Overgroups of regular unipotent elements in reductive groups

Michael Bate; Ben Martin; Gerhard  Röhrle

doi:10.1017/fms.2021.82

Overgroups of regular unipotent elements in reductive groups

Michael Bate^* (Corresponding Author), Ben Martin, Gerhard Röhrle

^*Corresponding author for this work

Mathematical Science

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Abstract

We study reductive subgroups H of a reductive linear algebraic group G - possibly nonconnected - such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.

Original language	English
Article number	e13
Number of pages	13
Journal	Forum of Mathematics, Sigma
Volume	10
Early online date	24 Feb 2022
DOIs	https://doi.org/10.1017/fms.2021.82
Publication status	Published - 24 Feb 2022

Bibliographical note

Acknowledgments. We thank Donna Testerman for comments on an earlier version of the paper, and the referee for several suggestions which have improved the exposition.

Keywords

G-irreducibility
G-complete reducibility
overgroups of regular unipotent elements
finite groups of Lie type

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10.1017/fms.2021.82Licence: CC BY

Bates_etal_FMS_Overgroups_Regular_Unipotent_VoR
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Cite this

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T1 - Overgroups of regular unipotent elements in reductive groups

AU - Bate, Michael

AU - Martin, Ben

AU - Röhrle, Gerhard

N1 - Acknowledgments. We thank Donna Testerman for comments on an earlier version of the paper, and the referee for several suggestions which have improved the exposition.

PY - 2022/2/24

Y1 - 2022/2/24

N2 - We study reductive subgroups H of a reductive linear algebraic group G - possibly nonconnected - such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.

AB - We study reductive subgroups H of a reductive linear algebraic group G - possibly nonconnected - such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.

KW - G-irreducibility

KW - G-complete reducibility

KW - overgroups of regular unipotent elements

KW - finite groups of Lie type

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U2 - 10.1017/fms.2021.82

DO - 10.1017/fms.2021.82

M3 - Article

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VL - 10

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e13

ER -

Overgroups of regular unipotent elements in reductive groups

Abstract

Bibliographical note

Keywords

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