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On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications. I. (English) Zbl 0629.22004
Let G be a connected Lie group with Lie algebra $${\mathfrak g}$$, and let $${\mathcal O}$$ be the coadjoint orbit through the element $$g\in {\mathfrak g}^ *$$. Further, let $${\mathfrak h}$$ be a real polarization at g, and let F be the corresponding G-invariant polarization of the symplectic manifold $${\mathcal O}$$. In this situation the space $${\mathcal E}^ 1_ F({\mathcal O})$$ of quantizable functions on $${\mathcal O}$$ defined by F is well-defined [cf., e.g., B. Kostant, Géom. sympl. Phys. math., Colloq. int. Aix-en- Provence 1974, 187-210 (1975; Zbl 0326.53047)]. Let $$H_ 0$$ be the analytic subgroup corresponding to $${\mathfrak h}$$, and set $$H=G_ gH_ 0$$, where $$G_ g$$ is the stabilizer of g in G, and suppose that there exists a unitary character $$\chi$$ : $$H\to {\mathbb{T}}$$ such that $$\chi (\exp X)=e^{i<g,X>}$$ for all $$X\in {\mathfrak h}$$. This means that the orbit $${\mathcal O}$$ is integral. Denote by $${\mathcal B}^ 1(G,\chi)$$ the set of all differential operators of order at most one in the homogeneous line bundle with base G/H defined by $$\chi$$. The main result of the paper is that the quantization map $\delta _{\chi}: {\mathcal E}^ 1_ F({\mathcal O})\quad \to \quad {\mathcal B}^ 1(G, \chi)$ is a Lie algebra isomorphism.
Suppose now that G is exponential and simply connected. In this case the space $${\mathcal B}(G,\chi)$$ is isomorphic to the space $${\mathcal B}^ 1({\mathbb{R}}^{d/2})$$ of all differential operators of order at most one on some space $${\mathbb{R}}^{d/2}$$. Using the isomorphism $$\delta _{\chi}$$ we deduce from this simple fact, that there exist global canonical coordinates on each coadjoint orbit $${\mathcal O}$$, i.e., coordinates $$(p_ 1,...,p_{d/2}$$, $$q_ 1,...,q_{d/2})$$ such that the canonical symplectic form $$\omega _{{\mathcal O}}$$ on $${\mathcal O}$$ is given by $\omega _{{\mathcal O}}=\sum ^{d/2}_{i=1}dp_ i \wedge dq_ i\quad.$ As an application it is shown how one can construct all strongly continuous, unitary, irreducible representations of G by Weyl quantization.

##### MSC:
 22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 43A85 Harmonic analysis on homogeneous spaces 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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