On good A1 subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups

Michael Bate; Sören Böhm; Ben Martin; Gerhard Röhrle

doi:10.2140/pjm.2025.336.29

On good A1 subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups

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

^*Corresponding author for this work

Mathematical Science

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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 under an extra hypothesis using so-called good A1 subgroups of G, introduced by Seitz. In the process we prove some new results about good A1 subgroups of G and their properties. 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
Pages (from-to)	29-61
Number of pages	35
Journal	Pacific Journal of Mathematics
Volume	336
Issue number	1-2
Early online date	26 May 2025
DOIs	https://doi.org/10.2140/pjm.2025.336.29
Publication status	Published - May 2025

Bibliographical note

We are grateful to M. Korhonen and D. Testerman for helpful comments on an earlier version of the manuscript, and to A. Thomas for providing the G2
example in Example 4.13. We thank the referee for a number of comments clarifying some points. Some of this work was completed during a visit to the Mathematisches Forschungsinstitut Oberwolfach: we thank them for their support.
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.

Data Availability Statement

No data availability statement.

Funding

The research of this work was supported in part by the DFG (Grant #RO 1072/22-1 (project number: 498503969) to G. Röhrle).

Funders	Funder number
Deutsche Forschungsgemeinschaft	498503969, RO 1072/22-1

Keywords

G-complete reducibility
G-irreducibility
distinguished unipotent elements
distinguished nilpotent elements
finite groups of Lie type
good A1 subgroups

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Cite this

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title = "On good A1 subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups",

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 under an extra hypothesis using so-called good A1 subgroups of G, introduced by Seitz. In the process we prove some new results about good A1 subgroups of G and their properties. We also formulate a counterpart of Korhonen{\textquoteright}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{\textquoteright}s theorem for Lie algebras.",

keywords = "G-complete reducibility, G-irreducibility, distinguished unipotent elements, distinguished nilpotent elements, finite groups of Lie type, good A1 subgroups",

author = "Michael Bate and S{\"o}ren B{\"o}hm and Ben Martin and Gerhard R{\"o}hrle",

note = "We are grateful to M. Korhonen and D. Testerman for helpful comments on an earlier version of the manuscript, and to A. Thomas for providing the G2 example in Example 4.13. We thank the referee for a number of comments clarifying some points. Some of this work was completed during a visit to the Mathematisches Forschungsinstitut Oberwolfach: we thank them for their support. 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.",

year = "2025",

month = may,

doi = "10.2140/pjm.2025.336.29",

language = "English",

volume = "336",

pages = "29--61",

journal = "Pacific Journal of Mathematics",

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publisher = "Mathematical Sciences Publishers",

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TY - JOUR

T1 - On good A1 subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups

AU - Bate, Michael

AU - Böhm, Sören

AU - Martin, Ben

AU - Röhrle, Gerhard

N1 - We are grateful to M. Korhonen and D. Testerman for helpful comments on an earlier version of the manuscript, and to A. Thomas for providing the G2 example in Example 4.13. We thank the referee for a number of comments clarifying some points. Some of this work was completed during a visit to the Mathematisches Forschungsinstitut Oberwolfach: we thank them for their support. 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.

PY - 2025/5

Y1 - 2025/5

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 under an extra hypothesis using so-called good A1 subgroups of G, introduced by Seitz. In the process we prove some new results about good A1 subgroups of G and their properties. 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.

AB - 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 under an extra hypothesis using so-called good A1 subgroups of G, introduced by Seitz. In the process we prove some new results about good A1 subgroups of G and their properties. 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.

KW - G-complete reducibility

KW - G-irreducibility

KW - distinguished unipotent elements

KW - distinguished nilpotent elements

KW - finite groups of Lie type

KW - good A1 subgroups

U2 - 10.2140/pjm.2025.336.29

DO - 10.2140/pjm.2025.336.29

M3 - Article

SN - 0030-8730

VL - 336

SP - 29

EP - 61

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 1-2

ER -

On good A1 subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups

Abstract

Bibliographical note

Data Availability Statement

Funding

Keywords

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