On the Moore Formula of Compact Nilmanifolds

Let $G$ be a connected and simply connected two-step nilpotent Lie group and $\Gamma$ a lattice subgroup of $G$. In this note, we give a new multiplicity formula, according to the sense of Moore, of irreducible unitary representations involved in the decomposition of the quasi-regular representation $\operatorname{Ind}_\Gamma^G(1)$. Extending then the Abelian case.