Abstract: Let $G$ be a connected and simply connected nilpotent Lie group with Lie algebra $\g$ and unitary dual $\widehat{G}$. The moment map for $\pi\in\widehat{G}$ 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 $\widehat{G}$ 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.
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