Non-G-completely reducible subgroups of the exceptional algebraic groups

Abstract

Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p>0 and let H be a subgroup of G. Then following Serre we say H is G-completely reducible or G-cr if, whenever H is contained in a parabolic subgroup P of G, then H is in a Levi subgroup of that parabolic. Building on work of Liebeck and Seitz, we find all triples (X,G,p) such that there exists a closed, connected, simple non-G-cr subgroup H<G with root system X.