On overgroups of distinguished unipotent elements in reductive groups and finite groups of Lie type

Michael Bate; Sören Böhm; Benjamin Martin; Gerhard Roehrle

doi:10.48550/arXiv.2407.16379

On overgroups of distinguished unipotent elements in reductive groups and finite groups of Lie type

Michael Bate, Sören Böhm, Benjamin Martin, Gerhard Roehrle

Mathematical Science

Research output: Working paper › Preprint

Abstract

Suppose $G$ is a simple algebraic group defined over an algebraically closed field of good characteristic $p$. In 2018 Korhonen showed that if $H$ is a connected reductive subgroup of $G$ which contains a distinguished unipotent element $u$ of $G$ of order $p$, then $H$ is $G$-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good $A_1$ subgroups of $G$, introduced by Seitz. We also formulate a counterpart of Korhonen's theorem for overgroups of $u$ which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of $u$ under a mild condition on $p$ depending on the rank of $G$, and we present an analogue of Korhonen's theorem for Lie algebras.

Original language	English
Publisher	ArXiv
Number of pages	17
DOIs	https://doi.org/10.48550/arXiv.2407.16379
Publication status	Published - 23 Jul 2024

Bibliographical note

We are grateful to M. Korhonen and D. Testerman for helpful comments on an earlier version of the manuscript. Some of this work was completed during a visit to the Mathematisches Forschungsinstitut Oberwolfach: we thank them for their support. We thank the referee for a number of comments clarifying some points. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.

Keywords

math.GR
math.RT
20G15, 14L24

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2407.16379v1Submitted manuscript, 310 KB

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abstract = "Suppose $G$ is a simple algebraic group defined over an algebraically closed field of good characteristic $p$. In 2018 Korhonen showed that if $H$ is a connected reductive subgroup of $G$ which contains a distinguished unipotent element $u$ of $G$ of order $p$, then $H$ is $G$-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good $A_1$ subgroups of $G$, introduced by Seitz. We also formulate a counterpart of Korhonen's theorem for overgroups of $u$ which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of $u$ under a mild condition on $p$ depending on the rank of $G$, and we present an analogue of Korhonen's theorem for Lie algebras.",

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AU - Martin, Benjamin

AU - Roehrle, Gerhard

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N2 - Suppose $G$ is a simple algebraic group defined over an algebraically closed field of good characteristic $p$. In 2018 Korhonen showed that if $H$ is a connected reductive subgroup of $G$ which contains a distinguished unipotent element $u$ of $G$ of order $p$, then $H$ is $G$-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good $A_1$ subgroups of $G$, introduced by Seitz. We also formulate a counterpart of Korhonen's theorem for overgroups of $u$ which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of $u$ under a mild condition on $p$ depending on the rank of $G$, and we present an analogue of Korhonen's theorem for Lie algebras.

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On overgroups of distinguished unipotent elements in reductive groups and finite groups of Lie type

Abstract

Bibliographical note

Keywords

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Cite this