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Stratonovich-Weyl correspondence for the Jacobi group. (English) Zbl 1304.22005

The Stratonovich-Weyl correspondence for the Jacobi group of index one, \(G^J_1=\mathrm{SU}(1,1)\ltimes H_1\), is studied in detail, where \(H_1\) denotes the 3-dimensional Heisenberg group. The general case of a quasi-Hermitian Lie group was presented in [B. Cahen, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 52, No. 1, 35–48 (2013; Zbl 1296.22007)], with a short exemplification for the Jacobi group of index \(n\), \(G^J_n\). The scheme for the construction of a holomorphic representation is taken from [K.-H. Neeb, Holomorphy and convexity in Lie theory. Berlin: de Gruyter (1999; Zbl 0936.22001)] and the previous paper of B. Cahen [Rend. Semin. Mat. Univ. Padova 129, 277–297 (2013; Zbl 1272.22007)], from where the theorems are extracted which give the Berezin transform \(S_{\chi}\), where the reproducing kernel Hilbert space \(\mathcal{H}_{\chi}\) has the reproducing kernel parametrized by \(\gamma\in\mathbb{R}\) and \(m\in\mathbb{Z}\). The Jacobi group is a group of Harish-Chandra type with associated homogeneous space the Siegel-Jacobi disk \(\mathcal{D}^J_1=\mathbb{C}\times\mathcal{D}_1\), where \(\mathcal{D}_1\) denotes the Siegel disk \(\{z\in\mathbb{C}| |z|<1\}\). The polar decomposition of \(S_{\chi}=(S_{\chi}S_{\chi}^*)^{1/2}W_{\chi}\) gives the Stratonovich-Weyl correspondence \(W_{\chi}\). The diffeomorphism \(\Psi_{\chi}\) from \(\mathcal{D}^J_1\) to a certain Kostant-Kirillov orbit \(\mathcal{O}(\xi_X)\) is described explicitly. The Berezin transform is extended such that it can be applied to the derived representation \(d\pi_{\chi}(X)\) for \(X\) in the Lie algebra of \(G^J_1\). This allows the author to obtain the Stratonovich-Weyl symbols for the derived representation of the Jacobi group \(G^J_1\).

MSC:

