代数群无限维抽象表示中的若干问题

本项目拟研究代数群的无限维抽象表示。这是一个新兴的研究方向，它与代数群的有理表示以及有限李型群的表示有着密切的联系。根据经典的Harish-Chandra理论，对诱导表示和限制表示的研究是一个基本的问题。目前，我们已经对由Borel子群的一维表示所诱导的表示研究得比较清楚，且得到了一大类无限维的不可约表示，不过尚未形成良好的理论框架。在半单李代数的BGG表示范畴中（或者代数群的有理表示范畴中），一个表示限制到Cartan子代数（或极大环面）的表示时，有很好的半单性。本项目将会对由极大环面的一维表示诱导的表示进行深入的研究。具体来讲，我们要确定其商表示的一些性质，并以此为基础建立一个合适的表示范畴，我们会期望此范畴具有某种好的“半单性”，同时在这个范畴的对象上能定义权的概念，从而使得此表示范畴是一个最高权范畴，具有诸多很好的性质。这些工作将为此方向的后续研究建立一个良好的理论框架和基础。

We try to study the infinite-dimensional abstract representations of algebraic groups in this project. This is a new research direction, which is closely related to the rational representations of algebraic groups and the representations of finite groups of Lie type. According to the classical Harish-Chandra theory, the study of induction and restriction of representations is a fundamental problem. At present, we have studied the representations which are induced by the one-dimensional representations of Borel subgroups. Precisely, we have obtained a large class of infinite-dimensional irreducible representations. However, it seems that there is no a good theoretical framework at now. In the BGG category of semisimple Lie algebras (resp. the category of rational representations of algebraic groups), a representation is semisimple when it is restricted as the representation of the Cartan subalgebras (resp. the maximal torus). In this project, we will study the representations which are induced by the one-dimensional representations of the maximal torus. Specifically, we need to determine some properties of its quotient representation, and then construct a suitable representation category. One will hope that this category has the "semisimple" property. Moreover, we expect that “weight” can be defined on each object of this category such that this representation category is a highest weight category, and has many good properties. These works will establish a good theoretical framework in the following research.