Decomposition of locally compact coset spaces

In a previous article of Wesolek and the author, it was shown that a compactly generated locally compact group G$G$ admits a finite normal series (Gi)$(G_i)$ in which the factors are compact, discrete or irreducible in the sense that no closed normal subgroup of G$G$ lies properly between Gi-1$G_{i-1}$ and Gi$G_{i}$. In the present article, we generalize this series to an analogous decomposition of the coset space G/H$G/H$ with respect to closed subgroups, where G$G$ is locally compact and H$H$ is compactly generated. This time, the irreducible factors are coset spaces Gi/Gi-1$G_{i}/G_{i-1}$ where Gi$G_{i}$ is compactly generated and there is no closed subgroup properly between Gi-1$G_{i-1}$ and Gi$G_{i}$. Such irreducible coset spaces can be thought of as a generalization of primitive actions of compactly generated locally compact groups; we establish some basic properties and discuss some sources of examples.