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Complete reducibility of subgroups of reductive algebraic groups over non-perfect fields IV: An ๐น4 example

  • Falk Bannuscher , Alastair Litterick and Tomohiro Uchiyama EMAIL logo
From the journal Journal of Group Theory

Abstract

Let ๐‘˜ be a non-perfect separably closed field. Let ๐บ be a connected reductive algebraic group defined over ๐‘˜. We study rationality problems for Serreโ€™s notion of complete reducibility of subgroups of ๐บ. In particular, we present the first example of a connected non-abelian ๐‘˜-subgroup ๐ป of ๐บ that is ๐บ-completely reducible but not ๐บ-completely reducible over ๐‘˜, and the first example of a connected non-abelian ๐‘˜-subgroup H โ€ฒ of ๐บ that is ๐บ-completely reducible over ๐‘˜ but not ๐บ-completely reducible. This is new: all previously known such examples are for finite (or non-connected) ๐ป and H โ€ฒ only.

Award Identifier / Grant number: 19K14516

Funding statement: While undertaking the work for this article, the second and third authors were supported by Alexander von Humboldt Fellowships. The third author also acknowledges the financial support of JSPS Grant-in-Aid for Early-Career Scientists (19K14516).

Acknowledgements

The authors would like to thank Gerhard Rรถhrle and Michael Bate for helpful comments.

  1. Communicated by: Timothy C. Burness

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Received: 2020-11-27
Revised: 2021-07-28
Published Online: 2021-11-30
Published in Print: 2022-05-01

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