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Journal of Lie Theory, Vol. 11, No. 1, pp. 135-154 (2001)
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Moment sets and the unitary dual of a nilpotent Lie group

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Ali Baklouti, Chal Benson, and Gail Ratcliff

Ali Baklouti

Faculté de Sciences de Sfax

Département de Mathématiques

3038 Sfax

Tunisia,

Chal Benson

University of Missouri-St. Louis

St. Louis MO 63121

U.S.A.,

and

Gail Ratcliff

Department t of Mathematics

and Computer Science

Univ of Missouri-St. Louis

St. Louis MO 63121

U.S.A.

**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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© 2001 ELibM for
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