Darstellungstheorietage and Nikolaus Conference 2021

In the study of representations of finite reductive groups, Jordan decomposition of characters associates to each character in a rational Lusztig series a unipotent character of the centralizer of a semi-simple element in the dual group. The talk will focus on how and in which sense one can construct a Jordan decomposition of characters of \({\bf G}^F\) which is equivariant with respect to Aut\(({\bf G}^F)\) whenever \(\bf G\) is a simple simply-connected algebraic group defined over a finite field through the Frobenius endomorphism $F$. The main difficulty comes from the disconnectedness of centralizers in the dual group \({\bf G}^*\). This is part of the project to determine the action of Aut\((G)\) on Irr\((G)\) for all finite (quasi-)simple groups \(G\).