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Global parametrization of scalar holomorphic coadjoint orbits of a quasi-Hermitian Lie group. (English) Zbl 1296.22007

In a series of papers, the author approached different aspects of Berezin’s quantization. In [B. Cahen, Math. Scand. 105, No. 1, 66–84 (2009; Zbl 1183.22006)] Berezin’s quantization on generalized flag manifolds \(M=G/H\) was studied, i.e. \(G\) is a compact, connected, simply-connected Lie group and \(H\) is a centralizer of a torus. The author has calculated the Berezin symbols of \(\pi(g)\), \(g\in G\) and \(d \pi (X)\), where \(X\) is in the Lie algebra \(\mathfrak{g}\) and \(\pi\) is a unitary irreducible representation of \(G\) holomorphically induced from a character of \(H\). In [Beitr. Algebra Geom. 51, No. 2, 301–311 (2010; Zbl 1342.22022)] the author studied the same problem on noncompact Hermitian symmetric spaces \(M=G/H\). The paper [Arch. Math., Brno 47, No. 1, 51–68 (2011; Zbl 1240.22011)] continues the investigation in the context of the Stratonovich-Weyl correspondence. In [Rend. Semin. Mat. Univ. Padova 129, 277–297 (2013; Zbl 1272.22007)] the author considers as group \(G\) a quasi-Hermitian Lie group and \(\pi\) a unitary highest weight representation realized in a reproducing Hilbert space of holomorphic functions; see K.-H. Neeb [Holomorphy and convexity in Lie theory. Berlin: de Gruyter (1999; Zbl 0936.22001)]. The class of quasi-Hermitian groups contains the compact, connected, simply-connected Lie groups, with their duals - the noncompact Hermitian groups, with associated homogeneous manifold the noncompact Hermitian symmetric spaces \(M=G/H\), but also semidirect products of semisimple Lie groups with nilpotent Lie groups, the simplest example being the Jacobi group \(G^J_n=H_n\ltimes \text{Sp}(n,\mathbb{R})\), where \(H_n\) denotes the \(2n+1\) dimensional Heisenberg group. The author has developed an extension of the Stratonovich-Weyl map in order to accommodate quantization of semidirect product type [Differ. Geom. Appl. 25, No. 2, 177–190 (2007; Zbl 1117.81087)]. In the paper under the review, the part concerning the quasi-Hermitian Lie groups \(G\) as groups of Harish-Chandra type is extracted from the quoted book of Karl-Herman Neeb, as well the holomorphic representation of \(G\). The associated homogeneous manifolds \(M=G/H\) are realized as \(\mathcal{D}=G0\) through the generalized Harish-Chandra embedding, where \(\mathcal{D}\) is not necessarily a bounded domain as in classical Berezin quantization, initially realized on Hermitian symmetric spaces, compact and noncompact, and \(\mathbb{C}^n\). An diffeomorphism \(\psi\) from \(\mathcal{D}\) to the coadjoint orbit \(\mathcal{O}(\xi_0)\) is established, generalizing a result from the quoted paper [Zbl 1272.22007]. Several possible applications of the parametrization of coadjoint orbits in deformation theory, harmonic analysis and mathematical physics are mentioned. In the paper under review the so called “scalar” elements \(\xi_0\) from the dual Lie algebra \(\mathfrak{g}^*\) are considered, which are fixed by \(H\) and regular in the sense that a certain Hermitian form is not isotropic. The diffeomorphism \(\psi\) is explicitly constructed in the case of the unitary group \(\text{SU}(p,q)\) - an example taken integrally from [Beitr. Algebra Geom. 51, No. 2, 301–311 (2010; Zbl 1342.22022)], and the case of the Jacobi group \(G^J_n\). The Stratonovich-Weyl correspondence for the Jacobi group of index \(n=1\) is studied in detail in [B. Cahen, “Stratonovich-Weyl correspondence for the Jacobi group”, Commun. Math. 22, No. 1, 31–48 (2014)].

MSC:

22E10 General properties and structure of complex Lie groups
22E15 General properties and structure of real Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
32M05 Complex Lie groups, group actions on complex spaces
32M10 Homogeneous complex manifolds
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
81S10 Geometry and quantization, symplectic methods
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References:

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