On the smoothness of centralizers in reductive groups

Herpel, Sebastian

doi:10.1090/S0002-9947-2012-05745-X

On the smoothness of centralizers in reductive groups
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Trans. Amer. Math. Soc. 365 (2013), 3753-3774 Request permission

Abstract:

Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$. In a recent paper, Bate, Martin, Röhrle 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 paper 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.

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