G-complete reducibility and saturation

Michael Bate; Sören Böhm; Alastair Litterick; Benjamin Martin; Gerhard Roehrle

doi:10.2140/pjm.2025.337.1

G-complete reducibility and saturation

Michael Bate
, Sören Böhm
, Alastair Litterick
, Benjamin Martin
, Gerhard Roehrle

Mathematical Science

University of York
Ruhr University Bochum
University of Essex

Research output: Contribution to journal › Article › peer-review

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Abstract

Let H ⊆ G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p > 0. In our first main theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V ). Our proof uses R.W. Richardson’s notion of reductive pairs to reduce to the GL(V ) case. We study Serre’s notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V ). We also study saturation of finite subgroups of Lie type in G. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case G = GL(V ).

Original language	English
Pages (from-to)	1-24
Number of pages	24
Journal	Pacific Journal of Mathematics
Volume	337
Issue number	1
DOIs	https://doi.org/10.2140/pjm.2025.337.1
Publication status	Published - 2 Jun 2025

Bibliographical note

Acknowledgments: We are grateful to J-P. Serre for some suggestions on an earlier version of the manuscript.

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.

Open Access made possible by participating institutions via Subscribe to Open.

Funding

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

Keywords

math.RT
math.GR
20G15, 14L24
G -complete reducibility
saturation
finite groups of Lie type

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10.2140/pjm.2025.337.1Licence: CC BY

2401.16927v1
https://creativecommons.org/licenses/by/4.0/
Submitted manuscript, 263 KBLicence: CC BY
Bate_etal_pjm_GCompleteReducibility_VOR
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Final published version, 453 KBLicence: CC BY

Cite this

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title = "G-complete reducibility and saturation",

abstract = "Let H ⊆ G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p > 0. In our first main theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V ). Our proof uses R.W. Richardson{\textquoteright}s notion of reductive pairs to reduce to the GL(V ) case. We study Serre{\textquoteright}s notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V ). We also study saturation of finite subgroups of Lie type in G. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case G = GL(V ). ",

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N1 - Acknowledgments: We are grateful to J-P. Serre for some suggestions on an earlier version of the manuscript. 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. Open Access made possible by participating institutions via Subscribe to Open.

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N2 - Let H ⊆ G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p > 0. In our first main theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V ). Our proof uses R.W. Richardson’s notion of reductive pairs to reduce to the GL(V ) case. We study Serre’s notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V ). We also study saturation of finite subgroups of Lie type in G. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case G = GL(V ).

AB - Let H ⊆ G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p > 0. In our first main theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V ). Our proof uses R.W. Richardson’s notion of reductive pairs to reduce to the GL(V ) case. We study Serre’s notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V ). We also study saturation of finite subgroups of Lie type in G. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case G = GL(V ).

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G-complete reducibility and saturation

Abstract

Bibliographical note

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Keywords

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