Previous |  Up |  Next

Article

Keywords:
Berezin quantization; Berezin transform; quasi-Hermitian Lie group; unitary representation; holomorphic representation; reproducing kernel Hilbert space; Jacobi group; Stratonovich-Weyl correspondence; coadjoint orbit
Summary:
We construct and study a Stratonovich-Weyl correspondence for the holomorphic representations of the Jacobi group.
References:
[1] Ali, S.T., Englis, M.: Quantization methods: a guide for physicists and analysts. Rev. Math. Phys., 17, 4, 2005, 391-490, DOI 10.1142/S0129055X05002376 | MR 2151954 | Zbl 1075.81038
[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, MR 1943284
[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, MR 1984098
[4] Berceanu, S.: A holomorphic representation of the Jacobi algebra. Rev. Math. Phys., 18, 2006, 163-199, DOI 10.1142/S0129055X06002619 | MR 2228923 | Zbl 1099.81036
[5] Berceanu, S., Gheorghe, A.: On the geometry of Siegel-Jacobi domains. Int. J. Geom. Methods Mod. Phys., 8, 2011, 1783-1798, MR 2876095 | Zbl 1250.22010
[6] Berezin, F.A.: Quantization. Math. USSR Izv., 8, 5, 1974, 1109-1165, Zbl 0312.53049
[7] Berezin, F.A.: Quantization in complex symmetric domains. Math. USSR Izv., 9, 2, 1975, 341-379, 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, DOI 10.1007/BF02570858 | MR 1048065 | Zbl 0695.10024
[9] Berndt, R., Schmidt, R.: Elements of the representation theory of the Jacobi group, Progress in Mathematics 163. 1998, Birkhäuser Verlag, Basel, MR 1634977
[10] Cahen, B.: Berezin quantization for discrete series. Beiträge Algebra Geom., 51, 2010, 301-311, MR 2682458
[11] Cahen, B.: Stratonovich-Weyl correspondence for compact semisimple Lie groups. Rend. Circ. Mat. Palermo, 59, 2010, 331-354, DOI 10.1007/s12215-010-0026-y | MR 2745515 | Zbl 1218.22008
[12] Cahen, B.: Stratonovich-Weyl correspondence for discrete series representations. Arch. Math. (Brno), 47, 2011, 41-58, MR 2813546 | Zbl 1240.22011
[13] Cahen, B.: Berezin Quantization and Holomorphic Representations. Rend. Sem. Mat. Univ. Padova, 129, 2013, 277-297, DOI 10.4171/RSMUP/129-16 | MR 3090642 | Zbl 1272.22007
[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, MR 3202747 | 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, DOI 10.1088/0305-4470/23/6/015 | MR 1048769 | Zbl 0706.60108
[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, DOI 10.1016/S0022-1236(03)00101-0 | MR 2004748 | Zbl 1035.32014
[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, DOI 10.1063/1.528967 | MR 1075750
[18] Folland, B.: Harmonic Analysis in Phase Space. 1989, Princeton Univ. Press, MR 0983366 | 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, MR 1187280
[20] Gracia-Bondìa, J.M., V¸rilly, J.C.: The Moyal Representation for Spin. Ann. Physics, 190, 1989, 107-148, MR 0994048
[21] Kirillov, A.A.: Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64. 2004, American Mathematical Society, Providence, Rhode Island, DOI 10.1090/gsm/064 | MR 2069175
[22] Neeb, K-H.: Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics, Vol. 28. 2000, Walter de Gruyter, Berlin, New-York, MR 1740617
[23] Nomura, T.: Berezin Transforms and Group representations. J. Lie Theory, 8, 1998, 433-440, MR 1650386 | 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, DOI 10.1512/iumj.1994.43.43023 | MR 1291529 | Zbl 0805.46053
[25] Peetre, J., Zhang, G.: A weighted Plancherel formula III. The case of a hyperbolic matrix ball. Collect. Math., 43, 1992, 273-301, MR 1252736
[26] Satake, I.: Algebraic structures of symmetric domains. 1971, Iwanami Sho-ten, Tokyo and Princeton Univ. Press, Princeton, NJ, MR 0591460
[27] Stratonovich, R.L.: On distributions in representation space. Soviet Physics. JETP, 4, 1957, 891-898, MR 0088173
[28] Unterberger, A., Upmeier, H.: Berezin transform and invariant differential operators. Commun. Math. Phys., 164, 3, 1994, 563-597, DOI 10.1007/BF02101491 | MR 1291245 | Zbl 0843.32019
[29] Zhang, G.: Berezin transform on compact Hermitian symmetric spaces. Manuscripta Math., 97, 1998, 371-388, DOI 10.1007/s002290050109 | MR 1654800 | Zbl 0920.22008
Partner of
EuDML logo