Decomposition of locally compact coset spaces

TL;DR

This paper extends a known finite normal series decomposition from locally compact groups to coset spaces, characterizing irreducible factors as minimal coset spaces with no intermediate closed subgroups, generalizing primitive actions.

Contribution

It generalizes the finite normal series decomposition from groups to coset spaces, introducing irreducible factors as minimal coset spaces with no intermediate closed subgroups.

Findings

Decomposition of coset spaces into irreducible factors.

Irreducible factors generalize primitive actions.

Basic properties and examples discussed.

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.