An orbit Cartan type decomposition of the inertia space of $\mathrm{SO}(2m)$ acting on $\mathbb R^{2m}$

Abstract

We study the inertia space of ${ℝ}^{2m}$ with the standard action of the special orthogonal group $SO\left(2m\right)$. In particular, we indicate a decomposition of the inertia space that induces the orbit Cartan type stratification of the inertia space recently defined by C. Farsi, M. Pflaum, and the first author for an arbitrary smooth $G$-manifold where $G$ is a compact Lie group.