Decomposition of locally compact coset spaces

Abstract

In a previous article by the author and P. Wesolek, it was shown that a compactly generated locally compact group $G$ admits a finite normal series $(G_i)$ in which the factors are compact, discrete or irreducible in the sense that no closed normal subgroup of $G$ lies properly between $G_{i-1}$ and $G_{i}$. In the present article, we generalize this series to an analogous decomposition of the coset space $G/H$ with respect to closed subgroups, where $G$ is locally compact and $H$ is compactly generated. This time, the irreducible factors are coset spaces $G_{i}/G_{i-1}$ where $G_{i}$ is compactly generated and there is no closed subgroup properly between $G_{i-1}$ and $G_{i}$. 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.