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Geometric quantization and the universal enveloping algebra of a nilpotent Lie group. (English) Zbl 0684.22004

The article builds upon the previous article [Math. Ann. 281, 633-669 (1988; Zbl 0629.22004)] by the author. In that article, he had studied the geometric quantization for integral orbits in the dual of an arbitrary connected Lie group G. To be more specific, if \({\mathfrak g}\) denotes the Lie algebra of G and \({\mathfrak g}^*\) its dual, and if \({\mathcal O}\) is an integral orbit in \({\mathfrak g}^*\) under the coadjoint action, then there is a canonical prequantization process in the sense of van Hove, Kostant and Souriau associated with the symplectic manifold \({\mathcal O}\). Moreover, if \({\mathcal O}\) admits a real polarization \({\mathfrak H}\) at a point \(g\in {\mathfrak g}^*\), then \({\mathfrak H}\) defines a G-invariant polarization F of \({\mathcal O}\), and there is a well-defined space \({\mathfrak E}^ 1_ F({\mathcal O})\) consisting of the smooth F-quantizable functions on \({\mathcal O}\). If \(H_ 0\) is the analytic subgroup of G corresponding to \({\mathfrak H}\) and \(H=G_ gH_ 0\) and \(\chi\) : \(H\to {\mathbb{T}}\) is the unitary character such that \(\chi\) (exp X)\(=e^{i<g,X>}\) for all \(X\in {\mathfrak H}\), then there is a canonical quantization homomorphism \(\delta_ X: {\mathfrak E}^ 1_ F({\mathcal O})\to {\mathfrak B}^ 1(G,\chi)\) into the space of “quantized” operators \({\mathfrak B}^ 1(G,\chi)\), which act on sections through the line bundle over G/H defined by \(\chi\).
The main results in the paper were that there exist global smooth, canonical coordinates for \({\mathcal O}\), and that \(\delta_{\chi}\) is in fact an isomorphism.
Now, in the paper under review, the author specializes these constructions to the case of nilpotent G. He shows that there even exists an algorithm for constructing polynomial global, canonical coordinates for the orbits (even simultaneously for all orbits in a stratum of what he calls a “fine \({\mathfrak F}\)-stratification” of \({\mathfrak g}^*)\), and that \(\delta_{\chi}\) is an isomorphism between the space of polynomial quantizable functions on \({\mathcal O}\) and the space of all polynomial quantized operators of order \(\leq 1\) contained in \({\mathfrak B}^ 1(G,\chi)\). Moreover, he shows how the quantization map connects with the universal enveloping algebra and Kirillov theory, thus giving in particular non-inductive proofs of two main facts in Kirillov theory. A comparison with the approach in [N. Wildberger, Quantization and harmonic analysis on nilpotent Lie groups, Ph. D. thesis, Yale Univ. (1983)], which unfortunately is not mentioned in the paper, would perhaps be of some interest.
Reviewer: D.Müller

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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