On the Smoothness of Centralizers in Reductive Groups

TL;DR

This paper extends known results on the smoothness of centralizers in reductive groups to all subgroup schemes, providing necessary and sufficient conditions on the characteristic of the field for their smoothness.

Contribution

It generalizes the smoothness results to non-smooth subgroup schemes and characterizes when all centralizers are smooth based on the field's characteristic.

Findings

Established a necessary and sufficient condition on the characteristic for all centralizers to be smooth.

Extended previous results from smooth subgroups to arbitrary subgroup schemes.

Connected the smoothness condition to standard hypotheses on reductive groups.

Abstract

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.