The structure of the space of coadjoint orbits of an exponential solvable Lie group

Source abstract

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 G . We construct a partition ℘ \wp of the dual g ∗ {\mathfrak {g}^{\ast } } of the Lie algebra g \mathfrak {g} of G G into finitely many Ad ∗ ⁡ ( G ) \operatorname {Ad}^{\ast } (G) -invariant algebraic sets with the following properties. For each Ω ∈ ℘ \Omega \in \wp , there is a subset Σ \Sigma of Ω \Omega which is a cross-section for the coadjoint orbits in Ω \Omega and such that the natural mapping Ω / Ad ∗ ⁡ ( G ) → Σ \Omega /\operatorname {Ad}^{\ast } (G) \to \Sigma is bicontinuous. Each Σ \Sigma is the image of an analytic Ad ∗ ⁡ ( G ) \operatorname {Ad}^{\ast }(G) -invariant function P P on Ω \Omega and is an algebraic subset of g ∗ {\mathfrak {g}^{\ast }} . The partition ℘ \wp has a total ordering such that for each Ω ∈ ℘ \Omega \in \wp , ∪ { Ω ′ : Ω ′ ≤ Ω } \cup \{ \Omega \prime :\Omega \prime \leq \Omega \} is Zariski open. For each Ω \Omega there is a cone W ⊂ g ∗ W \subset {\mathfrak {g}^{\ast } } , such that Ω \Omega is naturally a fiber bundle over Σ \Sigma with fiber W W and projection P P . There is a covering of Σ \Sigma by finitely many Zariski open subsets O O such that in each O O , there is an explicit local trivialization Θ : P − 1 ( O ) → W × O \Theta :{P^{ - 1}}(O) \to W \times O . Finally, we show that if Ω \Omega is the minimal element of ℘ \wp (containing the generic orbits), then its cross-section Σ \Sigma is a differentiable submanifold of g ∗ {\mathfrak {g}^{\ast } } . It follows that there is a dense open subset U U of G ∅ ^ G\hat \emptyset such that U U has the structure of a differentiable manifold and G ∅ ^ ∼ U G\widehat \emptyset \sim U has Plancherel measure zero.