Completeness of coherent state subsystems for nilpotent Lie groups

Let $G$ be a nilpotent Lie group and let $π$ be a coherent state representation of $G$. The interplay between the cyclicity of the restriction $π|_Γ$ to a lattice $Γ\leq G$ 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 $π|_Γ$.

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