In this paper we study the orbit closure problem for a reductive group $G\subseteq GL(X)$ acting on a finite dimensional vector space $V$ over ${\mathbb C}$. We assume that the center of $GL(X)$ lies within $G$ and acts on $V$ through a fixed non-trivial character. We study points $y,z\in V$ where (i) $z$ is obtained as the leading term of the action of a 1-parameter subgroup $\lambda (t)\subseteq G$ on $y$, and (ii) $y$ and $z$ have large distinctive stabilizers $K,H \subseteq G$. Let $O(z)$ (resp. $O(y)$) denote the $G$-orbits of $z$ (resp. $y$), and $\overline{O(z)}$ (resp. $\overline{O(y)}$) their closures, then (i) implies that $z\in \overline{O(y)}$. We address the question: under what conditions can (i) and (ii) be simultaneously satisfied, i.e, there exists a 1-PS $\lambda \subseteq G$ for which $z$ is observed as a limit of $y$. Using $\lambda$, we develop a leading term analysis which applies to $V$ as well as to ${\cal G}= Lie(G)$ the Lie algebra of $G$ and its subalgebras ${\cal K}$ and ${\cal H}$, the Lie algebras of $K$ and $H$ respectively. Through this we construct the Lie algebra $\hat{\cal K} \subseteq {\cal H}$ which connects $y$ and $z$ through their Lie algebras. We develop the properties of $\hat{\cal K}$ and relate it to the action of ${\cal H}$ on $\overline{N}=V/T_z O(z)$, the normal slice to the orbit $O(z)$. Next, we examine the possibility of {\em intermediate $G$-varieties} $W$ which lie between the orbit closures of $z$ and $y$, i.e. $\overline{O(z)} \subsetneq W \subsetneq O(y)$. These intermediate varieties are constructed using the grading obtained from $\lambda $ by its action on $V$ and ${\cal G}$. The paper hopes to contribute to the Geometric Complexity Theory approach of addressing problems in computational complexity in theoretical computer science.

翻译：本文研究约化群$G\subseteq GL(X)$作用于有限维复向量空间$V$时的轨道闭包问题。我们假设$GL(X)$的中心包含于$G$，并通过固定非平凡特征作用于$V$。研究满足以下条件的点$y,z\in V$：(i) $z$是通过1-参数子群$\lambda(t)\subseteq G$作用在$y$上的首项得到的；(ii) $y$与$z$具有较大的不同稳定子$K,H\subseteq G$。记$O(z)$（及$O(y)$）为$z$（及$y$）的$G$-轨道，$\overline{O(z)}$（及$\overline{O(y)}$）为其闭包，则条件(i)蕴含$z\in \overline{O(y)}$。本文探讨如下问题：在何种条件下可同时满足(i)与(ii)，即存在1-参数子群$\lambda\subseteq G$使得$z$可视为$y$的极限？利用$\lambda$，我们发展出一套首项分析方法，该方法同时适用于$V$、$G$的李代数${\cal G}=Lie(G)$及其子代数${\cal K}$与${\cal H}$（分别对应于$K$与$H$的李代数）。通过该方法，我们构造出连接$y$与$z$的李代数子代数$\hat{\cal K}\subseteq {\cal H}$，并研究$\hat{\cal K}$的性质及其与${\cal H}$在法向切片$\overline{N}=V/T_z O(z)$上作用的关联。进一步，我们考察介于$z$与$y$轨道闭包之间的{\em 中间$G$-簇}$W$的可能性，即满足$\overline{O(z)} \subsetneq W \subsetneq O(y)$的簇。这些中间簇通过$\lambda$在$V$及${\cal G}$上作用所诱导的分次结构构造而成。本文旨在为理论计算机科学中计算复杂性问题的几何复杂性理论研究方法提供新贡献。