的组表示的新结果。