Complete reducibility and separability

Abstract

Let G G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0 p > 0 . A subgroup of G G is said to be separable in G 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 G -complete reducibility for subgroups of G G . A separability hypothesis appears in many general theorems concerning G G -complete reducibility. We demonstrate that some of these results fail without this hypothesis. On the other hand, we prove that if G G is a connected reductive group and p p is very good for G G , then any subgroup of G G is separable; we deduce that under these hypotheses on G G , a subgroup H H of G G is G G -completely reducible provided Lie G G is semisimple as an H H -module. Recently, Guralnick has proved that if H H is a reductive subgroup of G G and C C is a conjugacy class of G G , then C ∩ H C\cap H is a finite union of H H -conjugacy classes. For generic p 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 n -tuples of elements from H H for n > 1 n > 1 .