Unipotent classes, nilpotent classes and representation theory of algebraic groups

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.

The proposal is concerned with several topics in the theory of algebraic groups: unipotent elements, nilpotent elements in Lie algebras, and representations. These are topics which play a major role in the many areas in which the theory of algebraic groups has an impact. For example, the relationship between the behaviours of algebraic groups of a given type in different characteristics, one of the central problems in this proposal, is a familiar theme in such diverse subjects as number theory and model theory. Indeed, there have been several such recent applications of algebraic group theory to model theory (and also vice versa). More will surely follow with better understanding of the topics of this proposal. The other sphere of impact is in the realm of current and future PhD students and postdoctoral researchers -- in other words, the future academic base of the country. There are quite a number of UK-based PhD students and postdocs working on algebraic groups, particularly on representation-theoretic aspects. This proposal is likely both to achieve new understanding, and also give rise to new problems and projects for students and postdocs. The representation-theoretic questions in the proposal seem particularly promising in this respect, as the links between representations and classes highlighted in this part of the proposal are currently rather uncharted territory.