Unipotent classes, nilpotent classes and representation theory of algebraic groups

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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.