Distinguished Orbits of Reductive Groups

TL;DR

This paper generalizes a classical theorem about closed orbits of reductive groups to include distinguished orbits, establishing a connection between real and complex settings and analyzing orbit behavior under gradient flow.

Contribution

It extends the Borel-Harish-Chandra theorem to distinguished orbits and compares real and complex group actions, also examining orbit collapse under gradient flow.

Findings

Distinguished orbits are characterized by critical points of the moment map.

The equivalence of distinguished orbits in real and complex groups is established.

Gradient flow causes the orbit to collapse to a single K-orbit in both settings.

Abstract

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.