Non-homeomorphic Nilmanifolds with Identical Unitary Spectrum

We show in every three-step nilpotent Lie groupGwith all coadjoint orbits flat the existence of a pair of discrete cocompact subgroupsΓ1andΓ2such thatΓ1\GandΓ2\Ghave the same unitary spectrum butΓ1is not isomorphic toΓ2. This result generalizes an example of Gornet. We mention without giving a proof a result that would enunciate that for a large category of subgroups of the four dimensional three-step chain group with non-flat coadjoint orbits, this phenomenon of non-isomorphic representation equivalence cannot occur. We also prove some short structural results for three-step nilpotent Lie groups with one-dimensional center.