Complete reducibility and separability

Let $G$ be a reductive linear algebraic group over an algebraically closed field of characteristic $p > 0$. A subgroup of $G$ is said to be separable in $G$ if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre’s concept of $G$-complete reducibility for subgroups of $G$. A separability hypothesis appears in many general theorems concerning $G$-complete reducibility. We demonstrate that some of these results fail without this hypothesis. On the other hand, we prove that if $G$ is a connected reductive group and $p$ is very good for $G$, then any subgroup of $G$ is separable; we deduce that under these hypotheses on $G$, a subgroup $H$ of $G$ is $G$-completely reducible provided Lie $G$ is semisimple as an $H$-module.

Recently, Guralnick has proved that if $H$ is a reductive subgroup of $G$ and $C$ is a conjugacy class of $G$, then $C\cap H$ is a finite union of $H$-conjugacy classes. For generic $p$ — when certain extra hypotheses hold, including separability — this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig’s deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick’s result is false if one considers conjugacy classes of $n$-tuples of elements from $H$ for $n > 1$.