Darstellungstheorietage and Nikolaus Conference 2021

Let \(G\) be a connected reductive algebraic group over an algebraically closed field \(k\), and let Lie\((G)\) be its associated Lie algebra. In 2011, Lusztig defined a partition of the unipotent variety of \(G\). This partition is very useful when working with representations of \(G\). Equivalently, one can consider certain subsets of the nilpotent variety of \(g\) called pieces. This approach appears in Lusztig’s article. The pieces for the exceptional groups of type \(G_2\), \(F_4\), \(E_6\), \(E_7\) and \(E_8\) in bad characteristic have not yet been determined. In this talk, I will give an introduction to both definitions of the nilpotent pieces and present a solution to this problem for groups of type \(G_2\) and \(F_4\).