On the smoothness of centralizers in reductive groups. (English) Zbl 1298.20057
Let \(G\) be a connected reductive linear algebraic group over an algebraically closed field \(k\), and let \(H\) be a closed subgroup subscheme of \(G\). This paper addresses the question of whether or not the (scheme-theoretic) centralizer of \(H\) is smooth. In the case that \(H\) itself is smooth, the question was addressed by the reviewer and co-authors [M. Bate et al., Math. Z. 269, No. 3-4, 809-832 (2011; Zbl 1242.20058)]. The results in this paper are a significant extension of this work.
The main result of the paper shows that all centralizers of closed subgroup subschemes of \(G\) are smooth if and only if the characteristic of \(k\) is zero or is “pretty good” for \(G\). The notion of a pretty good prime, also introduced in this paper, lies in between the existing notions of good and very good for primes and depends only on the root datum of \(G\). It allows one to distinguish between a reductive group and its semisimple part, which is important when answering this question. For example, all primes are pretty good for \(G=\text{GL}_2(k)\), but \(p=2\) is not pretty good for \(G=\text{SL}_2(k)\) (note that the centre of \(\text{SL}_2(k)\) is not smooth in characteristic \(2\)).
The author also relates his new idea of a pretty good prime to existing notions of standardness for reductive groups, including the “standard hypotheses” of J. C. Jantzen [Progress in Mathematics 228, 1-211 (2004; Zbl 1169.14319)], giving a unification of all of these ideas.
The main result of the paper shows that all centralizers of closed subgroup subschemes of \(G\) are smooth if and only if the characteristic of \(k\) is zero or is “pretty good” for \(G\). The notion of a pretty good prime, also introduced in this paper, lies in between the existing notions of good and very good for primes and depends only on the root datum of \(G\). It allows one to distinguish between a reductive group and its semisimple part, which is important when answering this question. For example, all primes are pretty good for \(G=\text{GL}_2(k)\), but \(p=2\) is not pretty good for \(G=\text{SL}_2(k)\) (note that the centre of \(\text{SL}_2(k)\) is not smooth in characteristic \(2\)).
The author also relates his new idea of a pretty good prime to existing notions of standardness for reductive groups, including the “standard hypotheses” of J. C. Jantzen [Progress in Mathematics 228, 1-211 (2004; Zbl 1169.14319)], giving a unification of all of these ideas.
Reviewer: Michael Bate (York)
Keywords:
reductive groups; linear algebraic groups; smooth centralizers; subgroup subschemes; pretty good primesReferences:
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