Quasi-regular representations on discrete coset spaces

For any coset $xH$, its preimage under $q$ is $xH$, which is open because left translation by $x$ is a homeomorphism and $H$ is open. Thus every singleton of the quotient topology on $X$ is open, so $X$ is discrete.

By [F2], ${\mathrm{\ell }}^2(X)$ is the Hilbert space of square-summable complex coordinate families on $X$. The action in [F3] is well defined on left cosets and satisfies the group-action law. For each $g\in G$, $\lambda (g)$ permutes the coordinate vectors, so it extends linearly to a norm-preserving bijection with inverse $\lambda (g^{-1})$; hence it is unitary and $g\mapsto \lambda (g)$ is a representation.

For each $xH\in X$, the orbit map $g\mapsto \lambda (g){\delta }_{xH}={\delta }_{gxH}$ is locally constant: at $g_0$ it is constant on the open neighborhood $g_{0}xH{x}^{-1}$ by [F4]. A finite linear combination of coordinate vectors therefore has a locally constant orbit map.

If $\eta$ is $G$-invariant, transitivity of the action in [F3] makes its coordinates equal to one constant $c$. If $X$ is infinite and $c\mathrm{\ne }0$, finite subsets of arbitrarily large cardinality have squared coordinate sum $n\mathrm{\mid }c{\mathrm{\mid }}^2$, contradicting square summability by [F5]; hence the invariant subspace is zero. If $X$ is finite, the constant function $1_X$ is square-summable, nonzero, and invariant. Therefore the invariant subspace is nonzero exactly when $G\mathrm{/}H$ is finite.

Given $\eta \in {\mathrm{\ell }}^2(X)$ and $ϵ>0$, choose a finite-support ${\eta }_0$ with $\mathrm{\parallel }\eta -{\eta }_0\mathrm{\parallel }<ϵ\mathrm{/}3$ by [F2, F5]. Near any $g_0\in G$, the orbit of ${\eta }_0$ is constant by step 1.3, and unitarity gives $\mathrm{\parallel }\lambda (g)\eta -\lambda (g_0)\eta \mathrm{\parallel }\le 2\mathrm{\parallel }\eta -{\eta }_0\mathrm{\parallel }+\mathrm{\parallel }\lambda (g){\eta }_0-\lambda (g_0){\eta }_0\mathrm{\parallel }<ϵ$. Thus every orbit map is continuous and ${\lambda }_{G\mathrm{/}H}$ is strongly continuous.

The coordinate vector ${\delta }_H$ has norm one by [F2]. For $h\in H$, $hH=H$, so $\lambda (h){\delta }_H={\delta }_H$ by [F3]; thus ${\delta }_H$ is $H$-fixed.