On complete reducibility in characteristic $p$

Let $G$ be a reductive group over a field $k$ which is algebraically closedof characteristic $p \neq 0$. We prove a structure theorem for a class ofsubgroup schemes of $G$, for $p$ bounded below by the Coxeter number of $G$. Asapplications, we derive semi-simplicity results, generalizing earlier resultsof Serre proven in 1998, and also obtain an analogue of Luna's \'etale slicetheorem for suitable bounds on $p$.Comment: Appendix is by Zhiwei Yu