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Unipotent classes, nilpotent classes and representation theory of algebraic groups

代数群的单能类、幂零类和表示论

基本信息

批准号：
EP/I033835/1
负责人：
Martin Liebeck
金额：
$ 3.06万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI033835%2F1
关键词：
Unipotent classes nilpotent representation theory

项目摘要

Our proposal concerns the unipotent and nilpotent classes in simple algebraic groups and Lie algebras, and their relationship with representation theory. Unipotent and nilpotent elements are fundamental to the theory of algebraic groups and finite groups of Lie type, and play a major role in both the structure and representation theory of the groups. When the characteristic of the field overwhich the algebraic group is defined is good , the theories of unipotent and nilpotent classes turn out to be very closelyrelated, and actually independent of the characteristic. However, when the characteristic is bad (meaning it is 2,3 or 5, dependingon the type of algebraic group), this is not the case; for example, in bad characteristic there are usually more classesthan in good - many more, in the case of classical groups. In this proposal one of our objectives is to understand the relationship between the classes in good characteristics and thosein bad characteristics. At the outset, it is not at all clear how to define what we mean by this in a precise way. We proposeto define bundles of classes within certain parabolic subgroups in a way that is characteristic-free. In good characteristica bundle will consist of just one class, while in bad characteristic it will consist of several. The bundles will exhaust all the classes, and in this way we will obtain a conceptual link between the theories in good and bad characteristics. Classes in the same bundle should share many common properties, so this link will be potentially very useful for applications. All this is currently conjectural, and we plan to establish it on a sound footing in this proposal.

我们的建议涉及单代数群和李代数中的幂单类和幂零类，以及它们与表示论的关系。幂幺元和幂零元是代数群和李型有限群理论的基础，在群的结构和表示理论中起着重要作用。当定义代数群的域的特征是好的时，幂单类和幂零类的理论变得非常密切相关，实际上与特征无关。然而，当特征是坏的（意味着它是2，3或5，取决于代数群的类型），情况并非如此;例如，在坏特征中通常有比好特征更多的类-在经典群的情况下更多。在这个建议中，我们的目标之一是了解具有好特性的类和具有坏特性的类之间的关系。一开始，我们根本不清楚如何准确地定义我们的意思。我们提出了在某些抛物子群中定义类丛的方法，这种方法是特征自由的。在好的特性中，束将仅由一个类组成，而在坏的特性中，它将由几个类组成。这些捆绑包将耗尽所有类，这样我们将获得好特征和坏特征理论之间的概念联系。同一个bundle中的类应该共享许多公共属性，所以这个链接对应用程序可能非常有用。所有这些目前都是理论性的，我们计划在本提案中将其建立在一个坚实的基础上。