Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 1

Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-split simple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In this paper, we present the first examples of nonabelian non-$G$-completely reducible $k$-subgroups of $G$ which are $G$-completely reducible over $k$. Our construction is based on that of subgroups of $G$ acting non-separably on the unipotent radical of a proper parabolic subgroup of $G$ in our previous work. We also present examples with the same property for a non-connected reductive group $G$. Along the way, several general results concerning complete reducibility over nonperfect fields are proved using the recently proved Tits center conjecture for spherical buildings. In particular, we show that under mild conditions a $k$-subgroup of $G$ is pseudo-reductive if it is $G$-completely reducible over $k$.