On complete reducibility in characteristic $p$

TL;DR

This paper establishes a structure theorem for certain subgroup schemes of reductive groups over algebraically closed fields of positive characteristic, leading to new semi-simplicity results and an analogue of Luna's étale slice theorem for sufficiently large primes.

Contribution

It provides a new structure theorem for subgroup schemes of reductive groups in characteristic p, extending semi-simplicity and geometric results to broader contexts.

Findings

Proves a structure theorem for subgroup schemes when p exceeds the Coxeter number.

Derives semi-simplicity results generalizing Serre's 1998 work.

Establishes an analogue of Luna's étale slice theorem for large p.

Abstract

Let $G$ be a reductive group over a field $k$ which is algebraically closed of characteristic $p \neq 0$. We prove a structure theorem for a class of subgroup schemes of $G$, for $p$ bounded below by the Coxeter number of $G$. As applications, we derive semi-simplicity results, generalizing earlier results of Serre proven in 1998, and also obtain an analogue of Luna's \'etale slice theorem for suitable bounds on $p$.