An orbit Cartan type decomposition of the inertia space of \(\mathrm{SO}(2m)\) acting on \(\mathbb R^{2m}\) (Q381212)

Let \(M\) be a smooth manifold and let \(G\) be a compact, connected Lie group acting smoothly on \(M\). Let \(G\) act by conjugation on itself and diagonally on \(G\times M\). The inertia space of \(M\) is the quotient space of the subspace \(\{ (g,x)\in G\times M\mid gx=x\}\) of \(G\times M\). The authors study the inertia space of \({\mathbb{R}}^{2m}\) with the standard action of the special orthogonal group \(\text{SO}(2m)\). They determine a decomposition of the inertia space of \({\mathbb{R}}^{2m}\) that induces a Whitney stratification. This stratification is called the orbit Cartan type stratification and it was first defined in [\textit{C. Farsi, M. Pflaum} and \textit{C. Seaton}, ``Inertia spaces of proper Lie group actions and their topological properties'', preprint, 2012, \url{arXiv 1207.0595v}]. This stratification gives the inertia space the structure of a differentiable stratified space.