Monogamous subvarieties of the nilpotent cone
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.20025
- arXiv:
- arXiv:2406.20025
- Bibcode:
- 2024arXiv240620025G
- Keywords:
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- Mathematics - Representation Theory;
- 17B45 (Primary) 17B50 (Secondary)
- E-Print:
- 15 pages