Moment sets and the unitary dual of a nilpotent Lie group

Abstract. Let G be a connected and simply connected nilpotent Lie group with Lie algebra g and unitary dual Ĝ. The moment map for pi ∈ G ̂ sends smooth vectors in the representation space of pi to g∗. The closure of the image of the moment map for pi is called its moment set. N. Wildberger has proved that the moment set for pi coincides with the closure of the convex hull of the corresponding coadjoint orbit. We say that G ̂ is moment separable when the moment sets differ for any pair of distinct irreducible unitary representations. Our main results provide sufficient and necessary conditions for moment separability in a restricted class of nilpotent groups. 1