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 (Formula presented.) admits a finite normal series (Formula presented.) in which the factors are compact, discrete or irreducible in the sense that no closed normal subgroup of (Formula presented.) lies properly between (Formula presented.) and (Formula presented.). In the present article, we generalize this series to an analogous decomposition of the coset space (Formula presented.) with respect to closed subgroups, where (Formula presented.) is locally compact and (Formula presented.) is compactly generated. This time, the irreducible factors are coset spaces (Formula presented.) where (Formula presented.) is compactly generated and there is no closed subgroup properly between (Formula presented.) and (Formula presented.). 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.