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 πjΓ to a lattice ≤ 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 πjΓ.