22E10 General properties and structure of complex Lie groups
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)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
81S10 Geometry and quantization, symplectic methods
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[1] Ali, S.T., Englis, M.: Quantization methods: a guide for physicists and analysts. Rev. Math. Phys., 17, 4, 2005, 391-490, · Zbl 1075.81038 · doi:10.1142/S0129055X05002376
[2] Arazy, J., Upmeier, H.: Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains. Function spaces, interpolation theory and related topics (Lund, 2000) 151–211. 2002, De Gruyter, Berlin, · Zbl 1027.32020
[3] Arazy, J., Upmeier, H.: Weyl Calculus for Complex and Real Symmetric Domains, Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001). Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 13, 3–4, 2002, 165-181, · Zbl 1150.43302
[4] Berceanu, S.: A holomorphic representation of the Jacobi algebra. Rev. Math. Phys., 18, 2006, 163-199, · Zbl 1099.81036 · doi:10.1142/S0129055X06002619
[5] Berceanu, S., Gheorghe, A.: On the geometry of Siegel-Jacobi domains. Int. J. Geom. Methods Mod. Phys., 8, 2011, 1783-1798, · Zbl 1250.22010 · doi:10.1142/S0219887811005920
[6] Berezin, F.A.: Quantization. Math. USSR Izv., 8, 5, 1974, 1109-1165, · Zbl 0312.53049 · doi:10.1070/IM1974v008n05ABEH002140
[7] Berezin, F.A.: Quantization in complex symmetric domains. Math. USSR Izv., 9, 2, 1975, 341-379, · Zbl 0324.53049 · doi:10.1070/IM1975v009n02ABEH001480
[8] Berndt, R., Böcherer, S.: Jacobi forms and discrete series representations of the Jacobi group. Math. Z., 204, 1990, 13-44, · Zbl 0695.10024 · doi:10.1007/BF02570858
[9] Berndt, R., Schmidt, R.: Elements of the representation theory of the Jacobi group, Progress in Mathematics 163. 1998, Birkhäuser Verlag, Basel, · Zbl 0931.11013
[10] Cahen, B.: Berezin quantization for discrete series. Beiträge Algebra Geom., 51, 2010, 301-311, · Zbl 1342.22022
[11] Cahen, B.: Stratonovich-Weyl correspondence for compact semisimple Lie groups. Rend. Circ. Mat. Palermo, 59, 2010, 331-354, · Zbl 1218.22008 · doi:10.1007/s12215-010-0026-y
[12] Cahen, B.: Stratonovich-Weyl correspondence for discrete series representations. Arch. Math. (Brno), 47, 2011, 41-58, · Zbl 1240.22011
[13] Cahen, B.: Berezin Quantization and Holomorphic Representations. Rend. Sem. Mat. Univ. Padova, 129, 2013, 277-297, · Zbl 1272.22007 · doi:10.4171/RSMUP/129-16
[14] Cahen, B.: Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica, 52, 2013, 35-48, · Zbl 1296.22007
[15] Cariñena, J.F., Gracia-Bondìa, J.M., Vàrilly, J.C.: Relativistic quantum kinematics in the Moyal representation. J. Phys. A: Math. Gen., 23, 1990, 901-933, · Zbl 0706.60108 · doi:10.1088/0305-4470/23/6/015
[16] Davidson, M., Òlafsson, G., Zhang, G.: Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials. J. Funct. Anal., 204, 2003, 157-195, · Zbl 1035.32014 · doi:10.1016/S0022-1236(03)00101-0
[17] Figueroa, H., Gracia-Bondìa, J.M., Vàrilly, J.C.: Moyal quantization with compact symmetry groups and noncommutative analysis. J. Math. Phys., 31, 1990, 2664-2671, · Zbl 0753.43002 · doi:10.1063/1.528967
[18] Folland, B.: Harmonic Analysis in Phase Space. 1989, Princeton Univ. Press, · Zbl 0682.43001
[19] Gracia-Bondìa, J.M.: Generalized Moyal quantization on homogeneous symplectic spaces, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), 93–114, Contemp. Math., 134. 1992, Amer. Math. Soc., Providence, RI, · Zbl 0788.58024
[20] Gracia-Bondìa, J.M., V\?rilly, J.C.: The Moyal Representation for Spin. Ann. Physics, 190, 1989, 107-148, · Zbl 0652.46028 · doi:10.1016/0003-4916(89)90262-5
[21] Kirillov, A.A.: Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64. 2004, American Mathematical Society, Providence, Rhode Island, · Zbl 1229.22003 · doi:10.1090/gsm/064
[22] Neeb, K-H.: Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics, Vol. 28. 2000, Walter de Gruyter, Berlin, New-York, · Zbl 0964.22004
[23] Nomura, T.: Berezin Transforms and Group representations. J. Lie Theory, 8, 1998, 433-440, · Zbl 0919.43008
[24] Ørsted, B., Zhang, G.: Weyl Quantization and Tensor Products of Fock and Bergman Spaces. Indiana Univ. Math. J., 43, 2, 1994, 551-583, · Zbl 0805.46053 · doi:10.1512/iumj.1994.43.43023
[25] Peetre, J., Zhang, G.: A weighted Plancherel formula III. The case of a hyperbolic matrix ball. Collect. Math., 43, 1992, 273-301, · Zbl 0836.43018
[26] Satake, I.: Algebraic structures of symmetric domains. 1971, Iwanami Sho-ten, Tokyo and Princeton Univ. Press, Princeton, NJ, · Zbl 0813.32028
[27] Stratonovich, R.L.: On distributions in representation space. Soviet Physics. JETP, 4, 1957, 891-898, · Zbl 0082.19302
[28] Unterberger, A., Upmeier, H.: Berezin transform and invariant differential operators. Commun. Math. Phys., 164, 3, 1994, 563-597, · Zbl 0843.32019 · doi:10.1007/BF02101491
[29] Zhang, G.: Berezin transform on compact Hermitian symmetric spaces. Manuscripta Math., 97, 1998, 371-388, · Zbl 0920.22008 · doi:10.1007/s002290050109
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