Unipotent and nilpotent elements in irreducible representations of simple algebraic groups
For any group G, a natural way to study a finite-dimensional representation f: G → GL(V) is through the properties of the linear maps f(g): V → V, for g ∈ G. What can we say about these linear maps, and what do their properties tell us about G and the representation f?
In this talk, we consider this question and related problems in the context of algebraic groups. Specifically, let G be a simple linear algebraic group over an algebraically closed field of characteristic p ≥ 0, and let f: G → GL(V) be a rational irreducible representation. For each unipotent element u ∈ G, what is the Jordan normal form of f(u)? For each nilpotent element e ∈ Lie(G), what is the Jordan normal form of df(e)?
Solutions to these question in specific cases have found many applications, one basic motivation being in the problem of determining the conjugacy classes of unipotent elements contained in maximal subgroups of simple algebraic groups. In characteristic zero, there is a fairly good answer by results of Jacobson-Morozov-Kostant, and the solutions in the unipotent case and the nilpotent case are essentially the same. I will focus on the case of positive characteristic p > 0, where much less is known and few general results are available. When G is simple of exceptional type, computations due to R. Lawther and D. Stewart describe the Jordan normal form of f(u) and df(e) when V is of minimal dimension (adjoint and minimal modules). I will discuss some recent results, mostly focusing on the case where G is simple of classical type.