Berezin quantization and holomorphic representations
Rendiconti del Seminario Matematico della Università di Padova, Volume 129  (2013), p. 277-298
@article{RSMUP_2013__129__277_0,
     author = {Cahen, Benjamin},
     title = {Berezin quantization and holomorphic representations},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     publisher = {Seminario Matematico of the University of Padua},
     volume = {129},
     year = {2013},
     pages = {277-298},
     zbl = {1272.22007},
     mrnumber = {3090642},
     language = {en},
     url = {http://www.numdam.org/item/RSMUP_2013__129__277_0}
}
Cahen, Benjamin. Berezin quantization and holomorphic representations. Rendiconti del Seminario Matematico della Università di Padova, Volume 129 (2013) , pp. 277-298. http://www.numdam.org/item/RSMUP_2013__129__277_0/

[1] S. T. Ali - M. Englis, Quantization methods: a guide for physicists and analysts, Rev. Math. Phys. 17, 4 (2005), pp. 391-490. | MR 2151954

[2] J. Arazy - H. Upmeier, 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, no 3-4 (2002), pp. 165-181. | MR 1984098

[3] J. Arazy - H. Upmeier, Invariant symbolic calculi and eigenvalues of invariant operators on symmeric domains, Function spaces, interpolation theory and related topics (Lund, 2000), pp. 151-211, de Gruyter, Berlin, 2002. | MR 1943284

[4] D. Arnal - J.-C. Cortet, Nilpotent Fourier Transform and Applications, Lett. Math. Phys. 9 (1985), pp. 25-34. | MR 774736

[5] L. Auslander - B. Kostant, Polarization and Unitary Representations of Solvable lie Groups, Invent. Math. 14 (1971), pp. 255-354. | MR 293012

[6] I. Beltiţă - D. Beltiţă, Magnetic pseudo-differential Weyl calculus on nilpotent Lie groups, Ann. Global Anal. Geom. 36, 3 (2009), pp. 293-322. | MR 2544305

[7] P. Bernat - N. Conze - M. Duflo - M. Levy-Nahas - M. Rais - P. Renouard - M. Vergne, Representations des groupes de Lie résolubles, Dunod, Paris 1972. | MR 444836

[8] F. A. Berezin, Quantization, Math. USSR Izv. 8, 5 (1974), pp. 1109-1165. | MR 395610

[9] F. A. Berezin, Quantization in complex symmetric domains, Math. USSR Izv. 9, 2 (1975), pp. 341-379.

[10] C. Brif - A. Mann, Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries, Phys. Rev. A 59, 2 (1999), pp. 971-987. | MR 1679730

[11] B. Cahen, Deformation Program for Principal Series Representations, Lett. Math. Phys. 36 (1996), pp. 65-75. | MR 1371298

[12] B. Cahen, Quantification d'une orbite massive d'un groupe de Poincaré généralisé, C. R. Acad. Sci. Paris t. 325, série I (1997), pp. 803-806. | MR 1483721

[13] B. Cahen, Weyl quantization for semidirect products, Differential Geom. Appl. 25 (2007), pp. 177-190. | MR 2311733

[14] B. Cahen, Weyl quantization for principal series, Beiträge Algebra Geom. 48, 1 (2007), pp. 175-190. | MR 2326408

[15] B. Cahen, Contraction of compact semisimple Lie groups via Berezin quantization, Illinois J. Math. 53, 1 (2009), pp. 265-288. | MR 2584946

[16] B. Cahen, Berezin quantization on generalized flag manifolds, Math. Scand. 105 (2009), pp. 66-84. | MR 2549798

[17] B. Cahen, Contraction of discrete series via Berezin quantization, J. Lie Theory, 19 (2009), pp. 291-310. | MR 2572131

[18] B. Cahen, Berezin quantization for discrete series, Beiträge Algebra Geom. 51 (2010), pp. 301-311. | MR 2682458

[19] B. Cahen, Stratonovich-Weyl correspondence for compact semisimple Lie groups, Rend. Circ. Mat. Palermo, 59 (2010), pp. 331-354. | MR 2745515

[20] B. Cahen, Stratonovich-Weyl correspondence for discrete series representations, Arch. Math. (Brno), 47 (2011), pp. 41-58. | MR 2813546

