Generic Stabilizers for Simple Algebraic Groups
Generic Stabilizers for Simple Algebraic Groups
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DOI：
10.1307/mmj/20217216
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发表时间：
2021-05
影响因子：
0.9
通讯作者：
S. Garibaldi;R. Guralnick
中科院分区：
文献类型：
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作者：
S. Garibaldi;R. Guralnick
在代数闭域$k$上，证明了与一般向量的代数群$G$中的稳定子有关的无数结果。我们的结果是在群方案的水平上的，它比同时考虑$G$的李代数和$k$点的群$G(K)$都包含更多的信息。对于$G$简单和$V$忠实且不可约的情形，我们证明了在一般位置存在稳定器，有时称为主轨道型。我们确定了一般位置的稳定器光滑的$G$和$V$，或$V/G&dim G$，或在V$中有一个$v的稳定子在$G$中是平凡的。
We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of $G$ and the group $G(k)$ of $k$-points. For $G$ simple and $V$ faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those $G$ and $V$ for which the stabilizer in general position is smooth, or $\dim V/G<\dim G$, or there is a $v \in V$ whose stabilizer in $G$ is trivial.