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.

首席研究员：乔治·卢斯蒂格（George Lusztig）的提案编号：0243345 INSTITITION：MASSACHUSETTS技术学会学院的代表理论陈述群体中的理论陈述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。代表理论中最困难的领域之一是P-ADIC领域的群体，该群体与数字理论具有牢固的联系。这些组研究的主要工具之一是使用仿射Hecke代数。 G. Lusztig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。还建议建立相应的渐近hecke代数的存在。这应该提供有关P-ADIC领域组的表示理论的新信息。还建议在有限场被某些有限环取代的情况下，进一步研究deligne-lusztig理论的类似物。这再次应在P-ADIC领域的组代表理论中应用。提议进一步研究量化的代数的规范底座，从变形绳索的角度进行。还建议继续研究还原群体的角色滑轮。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矩阵来研究对称性的概念，矩阵更适合计算。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。