Representations of affine Kac-Moody algebras and representations of Groups over a 2-dimensional local field

仿射 Kac-Moody 代数的表示和二维局部域上群的表示

• 批准号: 0600903
• 负责人: Dennis Gaitsgory
• 金额: --
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600903&HistoricalAwards=false
• 关键词: Representations; affine; Kac; Moody; algebras

Let G be a complex semi-simple group, and let g be its Lie algebra.Our main object of study is the Kac-Moody extension g' of the loopalgebra g((t)); more specifically, the category of representations ofg' at the critical level, denoted g'_(crit)-mod. Our goal is to develop alocalization theory for this category, i.e., to relate it to other categoriesthat are constructed locally out of spaces, endowed with an actionof the loop group G((t)), such as the affine Grassmannian. Ultimately,we believe that the structure of g'_(crit)-mod is governed by a patternof local Langlands correspondence, i.e., its extrinsic structure can bedescribed solely in terms of the space of local systems on the formalpunctured disc with respect to the Langlands dual group.The second (and, so far, unrelated) part of the project is the developmentof representation theory for groups associated to G and 2-dimensional localfields.The object of study of the current project is representation theory.Starting from the 1930's it was realized that there is fairly small listof types of possible fundamental symmetries that occur in Nature.These types are called "root data", and the corresponding concretemathematical objects are called semi-simple groups. In order to havea representation theory one has to pick one of those semi-simplegroups and couple it with a data of algebraic or analytic nature, suchas what is known in mathematics as a "field" or "ring", e.g., real orcomplex numbers. Having a representation theory, the objects thatone deals with look fairly complicated, and most of the questions thatone can ask are not amenable to complete answers. However, wehope to find a point of view on representations, which, on the onehand, would answer some deep questions about their structure, and,on the other hand, will ultimately reduce to questions about theinitial data, i.e., the root system.

设G是一个复半单群，g是它的李代数.我们主要研究的是循环代数g（（t））的Kac-Moody扩张g'，更具体地说，是g'在临界水平上的表示范畴，记为g '_（crit）-mod.我们的目标是发展这一范畴的局部化理论，即：把它与其他局部构造于空间之外的范畴联系起来，赋予循环群G（（t））的作用，如仿射格拉斯曼。最后，我们认为g '_（crit）-mod的结构受局部Langlands对应模式的支配，即，它的外在结构只能用正规穿孔圆盘上局部系统相对于Langlands对偶群的空间来描述（到目前为止，该项目的一部分是发展与G和2相关的群的表示理论，维局部场。当前项目的研究对象是表示论。从20世纪30年代开始，人们意识到，自然界中可能出现的基本对称性的一个小列表。这些类型被称为“根数据”，相应的具体数学对象被称为半单群。为了有一个表示理论，人们必须选择其中一个半单群，并将其与代数或分析性质的数据耦合，例如数学中称为“域”或“环”的数据，例如，真实的或复数。有了表征理论，人们所处理的对象看起来相当复杂，而且人们所能提出的大多数问题都无法得到完整的答案。然而，我们希望找到一个关于表征的观点，一方面，这将回答一些关于其结构的深层次问题，另一方面，最终将归结为关于初始数据的问题，即，根系统。