Let $G$ be a finitely generated nilpotent group of unitary operators on a Hilbert space ${\cal H}$. We prove that ${\cal H}$ is decomposable into a direct sum ${\cal H}=\bigoplus_{\alpha\in A}{\cal L}_{\alpha}$ of pairwise orthogonal closed subspaces so that elements of $G$ permute the subspaces ${\cal L}_{\alpha}$, and if $T({\cal L}_{\alpha})={\cal L}_{\alpha}$, then the action of $T$ on ${\cal L}_{\alpha}$ is either scalar or has continuous spectrum. We also provide examples showing that analogous results do not hold for solvable non-nilpotent groups.