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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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References:
[1] Arnal, D., Cortet, J.C., Molin, P., Pinczon, G.: Covariance and geometrical invariance in *-quantization. J. Math. Phys.24, 276-283 (1983) · Zbl 0515.22015 · doi:10.1063/1.525703
[2] Auslander, L., Kostant, B.: Polarization and unitary representations of solvable Lie groups. Invent. Math.14, 255-354 (1971) · Zbl 0233.22005 · doi:10.1007/BF01389744
[3] Bourbaki, N.: Intégration, Chaps. VII?VIII. Mesure de Haar. Convolution et représentations. Paris: Hermann 1963
[4] Dieudonné, J.: Élements d’analyse. Tome III. Chapitres XVI et XVII. Paris: Gauthier-Villars 1970
[5] Ginzburg V. A.: Method of in orbits in the representation theory of complex Lie groups. J. Funct. Anal. Appl.15, 18-28 (1981) · Zbl 0467.22008 · doi:10.1007/BF01082375
[6] Kostant, B.: Quantization and unitary representations. In: Lecture notes in modern analysis and applications. III. Lecture Notes in Mathematics. Berlin Heidelberg New York: Springer 1970
[7] Kostant, B.: On the definition of quantization. In: Géométrie symplectique et physique mathématique, Aix-en-Provence, 24-28 juin 1974, Éditions du Centre National de la Recherche Scientifique, Paris, 1975 pp. 187-210
[8] Pedersen, N. V.: Geometric quantization and the universal enveloping algebra of a nilpotent Lie group. Preprint, Univ. of Copenhagen, 1988
[9] Poulsen, N.S.: OnC ?-vectors and intertwining bilinear forms for representations of Lie groups. J. Funct. Anal.9, 87-120 (1972) · Zbl 0237.22013 · doi:10.1016/0022-1236(72)90016-X
[10] Pukanszky, L.: On the theory of exponential groups. Trans. Am. Math. Soc.126, 487-507 (1967) · Zbl 0207.33605 · doi:10.2307/1994311
[11] Reed, M. Simon, B.: Methods of modern mathematical physics I. New York: Academic Press 1972 · Zbl 0242.46001
[12] Simms, D. J., Woodhouse, N.M.J.: Lectures on geometric quantization. (Lecture Notes in Physics, Vol. 53). Berlin Heidelberg New York: Springer 1976 · Zbl 0343.53023
[13] Vergne, M.: La structure de Poisson sur l’algèbre symmétrique d’une algèbre de Lie nilpotente. Bull. Soc. Math. France100, 301-335 (1972) · Zbl 0256.17002
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