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 (Proc. Amer. Math. Soc., 2013) 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)=(C2​,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.Comment: 33 page