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 $\C$. 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 $O(z)$. We examine the case of {\em alignment} when a semisimple element belongs to both ${\cal H}$ and ${\cal K}$, and the conditions for the same. We illustrate some consequences of alignment. 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 have a direct bearing on representation theoretic as well as geometric properties which connect $z$ and $y$.

翻译：本文研究约化群$G\subseteq GL(X)$作用于$\C$上有限维向量空间$V$时的轨道闭包问题。假设$GL(X)$的中心包含于$G$，并通过固定非平凡特征作用于$V$。我们研究点$y,z\in V$满足以下条件：(i) $z$是通过$y$作用的单参数子群$\lambda (t)\subseteq G$的首项项；(ii) $y$和$z$具有大且独特的稳定化子$K,H \subseteq G$。记$O(z)$（$O(y)$）为$z$（$y$）的$G$-轨道，$\overline{O(z)}$（$\overline{O(y)}$）为其闭包，则(i)意味着$z\in \overline{O(y)}$。我们探讨以下问题：在何种条件下(i)和(ii)可同时成立，即存在单参数子群$\lambda \subseteq G$使得$z$可作为$y$的极限被观测到。利用$\lambda$，我们发展适用于$V$以及${\cal G}= Lie(G)$（$G$的李代数）及其子代数${\cal K}$和${\cal H}$（分别对应$K$和$H$的李代数）的首项项分析。由此构造李代数$\hat{\cal K} \subseteq {\cal H}$，通过李代数连接$y$和$z$。我们发展$\hat{\cal K}$的性质，并将其与${\cal H}$在$\overline{N}=V/T_z O(z)$（轨道$O(z)$的法向切片）上的作用相关联。我们考察当半单元素同时属于${\cal H}$和${\cal K}$时的**对齐**情况及其条件，并阐述对齐的部分推论。最后，我们探讨位于$z$和$y$轨道闭包之间的**中间$G$-簇**$W$（即$\overline{O(z)} \subsetneq W \subsetneq O(y)$）存在的可能性，这些簇对连接$z$与$y$的表示论性质与几何性质具有直接影响。