The subgroup structure of pseudo-reductive groups Article Swipe

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Michael Bate , Benjamin Martin , Gerhard Röhrle , Damian Sercombe ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2406.11286

Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $π':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $π'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $π'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$.

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2406.11286
PDF: https://arxiv.org/pdf/2406.11286
OA Status: green
Related Works: 10
OpenAlex ID: https://openalex.org/W4399795010

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DOI: https://doi.org/10.48550/arxiv.2406.11286

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Title: The subgroup structure of pseudo-reductive groups

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Type: preprint

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Language: en

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Publication year: 2024

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Publication date: 2024-06-17

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Authors: Michael Bate, Benjamin Martin, Gerhard Röhrle, Damian Sercombe

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Landing page: https://arxiv.org/abs/2406.11286

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PDF URL: https://arxiv.org/pdf/2406.11286

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