Unipotent classes, nilpotent classes and representation theory of algebraic groups

本文讨论了简单代数群和李代数中的幂偶类和幂零类，以及它们与表示理论的关系。幂偶元和幂零元是代数群和有限李型群理论的基础，在群的结构和表示理论中都起着重要的作用。当代数群所定义的域的特征很好时，幂偶类和幂零类的理论是密切相关的，实际上是独立于特征的。然而，当特征很差时（即是2、3或5，取决于代数群的类型），情况就不是这样了；例如，在坏的特征中，通常比好的特征中有更多的类别——在经典群体中，要多得多。在这个建议中，我们的目标之一是理解具有好特征的阶级和具有坏特征的阶级之间的关系。一开始，我们根本不清楚如何以一种精确的方式来定义我们的意思。我们提出用一种无特征的方式来定义某些抛物子群内的类束。在好的特性中，一个包只包含一个类，而在坏的特性中，它将包含几个类。这些包将耗尽所有的类，通过这种方式，我们将在好特性和坏特性的理论之间获得概念上的联系。同一个bundle中的类应该共享许多公共属性，因此这个链接对应用程序可能非常有用。所有这些目前都是推测性的，我们计划在本提案中将其建立在坚实的基础上。

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.