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，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扩张g‘，更具体地说，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个范畴的局部化理论，即把它与其他局部构造于空间外的范畴联系起来，赋予G((T))环群的作用，如仿射Grassman范畴。最终，我们认为g‘_(Crit)-mod的结构是由局部朗兰兹对应的模式决定的，即它的外在结构可以仅用形式穿孔圆盘上局部系统相对于朗兰兹对偶群的空间来描述。项目的第二部分(也是到目前为止无关的)是与G和二维局域有关的群的表示理论的发展。本课题的研究对象是表示理论。从1930年的S开始，人们意识到在自然界中存在的可能的基本对称类型相当少。这些类型被称为根数据，相应的具体数学对象称为半单群。为了有一个表示理论，人们必须选择这些半简单群中的一个，并将其与代数或分析性质的数据相耦合，例如在数学上被称为“场”或“环”的东西，例如，实数或复数。有了表征理论，一个人所处理的对象看起来相当复杂，而且一个人能问的大多数问题都不能得到完整的答案。然而，我们希望找到一种关于表示的观点，一方面，这将回答关于其结构的一些深层次问题，另一方面，最终将归结为关于初始数据，即根系的问题。

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.