单代数群的通有稳定子
Generic stabilizers for simple algebraic groups
摘要：
我们证明了无数的结果有关的稳定剂的代数群$G$的一般向量的表示$V$的$G$在代数闭域$k$。我们的结果是在群方案的水平上，这比同时考虑$G$的李代数和$k$ -点的群$G(k)$携带更多的信息。对于$G$简单和$V$忠实和不可约的，我们证明了存在一个稳定的一般位置，有时称为主轨道型。我们确定那些$G$和$V$的稳定子在一般位置是光滑的，或$\dim V/G < \dim G$，或有一个$v \in V$，其稳定子在$G$是平凡的。
英文摘要：
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.
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