[21] B. Cahen, Weyl quantization for the semi-direct product of a compact Lie group and a vector space, Comment. Math. Univ. Carolin. 50, 3 (2009), pp. 325-347. | MR 2573408

[22] B. Cahen, Weyl quantization for Cartan motion groups, Comment. Math. Univ. Carolin. 52, 1 (2011), pp. 127-137. | MR 2828363

[23] M. Cahen - S. Gutt - J. Rawnsley, Quantization on Kähler manifolds I, Geometric interpretation of Berezin quantization, J. Geom. Phys. 7 (1990), pp. 45-62. | MR 1094730

[24] J. F. Cariñena - J. M. Gracia-Bondìa - J. C. Vàrilly, Relativistic quantum kinematics in the Moyal representation, J. Phys. A: Math. Gen. 23 (1990), pp. 901-933.

[25] P. Cotton - A. H. Dooley, Contraction of an Adapted Functional Calculus, J. Lie Theory, 7 (1997), pp. 147-164. | MR 1473162

[26] M. Davidson - G. Òlafsson - G. Zhang, Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials, J. Funct. Anal. 204 (2003), pp. 157-195. | MR 2004748

[27] H. Figueroa - J. M. Gracia-Bondìa - J. C. Vàrilly, Moyal quantization with compact symmetry groups and noncommutative analysis, J. Math. Phys. 31 (1990), pp. 2664-2671. | MR 1075750

[28] B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989. | MR 983366

[29] J. M. Gracia-Bondìa, Generalized Moyal quantization on homogeneous symplectic spaces, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), pp. 93-114, Contemp. Math., 134, Amer. Math. Soc., Providence, RI, 1992. | MR 1187280

[30] J. M. Gracia-Bondìa - J. C. Vàrilly, The Moyal Representation for Spin, Ann. Physics, 190 (1989), pp. 107-148. | MR 994048

[31] M. Gotay, Obstructions to Quantization, in: Mechanics: From Theory to Computation (Essays in Honor of Juan-Carlos Simo), J. Nonlinear Science Editors, Springer New-York, 2000, pp. 271-316. | MR 1766355

[32] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, American Mathematical Society, Providence, Rhode Island 2001. | MR 1834454

[33] A. W. Knapp, Representation theory of semi-simple groups. An overview based on examples, Princeton Math. Series t. 36 (1986). | MR 855239

[34] A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics Vol. 64, American Mathematical Society, Providence, Rhode Island, 2004. | MR 2069175

[35] B. Kostant, Quantization and unitary representations, in: Modern Analysis and Applications, Lecture Notes in Mathematics 170, Springer-Verlag, Berlin, Heidelberg, New-York, 1970, pp. 87-207. | MR 294568

[36] K-H. Neeb, Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics, Vol. 28, Walter de Gruyter, Berlin, New-York 2000. | MR 1740617

[37] T. Nomura, Berezin Transforms and Group representations, J. Lie Theory, 8 (1998), pp. 433-440. | MR 1650386

[38] B. Ørsted - G. Zhang, Weyl Quantization and Tensor Products of Fock and Bergman Spaces, Indiana Univ. Math. J. 43, 2 (1994), pp. 551-583. | MR 1291529

[39] J. Peetre - G. Zhang, A weighted Plancherel formula III. The case of a hyperbolic matrix ball, Collect. Math. 43 (1992), pp. 273-301. | MR 1252736

[40] N. V. Pedersen, Matrix coefficients and a Weyl correspondence for nilpotent Lie groups, Invent. Math. 118 (1994), pp. 1-36. | MR 1288465

[41] I. Satake, Algebraic structures of symmetric domains, Iwanami Sho-ten, Tokyo and Princeton Univ. Press, Princeton, NJ, 1971. | MR 591460

[42] R. L. Stratonovich, On distributions in representation space, Soviet Physics. JETP, 4 (1957), pp. 891-898. | MR 88173

[43] A. Unterberger - H. Upmeier, Berezin transform and invariant differential operators, Commun. Math. Phys. 164, 3 (1994), pp. 563-597. | MR 1291245

[44] N. J. Wildberger, Convexity and unitary representations of a nilpotent Lie group, Invent. Math. 89 (1989), pp. 281-292. | MR 1016265

[45] G. Zhang, Berezin transform on compact Hermitian symmetric spaces, Manuscripta Math. 97 (1998), pp. 371-388. | MR 1654800