Journal of Lie Theory, Vol. 11, No. 1, pp. 135-154 (2001)

Moment sets and the unitary dual of a nilpotent Lie group

Ali Baklouti, Chal Benson, and Gail Ratcliff

Ali Baklouti
Faculté de Sciences de Sfax
Département de Mathématiques
3038 Sfax
Chal Benson
University of Missouri-St. Louis
St. Louis MO 63121
Gail Ratcliff
Department t of Mathematics
and Computer Science
Univ of Missouri-St. Louis
St. Louis MO 63121

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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