Description of infinite orbits on multiple projective spaces

Abstract

Let $G$ be the general linear group of the degree $n\geq 2$ over the field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. In this article, we give a description of orbit decomposition of the multiple projective space $G^m/P^m$ under the diagonal action of $G$ where $P$ is the maximal parabolic subgroup of $G$ such that $G/P\cong\mathbb{P}^{n-1}\mathbb{K}$. We also construct representatives of orbits. If $m\geq 4$, the number of orbits is infinite, and we give a description of those uncountably many orbits.