Abstract
Let G be a simple algebraic group over an algebraically closed field K of characteristic
{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)}
, the subgroup X cannot be contained in a proper parabolic subgroup of G. In the case where u has order
{>p}
, we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.
The centralizer of a regular unipotent element in a semisimple algebraic group
