Minimal Projective Orbits of Semi-simple Lie Groups

Abstract

Let $G$ be a Lie group $G$ with representation $\rho$ on a real simple $G$-module $\mathbb{V}$. We will call the orbits of the induced action of $\rho$ on the projectivization $P\mathbb{V}$ the projective orbits, and projective orbits of lowest possible dimension will be called minimal. We show that when $G$ is semi-simple and non-compact, there exists a compact subgroup $K\subset G$ such that the minimal orbits of $G$ are in bijection with the minimal $K$-orbits on a $K$-invariant proper subspace $\mathbb{W}\subset \mathbb{V}$. In the case that $G$ is split-real, $K$ is the trivial subgroup and there is a unique closed projective orbit, which is moreover of minimal dimension.