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Separation of coadjoint orbits of an exponential group
Abstract
Looking to the separation of irreducible unitary representations of an exponential Lie group G through the image of their moment map, we propose here a new way: instead to extend the moment map to the universal enveloping algebra of G, we define a non linear mapping Phi from the dual of the Lie algebra g of G to the dual g(+*) of a larger solvable group G(+), and we extend the representation from G to G+, in such a manner that the corresponding coadjoint orbits in 9(+*) have distinct closed convex hull. This allows us to separate the irreducible unitary representations of G. (c) 2007 Elsevier Masson SAS. Tous droits reserves.
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.
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.
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+.
Given an exponential Lie group G, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Hölder basis. Thus we obtain a stratification of g* into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section ∑ ⊂ Ω for coadjoint orbits in Ω, so that each pair (Ω ∑) behaves predictably under the associated restriction maps on g*. The cross-section mapping σ : Ω → ∑ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with ℓ ε Ω. For each Ω, algebras ε0(Ω) and ε1(Ω) of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way. Now let 2d > 0 be the dimension of coadjoint orbits in Ω. An explicit algorithm is given for the construction of complex-valued real analytic functions {q1, q2, ⋯ , qd} and {p1, p2, ⋯ , pd} such that on each coadjoint orbit O in Ω, the canonical 2-form is given by ∑d pk ̂ dqk. The functions {q1, q 2, ⋯ , qd} belong to ε0(Ω), and the functions {p1, p2, ⋯ , pd} belong to ε1(Ω). The associated geometric polarization on each orbit O coincides with the complex Vergne polarization, and a global Darboux chart on O is obtained in a simple way from the coordinate functions (p1, ⋯ , pd, q1, ⋯ , q d) (restricted to O). Finally, the linear evaluation functions ℓ → ℓ(X) are shown to be quantizable as well.
The moment map of symplectic geometry is extended to associate to any unitary representation of a nilpotent Lie group aG-invariant subset of the dual of the Lie algebra. We prove that this subset is the closed conex hull of the Kirillov orbit of the representation.
In this paper we address the problem of describing in explicit algebraic terms the collective structure of the coadjoint orbits of a connected, simply connected exponential solvable Lie group G. We construct a partition ℘ of the dual g* of the Lie algebra g of G into finitely many $\operatorname{Ad}^\ast(G)$-invariant algebraic sets with the following properties. For each Ω ∈ ℘, there is a subset Σ of Ω which is a cross-section for the coadjoint orbits in Ω and such that the natural mapping $\Omega/\operatorname{Ad}^\ast (G) \rightarrow \Sigma$ is bicontinuous. Each Σ is the image of an analytic $\operatorname{Ad}^\ast (G)$-invariant function P on Ω and is an algebraic subset of g*. The partition ℘ has a total ordering such that for each $\Omega \in \wp, \bigcup\{\Omega': \Omega' \leq \Omega\}$ is Zariski open. For each Ω there is a cone $W \subset \mathfrak{g}^\ast$, such that Ω is naturally a fiber bundle over Σ with fiber W and projection P. There is a covering of Σ by finitely many Zariski open subsets O such that in each O, there is an explicit local trivialization Θ: P-1(O) → W × O. Finally, we show that if Ω is the minimal element of ℘ (containing the generic orbits), then its cross-section Σ is a differentiable submanifold of g*. It follows that there is a dense open subset U of $G^\wedge$ such that U has the structure of a differentiable manifold and $G^\wedge \sim U$ has Plancherel measure zero.
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.
On each orbit W of the coadjoint representation of any nilpotent (connected, simply connected) Lie group G, we construct *-products and associated Von Neumann algebras G. G acts canonically on G by automorphisms. In the unique faithful, irreducible representation of G, this action is implemented by the unitary irreducible representation of G corresponding to W by the Kirillov method. This construction is uniquely determined by W and gives the classification of all unitary irreducible representations of G.
Séparation des représentations unitaires des groupes de Lie nilpotents, in : Lie Theory and its Applications in Physics II
- A Baklouti
- J Ludwig
- M Selmi
- A Baklouti
- J Ludwig
- M Selmi