Unipotent elements forcing irreducibility in linear algebraic groups

Abstract

Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic p > 0 {p>0} . We consider connected reductive subgroups X of G that contain a given distinguished unipotent element u of G. A result of Testerman and Zalesski [D. Testerman and A. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 2013, 1, 13–28] shows that if u is a regular unipotent element, then X cannot be contained in a proper parabolic subgroup of G. We generalize their result and show that if u has order p, then except for two known examples which occur in the case ( G , p ) = ( C 2 , 2 ) {(G,p)=(C_{2},2)} , the subgroup X cannot be contained in a proper parabolic subgroup of G. In the case where u has order > p {>p} , we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.