Monogamous subvarieties of the nilpotent cone

Simon M. Goodwin; Rachel Pengelly; David I. Stewart; Adam R. Thomas

Monogamous subvarieties of the nilpotent cone

Simon M. Goodwin, Rachel Pengelly, David I. Stewart, Adam R. Thomas

Mathematics

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Abstract

Let G be a reductive algebraic group over an algebraically closed field k of prime characteristic not 2, whose Lie algebra is denoted g. We call a subvariety X of the nilpotent cone N⊂g monogamous if for every e∈X, the sl₂-triples (e,h,f) with f∈X are conjugate under the centraliser C_G(e). Building on work by the first two authors, we show there is a unique maximal closed G-stable monogamous subvariety V⊂N and that it is an orbit closure, hence irreducible. We show that V can also be characterised in terms of Serre's G-complete reducibility.

Original language	English
Journal	Pacific Journal of Mathematics
Publication status	Accepted/In press - 13 Sept 2024

Bibliographical note

Not yet published as of 07/10/2024.

Part of Special issue in memory of Gary Seitz

Access to Document

GoodwinS2024Monogamous
This is the accepted author manuscript of an article to be published in the Pacific Journal of Mathematics: final published version will be available at https://msp.org/pjm/2024/329-2/
Accepted author manuscript, 218 KBLicence: Other (please specify with Rights Statement)

Cite this

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abstract = "Let G be a reductive algebraic group over an algebraically closed field k of prime characteristic not 2, whose Lie algebra is denoted g. We call a subvariety X of the nilpotent cone N⊂g monogamous if for every e∈X, the sl2-triples (e,h,f) with f∈X are conjugate under the centraliser CG(e). Building on work by the first two authors, we show there is a unique maximal closed G-stable monogamous subvariety V⊂N and that it is an orbit closure, hence irreducible. We show that V can also be characterised in terms of Serre's G-complete reducibility.",

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AU - Goodwin, Simon M.

AU - Pengelly, Rachel

AU - Stewart, David I.

AU - Thomas, Adam R.

N1 - Not yet published as of 07/10/2024. Part of Special issue in memory of Gary Seitz

PY - 2024/9/13

Y1 - 2024/9/13

N2 - Let G be a reductive algebraic group over an algebraically closed field k of prime characteristic not 2, whose Lie algebra is denoted g. We call a subvariety X of the nilpotent cone N⊂g monogamous if for every e∈X, the sl2-triples (e,h,f) with f∈X are conjugate under the centraliser CG(e). Building on work by the first two authors, we show there is a unique maximal closed G-stable monogamous subvariety V⊂N and that it is an orbit closure, hence irreducible. We show that V can also be characterised in terms of Serre's G-complete reducibility.

AB - Let G be a reductive algebraic group over an algebraically closed field k of prime characteristic not 2, whose Lie algebra is denoted g. We call a subvariety X of the nilpotent cone N⊂g monogamous if for every e∈X, the sl2-triples (e,h,f) with f∈X are conjugate under the centraliser CG(e). Building on work by the first two authors, we show there is a unique maximal closed G-stable monogamous subvariety V⊂N and that it is an orbit closure, hence irreducible. We show that V can also be characterised in terms of Serre's G-complete reducibility.

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M3 - Article

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

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