  Let $G$ be a nilpotent Lie group and let $\pi$ be a coherent state
representation of $G$. The interplay between the cyclicity of the restriction
$\pi|_{\Gamma}$ to a lattice $\Gamma \leq G$ and the completeness of subsystems
of coherent states based on a homogeneous $G$-space is considered. In
particular, it is shown that necessary density conditions for Perelomov's
completeness problem can be obtained via density conditions for the cyclicity
of $\pi|_{\Gamma}$.
