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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:

[1] Arnal, D., Cortet, J.-C.: Nilpotent Fourier transform and applications. Lett. Math. Phys. 9 (1985), 25-34. · Zbl 0616.46041
[2] Bar-Moshe, D.: A method for weight multiplicity computation based on Berezin quantization. SIGMA 5 (2009), 091, 1-12. · Zbl 1188.22009
[3] Bar-Moshe, D., Marinov, M. S.: Realization of compact Lie algebras in Kähler manifolds. J. Phys. A: Math. Gen. 27 (1994), 6287-6298. · Zbl 0843.58056
[4] Berezin, F. A.: Quantization. Math. USSR Izv. 8, 5 (1974), 1109-1165. · Zbl 0312.53049
[5] Berezin, F. A.: Quantization in complex symmetric domains. Math. USSR Izv. 9, 2 (1975), 341-379. · Zbl 0324.53049
[6] Berceanu, S.: A holomorphic representation of the Jacobi algebra. Rev. Math. Phys. 18 (2006), 163-199. · Zbl 1099.81036
[7] Berceanu, S., Gheorghe, A.: On the geometry of Siegel-Jacobi domains. Int. J. Geom. Methods Mod. Phys. 8 (2011), 1783-1798. · Zbl 1250.22010
[8] Bernatska, J., Holod, P.: Geometry and topology of coadjoint orbits of semisimple Lie groups. Mladenov, I. M., de León, M. (eds) Proceedings of the 9th international conference on ’Geometry, Integrability and Quantization’, June 8-13, 2007, Varna, Bulgarian Academy of Sciences, Sofia, 2008, 146-166. · Zbl 1208.22009
[9] Berndt, R., Böcherer, S.: Jacobi forms and discrete series representations of the Jacobi group. Math. Z. 204 (1990), 13-44. · Zbl 0695.10024
[10] Berndt, R., Schmidt, R.: Elements of the Representation Theory of the Jacobi Group. Progress in Mathematics 163, Birkhäuser Verlag, Basel, 1988. · Zbl 1235.11046
[11] Cahen, B.: Deformation program for principal series representations. Lett. Math. Phys. 36 (1996), 65-75. · Zbl 0843.22020
[12] Cahen, B.: Contraction de \(SU(1,1)\) vers le groupe de Heisenberg. Mathematical works, Part XV, Séminaire de Mathématique Université du Luxembourg, Luxembourg, (2004), 19-43. · Zbl 1074.22005
[13] Cahen, B.: Weyl quantization for semidirect products. Diff. Geom. Appl. 25 (2007), 177-190. · Zbl 1117.81087
[14] Cahen, B.: Multiplicities of compact Lie group representations via Berezin quantization. Mat. Vesnik 60 (2008), 295-309. · Zbl 1199.22016
[15] Cahen, B.: Contraction of compact semisimple Lie groups via Berezin quantization. Illinois J. Math. 53, 1 (2009), 265-288. · Zbl 1185.22008
[16] Cahen, B.: Berezin quantization on generalized flag manifolds. Math. Scand. 105 (2009), 66-84. · Zbl 1183.22006
[17] Cahen, B.: Contraction of discrete series via Berezin quantization. J. Lie Theory 19 (2009), 291-310. · Zbl 1185.22007
[18] Cahen, B.: Berezin quantization for discrete series. Beiträge Algebra Geom. 51 (2010), 301-311. · Zbl 1342.22022
[19] Cahen, B.: Stratonovich-Weyl correspondence for discrete series representations. Arch. Math. (Brno) 47 (2011), 41-58. · Zbl 1240.22011
[20] Cahen, B.: Berezin quantization and holomorphic representations. Rend. Sem. Mat. Univ. Padova, to appear. · Zbl 1272.22007
[21] Cahen, M., Gutt, S., Rawnsley, J.: Quantization on Kähler manifolds I, Geometric interpretation of Berezin quantization. J. Geom. Phys. 7 (1990), 45-62. · Zbl 0719.53044
[22] Cahen, M., Gutt, S., Rawnsley, J.: Quantization on Kähler manifolds III. Lett. Math. Phys. 30 (1994), 291-305. · Zbl 0826.53052
[23] Cotton, P., Dooley, A. H.: Contraction of an adapted functional calculus. J. Lie Theory 7 (1997), 147-164. · Zbl 0882.22015
[24] Folland, B.: Harmonic Analysis in Phase Space. Princeton Univ. Press, Princeton, 1989. · Zbl 0682.43001
[25] Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Graduate Studies in Mathematics 34, American Mathematical Society, Providence, Rhode Island, 2001. · Zbl 0993.53002
[26] Kirillov, A. A.: Lectures on the Orbit Method, Graduate Studies in Mathematics. 64, American Mathematical Society, Providence, Rhode Island, 2004. · Zbl 1229.22003
[27] Kostant, B.: Quantization and unitary representations. Modern Analysis and Applications, Lecture Notes in Mathematics 170, Springer-Verlag, Berlin, Heidelberg, New York, (1970), 87-207. · Zbl 0223.53028
[28] Neeb, K-H.: Holomorphy and Convexity in Lie Theory. de Gruyter Expositions in Mathematics 28, Walter de Gruyter, Berlin, New York, 2000. · Zbl 0964.22004
[29] Satake, I: Algebraic Structures of Symmetric Domains. Iwanami Sho-ten, Tokyo and Princeton Univ. Press, Princeton, NJ, 1971.
[30] Skrypnik, T. V.: Coadjoint orbits of compact Lie groups and generalized stereographic projection. Ukr. Math. J. 51 (1999), 1939-1944. · Zbl 0937.39008
[31] Varadarajan, V. S.: Lie groups, Lie Algebras and Their Representations. Graduate Texts in Mathematics 102, Springer-Verlag, Berlin, 1984. · Zbl 0955.22500
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