Geodesic
Moment Sets and the Unitary Dual of a Nilpotent Lie Group
Journal of Lie theory, Tome 11 (2001) no. 1, pp. 135-154.

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Let G be a connected and simply connected nilpotent Lie group with Lie algebra L(G) and unitary dual D(G). The moment map for π of D(G) sends smooth vectors in the representation space of π to L(G)*. The closure of the image of the moment map for π is called its moment set. N. Wildberger has proved that the moment set for π coincides with the closure of the convex hull of the corresponding coadjoint orbit. We say that D(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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     author = {A. Baklouti and C. Benson and G. Ratcliff },
     title = {Moment {Sets} and the {Unitary} {Dual} of a {Nilpotent} {Lie} {Group}},
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A. Baklouti; C. Benson; G. Ratcliff . Moment Sets and the Unitary Dual of a Nilpotent Lie Group. Journal of Lie theory, Tome 11 (2001) no. 1, pp. 135-154. http://geodesic.mathdoc.fr/item/JLT_2001_11_1_JLT_2001_11_1_a7/