Classification of All Parabolic Subgroup-Schemes of a Reductive Linear Algebraic Group Over an Algebraically Closed Field

Let $G$ be a reductive linear algebraic group over an algebraically closed field $K$. The classification of all parabolic subgroups of $G$ has been known for many years. In that context subgroups of $G$ have been understood as varieties, i.e. as reduced schemes. Also several nontrivial nonreduced subgroup schemes of $G$ are known, but until now nobody knew how many there are and what there structure is. Here I give a classification of all parabolic subgroup schemes of $G$ in $\operatorname {char}(K) > 3$ .