Closed Orbits and Descents for Enhanced Standard Representations of Classical Groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917409778040832 |
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| author | Liang, Chen |
| author_facet | Liang, Chen |
| contents | Let $G=\mathrm{GL}_n(\mathbb{F})$, $\mathrm{O}_n(\mathbb{F})$, or $\mathrm{Sp}_{2n}(\mathbb{F})$ be one of the classical groups over an algebraically closed field $\mathbb{F}$ of characteristic $0$, let $\breve{G}$ be the MVW-extension of $G$, and let $\mathfrak{g}$ be the Lie algebra of $G$. In this paper, we classify the closed orbits in the enhanced standard representation $\mathfrak{g}\times E$ of $G$, where $E$ is the natural representation if $G=\mathrm{O}_n(\mathbb{F})$ or $\mathrm{Sp}_{2n}(\mathbb{F})$, and is the direct sum of the natural representation and its dual if $G=\mathrm{GL}_n(\mathbb{F})$. Additionally, for every closed $G$-orbit in $\mathfrak{g}\times E$, we prove that it is $\breve{G}$-stable, and determine explicitly the corresponding stabilizer group as well as the action on the normal space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21911 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Closed Orbits and Descents for Enhanced Standard Representations of Classical Groups Liang, Chen Representation Theory Commutative Algebra Algebraic Geometry Let $G=\mathrm{GL}_n(\mathbb{F})$, $\mathrm{O}_n(\mathbb{F})$, or $\mathrm{Sp}_{2n}(\mathbb{F})$ be one of the classical groups over an algebraically closed field $\mathbb{F}$ of characteristic $0$, let $\breve{G}$ be the MVW-extension of $G$, and let $\mathfrak{g}$ be the Lie algebra of $G$. In this paper, we classify the closed orbits in the enhanced standard representation $\mathfrak{g}\times E$ of $G$, where $E$ is the natural representation if $G=\mathrm{O}_n(\mathbb{F})$ or $\mathrm{Sp}_{2n}(\mathbb{F})$, and is the direct sum of the natural representation and its dual if $G=\mathrm{GL}_n(\mathbb{F})$. Additionally, for every closed $G$-orbit in $\mathfrak{g}\times E$, we prove that it is $\breve{G}$-stable, and determine explicitly the corresponding stabilizer group as well as the action on the normal space. |
| title | Closed Orbits and Descents for Enhanced Standard Representations of Classical Groups |
| topic | Representation Theory Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2507.21911 |