Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example

Falk Bannuscher; Alastair Litterick; Tomohiro Uchiyama

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example

Group Theory 2021-11-09 v2

Authors: Falk Bannuscher , Alastair Litterick , Tomohiro Uchiyama

Abstract

Let $k$ be a nonperfect separably closed field. Let $G$ be a connected reductive algebraic group defined over $k$ . We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$ . In particular, we present the first example of a connected nonabelian $k$ -subgroup $H$ of $G$ that is $G$ -completely reducible but not $G$ -completely reducible over $k$ , and the first example of a connected nonabelian $k$ -subgroup $H'$ of $G$ that is $G$ -completely reducible over $k$ but not $G$ -completely reducible. This is new: all previously known such examples are for finite (or non-connected) $H$ and $H'$ only.

Keywords

reductive algebraic groups complete reducibility nonperfect fields rationality problems f4 serre's notion

Cite

@article{arxiv.2011.13659,
  title  = {Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example},
  author = {Falk Bannuscher and Alastair Litterick and Tomohiro Uchiyama},
  journal= {arXiv preprint arXiv:2011.13659},
  year   = {2021}
}

Comments

arXiv admin note: text overlap with arXiv:1703.00103, arXiv:1612.05863. To appear in Journal of Group Theory

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example

Abstract

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