可约群的可区别轨道
Distinguished Orbits of Reductive Groups
摘要：
本文证明了Borel-Harish-Chandra关于约化群的线性作用的闭轨道的一个定理的推广。考虑一个在实向量空间$V$上线性有理作用的实约化代数群$G$. $G$可以看作作用于$V^\mathbb C := V \otimes \mathbb C$的复约化群$G^\mathbb C$的实点。Borel-Harish-Chandra证明了$G^\mathbb C \cdot v \cap V$是$G$ -轨道的有限并，而且$G^\mathbb C \cdot v$是闭的当且仅当$G\cdot v$是闭的.我们表明，同样的结果不仅适用于封闭的轨道，但所谓的区别轨道。如果一个轨道包含射影空间上矩映射的范数平方的临界点，则称该轨道是可区别的。我们的主要结果比较了复杂的和真实的设置，显示$G\cdot v$是区分当且仅当$G^\mathbb C \cdot v$是区分。此外，我们表明，如果一个轨道的区别，然后在负梯度流的范数平方的时刻地图的整个$G$ -轨道崩溃到一个单一的$K$ -轨道。这一结果在复杂和真实环境中都成立。最后，我们将应用于李群的左不变几何的研究。
英文摘要：
We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group $G$ acting linearly and rationally on a real vector space $V$. $G$ can be viewed as the real points of a complex reductive group $G^\mathbb C$ which acts on $V^\mathbb C := V \otimes \mathbb C$. Borel-Harish-Chandra show that $G^\mathbb C \cdot v \cap V$ is a finite union of $G$-orbits; moreover, $G^\mathbb C \cdot v$ is closed if and only if $G\cdot v$ is closed. We show that the same result holds not just for closed orbits but for the so-called distinguished orbits. An orbit is called distinguished if it contains a critical point of the norm squared of the moment map on projective space. Our main result compares the complex and real settings to show $G\cdot v$ is distinguished if and only if $G^\mathbb C \cdot v$ is distinguished. In addition, we show that if an orbit is distinguished, then under the negative gradient flow of the norm squared of the moment map the entire $G$-orbit collapses to a single $K$-orbit. This result holds in both the complex and real settings. We finish with some applications to the study of the left-invariant geometry of Lie groups.
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