Geometric methods in representation theory

表示论中的几何方法

• 批准号: 0243345
• 负责人: George Lusztig
• 金额: $ 60万
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0243345&HistoricalAwards=false
• 关键词: Geometric; methods; representation; theory

Principal Investigator: George Lusztig Proposal Number: 0243345Institution: Massachusetts Institute of TechnologyABSTRACTGeometric methods in representation theoryRepresentation 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 is the use of 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 further investigate the analogue of the Deligne-Lusztig theory in the case where finite fields are replaced by certain finite rings. This again should have applications to the representation theory of groups over p-adic fields. It is proposed to further investigate the canonical bases of quantized enveloping algebras from the point of view of perverse sheaves. It is also proposed to continue the study of character sheaves on reductive groups. 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 applications 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.

主要研究者：乔治卢斯蒂格提案编号：0243345机构：马萨诸塞州理工学院摘要表示论中的几何方法李群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。表示论中最困难的领域之一是p-adic域上的群，它与数论有很强的联系。其中一个主要的工具，在研究这些群体是使用仿射赫克代数。G. Lusztig建议继续研究仿射Hecke代数与不等参数，特别是建立一个几何解释其规范的基础。并给出了相应的渐近Hecke代数的存在性.这应该给新的信息的代表性理论的群体在p-adic领域。还建议进一步研究类似的Deligne-Lusztig理论的情况下，有限域被某些有限环。这也应该有应用程序的代表性理论的群体对p-adic领域。本文从反常层的观点出发，进一步研究量子化包络代数的标准基。还建议继续研究约化群上的特征标层。这些主题的进展预计将应用于数学和理论物理的各个部分。群表示理论试图通过矩阵来研究对称性的想法，这更适合于计算。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。