Equivalence of Quasi-regular Representations of Two and Three-Step Nilpotent Lie Groups

We characterize all pairs of cocompact, discrete subgroups Γ1 and Γ2 of a two-step nilpotent Lie group M such that the quasi-regular representations of M on L2(Γ1\M) and L2(Γ2\M) are unitarily equivalent. This characterization is used to construct new examples. We also describe a pair of nonisomorphic, cocompact, discrete subgroups Γ1 and Γ2 of a three-step nilpotent Lie group M such that the quasi-regular representations of M on L2(Γ1\M) and L2(Γ2\M) are unitarily equivalent.