Abstract

Linear algebraic groups are subgroups of the general linear group which are closed under the Zariski topology. The concept is similar to that of Lie groups, although fields of prime characteristic are admitted, and indeed arguably provide the most interesting problems and applications. A very important class of linear algebraic groups are reductive groups, which include all the classical groups. This talk is concerned with the conjugacy classes of unipotent elements in a reductive group, i.e. the matrices which have all eigenvalues all equal to 1. These classes are very important in many topics and we have a nice classification of them when the characteristic of the field is not too small. When it is too small things can get a bit ugly, but a unified geometric picture has been proposed by G. Lusztig in a series of conjectures, some of which he proved in a case-by-case manner. In this talk I will explain a uniform approach to these conjectures using geometric invariant theory, which has yielded a short case-free proof. This is joint work with Professor A. Premet (Manchester).