On the Smoothness of Centralizers in Reductive Groups

Let G be a connected reductive algebraic group over an algebraically closed field k. In a recent paper, Bate, Martin, R\"ohrle and Tange show that every (smooth) subgroup of G is separable provided that the characteristic of k is very good for G. Here separability of a subgroup means that its scheme-theoretic centralizer in G is smooth. Serre suggested extending this result to arbitrary, possibly non-smooth, subgroup schemes of G. The aim of this note is to prove this more general result. Moreover, we provide a condition on the characteristic of k that is necessary and sufficient for the smoothness of all centralizers in G. We finally relate this condition to other standard hypotheses on connected reductive groups.