标题： 闭轨和下降对于经典群的增强标准表示 显示英文标题

标题： Closed Orbits and Descents for Enhanced Standard Representations of Classical Groups

摘要： 设 $G=\mathrm{GL}_n(\mathbb{F})$, $\mathrm{O}_n(\mathbb{F})$或 $\mathrm{Sp}_{2n}(\mathbb{F})$为代数闭域 $\mathbb{F}$在特征 $0$下的典型群，令 $\breve{G}$为 $G$的 MVW 扩展，令 $\mathfrak{g}$为 $G$的李代数。 在本文中，我们对增强标准表示$\mathfrak{g}\times E$中的闭轨进行分类，其中$G$是自然表示，若$E$为$G=\mathrm{O}_n(\mathbb{F})$或$\mathrm{Sp}_{2n}(\mathbb{F})$，或是自然表示与其对偶的直和若$G=\mathrm{GL}_n(\mathbb{F})$。 此外，对于$\mathfrak{g}\times E$中的每个闭合$G$-轨道，我们证明它是$\breve{G}$-稳定的，并明确确定相应的稳定子群以及在法空间上的作用。 显示英文摘要

摘要： 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.