On the Smoothness of Centralizers in Reductive Groups
By Sebastian Herpel. 2010
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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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/93151c8d332ac5d4bebd57a7a9ed3cdb4c979ccf
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Generalised Gelfand-Graev Representations in Small Characteristics (2014, background): “for all x ∈h}. If h={x}then we simply write C G(x)(resp. cg(x)) for C G(h)(resp. Cg(h)). We say the centraliser C G(h) is smooth if Lie(C G(h)) = cg(h) (or equivalently that dimC G(h) = dimcg(h)). In [Her13] Herpel has considered the smoothness of subgroup schemes of connected reductive algebraic gr”