Harmonic analysis on two and three-step nilmanifolds

Let G be a connected, simply connected nilpotent Lie group of step 1, 2 or 3, which contains a discrete cocompact subgroup Γ. Let τ=indΓG1 be the quasi-regular representa-tion of G. Our main goals in this paper are twofold. The first is to give an orbital description of the decomposition of τ into irreducibles. The second main goal is to give an explicit intertwining operator between τ and its disintegration. As a straight application, we give a new multiplicity formula which is a partial answer to a question proposed by J. Brezin.