Separability and complete reducibility of subgroups of the Weyl group of a simple algebraic group of type $E_7$

Abstract

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $k$. The aim of this paper is to present a method to find triples $(G,M,H)$ with the following three properties. Property 1: $G$ is simple and $k$ has characteristic 2. Property 2: $H$ and $M$ are closed reductive subgroups of $G$ such that $H<M<G$, and $(G,M)$ is a reductive pair. Property 3: $H$ is $G$-completely reducible, but not $M$-completely reducible. We exhibit our method by presenting a new example of such a triple in $G=E_7$. Then we consider a rationality problem and a problem concerning conjugacy classes as important applications of our construction.