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

设G是一个复半单群，设G是它的李代数。我们的主要研究对象是环代数g（(t)）的Kac-Moody扩展g'；更具体地说，g‘在临界水平上的表示类别，表示为g’_（临界）-mod。我们的目标是为这个范畴发展局部化理论，也就是说，将它与其他由空间局部构造的范畴联系起来，这些范畴被赋予了一个环群G（(t)）的作用，比如仿射格拉斯曼。最后，我们认为g'_(crit)-mod的结构是由一种局部朗兰兹对应模式控制的，即它的外在结构可以用形式穿孔盘上局部系统的空间来描述。该项目的第二个部分（到目前为止，是不相关的）是开发与G和二维局部域相关的组的表示理论。本课题的研究对象是表征理论。从20世纪30年代开始，人们意识到，自然界中可能存在的基本对称性类型相当少。这些类型被称为“根数据”，相应的具体数学对象被称为半简单群。为了有一个表示理论，我们必须从这些半单群中选择一个，并将其与代数或解析性质的数据相结合，例如在数学中称为“域”或“环”的数据，例如实数或复数。有了表征理论，人们处理的对象看起来相当复杂，人们可以提出的大多数问题都无法得到完整的答案。然而，我们希望找到一种关于表征的观点，一方面，它将回答关于它们结构的一些深层问题，另一方面，它将最终简化为关于初始数据（即根系统）的问题。

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.