Monogamous subvarieties of the nilpotent cone
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909624271110144 |
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| author | Goodwin, Simon M. Pengelly, Rachel Stewart, David I. Thomas, Adam R. |
| author_facet | Goodwin, Simon M. Pengelly, Rachel Stewart, David I. Thomas, Adam R. |
| contents | Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of prime characteristic not $2$, whose Lie algebra is denoted $\mathfrak{g}$. We call a subvariety $\mathfrak{X}$ of the nilpotent cone $N \subset \mathfrak{g}$ monogamous if for every $e\in \mathfrak{X}$, the $\mathfrak{sl}_2$-triples $(e,h,f)$ with $f\in \mathfrak{X}$ 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 $V \subset N$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_20025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monogamous subvarieties of the nilpotent cone Goodwin, Simon M. Pengelly, Rachel Stewart, David I. Thomas, Adam R. Representation Theory 17B45 (Primary) 17B50 (Secondary) Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of prime characteristic not $2$, whose Lie algebra is denoted $\mathfrak{g}$. We call a subvariety $\mathfrak{X}$ of the nilpotent cone $N \subset \mathfrak{g}$ monogamous if for every $e\in \mathfrak{X}$, the $\mathfrak{sl}_2$-triples $(e,h,f)$ with $f\in \mathfrak{X}$ 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 $V \subset N$ 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. |
| title | Monogamous subvarieties of the nilpotent cone |
| topic | Representation Theory 17B45 (Primary) 17B50 (Secondary) |
| url | https://arxiv.org/abs/2406.20025 |