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S'eparation des repr'esentations uni-taires des groupes de Lie nilpotents

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... It is shown in [BLS98] that such generalized moment sets always separate the unitary dual G of any connected and simply connected nilpotent Lie group. ...
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Let G be a connected and simply connected nilpotent Lie group with Lie algebra g and unitary dual b G. The moment map for 2 b G sends smooth vectors in the representation space of to g . The closure of the image of the moment map for is called its moment set. N. Wildberger has proved that the moment set for coincides with the closure of the convex hull of the corresponding coadjoint orbit. We say that b G is moment separable when the moment sets dier for any pair of distinct irreducible unitary representations. Our main results provide sucient and necessary conditions for moment separability in a restricted class of nilpotent groups.
... Nevertheless, as shown in [8], the moment set does not characterize the representation even for nilpotent connected and simply connected Lie groups. In [4], Baklouti, Ludwig and Selmi extended the moment map to the dual of the (complex) universal enveloping algebra U(g) of the complexification g C of g as follows: For all A in U(g) and in H ∞ \ {0}, ...
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We show that every unitary representation π of a connected Lie group G is characterized up to quasi-equivalence by its complete moment set.Moreover, irreducible unitary representations π of G are characterized by their moment sets.
... A counterexample is given by Wildberger: G can be not moment separable even for a nilpotent connected, simply connected Lie group. Thus, A. Baklouti, J. Ludwig and M. Selmi in [7] extended the moment map to the dual of the universal enveloping algebra U(g C ) of the complexification g C of g and in [4], it is shown that the generalized moment set characterizes the unitary irreducible representation for an exponential Lie group. We finally prove that this holds for any connected and simply connected Lie group (see [1,2]). ...
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The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R⋉N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G− of G by a discrete subgroup, then we can extend G to an overgroup G+, extend simultaneously each unitary irreducible representation of G to G+ and separate the representations of G by moment sets for G+.RésuméLe groupe de diamant G est un groupe de Lie résoluble non exponentiel, produit semi direct de R avec le groupe de Heisenberg Hn. On considère ce groupe comme un premier exemple d'un produit semi direct de la forme R⋉N où N est nilpotent connexe et simplement connexe.Par un calcul simple, on montre que les ensembles moment de G séparent les orbites coadjointes et leurs représentations unitaires irréductibles génériques.Alors, on s'interesse au problème de la séparation de toutes les représentations unitaires et irréductibles. D' abord, on montre que les ensembles moment caractérisent les représentations unitaires et irréductibles du groupe quotient G− de G par un sous groupe discret. Ensuite, on construit un surgroupe G+ et on prolonge chaque représentation unitaire et irréductible de G à G+. Enfin, on sépare toutes les représentations unitaires et irréductibles de G par les ensembles moment de G+.
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Describing the Gelfand construction for the analytic states on an universal enveloping algebra, we characterize pure states and re-find the main result of a preceding work with L. Abdelmoula and J. Ludwig on the separation of unitary irreducible representations of a connected Lie group by their generalized moment sets. Mathematics Subject Classification 2010: 22D10, 22D20, 52A05.
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Let G be a connected and simply connected solvable Lie group. The moment map for π in Gˆ, unitary dual of G , sends smooth vectors in the representation space of π to g⁎,g⁎, dual space of g.g. The closure of the image of the moment map for π is called its moment set , denoted by IπIπ. Generally, the moment set IπIπ, π∈Gˆ does not characterize π , even for generic representations. However, we say that Gˆ is moment separable when the moment sets differ for any pair of distinct irreducible unitary representations. In the case of an exponential solvable Lie group G , D. Arnal and M. Selmi exhibited an accurate construction of an overgroup G+G+, containing G as a subgroup and an injective map Φ from Gˆ into G+ˆ in such a manner that Φ(Gˆ) is moment separable and IΦ(π)IΦ(π) characterizes π , π∈Gˆ. In this work, we provide the existence of a quadratic overgroup for the diamond Lie group, which is the semi-direct product of RnRn with (2n+1)(2n+1)-dimensional Heisenberg group for some n⩾1n⩾1.
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Let $G$ be a connected and simply connected Lie group with Lie algebra $\mathfrak g $ . We say that a subset $X$ in the set $\mathfrak g ^\star / G$ of coadjoint orbits is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in $X$ . In this paper, we define a class of solvable Lie groups, and we give an explicit construction of an overgroup $G^+$ and a quadratic map $\varphi $ sending each generic orbit in $\mathfrak g ^\star $ to a $G^+$ -orbit in $\mathfrak{g ^+}^\star $ , in such a manner that the set $\varphi (\mathfrak g ^\star _{gen}){/ G^+}$ is convex hull separable. We then call $G^+$ a weak quadratic overgroup for $G$ . Thanks to this construction, we prove that any nilpotent Lie group, with dimension at most 7 admits such a weak quadratic overgroup. Finally, we produce different examples of solvable Lie groups, having weak quadratic overgroups, but which are not in our class of Lie groups and for which usual constructions fail to hold.
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Soient $G$ un groupe de Lie connexe et $ \pi$ une représentation unitaire de $G$ dans un espace de Hilbert ${\mathcal H}_{\pi}$ non séparable. Pour chaque ensemble infini $I$, on définit la représentation unitaire $\hat{\pi}_I=(\# I) \pi$ de $G$ dans l'espace ${\mathcal H}_{\hat{\pi}_I}=(\# I){\mathcal H}_{\pi}$. Alors, on montre que l'ensemble moment généralisé de $\hat{\pi}_I$ caractérise $\pi$ à quasi-équivalence près.
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