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

Falk Bannuscher; Alastair Litterick; Tomohiro Uchiyama

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arXiv · 2011.13659

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

Falk Bannuscher·Alastair Litterick·Tomohiro Uchiyama

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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.

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BibTeX RIS

Falk Bannuscher, Alastair Litterick, Tomohiro Uchiyama. 2020-11-27. Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example. https://arxiv.org/abs/2011.13659

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