Berezin correspondence and Stratonovich-Weyl correspondence for the semi-direct product of the Heisenberg group and \(SU (p,q)\). (English) Zbl 07898737
Summary: We study the Berezin correspondence and the Stratonovich-Weyl correspondence associated with a holomorphic representation of the semi-direct product \(H \rtimes SU (p, q)\), where \(H\) is the \((2p + 2q + 1)\)-dimensional Heisenberg group.
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 32M10 | Homogeneous complex manifolds |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
Keywords:
Berezin quantization; Berezin correspondence; Berezin transform; quasi-Hermitian Lie group; unitary representation; holomorphic representation; reproducing Kernel Hilbert space; Stratonovich-Weyl correspondenceReferences:
| [1] | J. Arazy and H. Upmeier. Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains. Function Spaces, Interpolation Theory and Related Topics (Lund, 2000). de Gruyter, Berlin, 2002, pp.151-211. · Zbl 1027.32020 |
| [2] | D. Arnal, M. Cahen and S. Gutt. Representation of compact Lie groups and quantization by deformation. Acad. Roy. Belg. Bull. Cl. Sci. (1988), 123-141. · Zbl 0681.58016 |
| [3] | D. Arnal and J. C. Cortet. Nilpotent Fourier transform and applications. Lett. Math. Phys. 9 (1985), 25-34. · Zbl 0616.46041 |
| [4] | F. A. Berezin. Covariant and contravariant symbols of operators. Math. USSR Izv. 6(5) (1972), 1117-1151. · Zbl 0259.47004 |
| [5] | F. A. Berezin. Quantization. Math. USSR Izv. 8(5) (1974), 1109-1165. · Zbl 0312.53049 |
| [6] | F. A. Berezin. Quantization in complex symmetric domains. Math. USSR Izv. 9(2) (1975), 341-379. · Zbl 0324.53049 |
| [7] | B. Cahen. Contractions of SU(1, n) and SU(n+1) via Berezin quantization. J. Anal. Math. 97 (2005), 83-102. |
| [8] | B. Cahen. Berezin quantization on generalized flag manifolds. Math. Scand. 105 (2009), 66-84. · Zbl 1183.22006 |
| [9] | B. Cahen. Contraction of compact semisimple Lie groups via Berezin quantization. Illinois J. Math. 53 (2009), 265-288. · Zbl 1185.22008 |
| [10] | B. Cahen. Contraction of discrete series via Berezin quantization. J. Lie Theory 19 (2009), 291-310. · Zbl 1185.22007 |
| [11] | B. Cahen. Stratonovich-Weyl correspondence for discrete series representations. Arch. Math. (Brno) 47 (2011), 41-58. |
| [12] | B. Cahen. Berezin quantization and holomorphic representations. Rend. Semin. Mat. Univ. Padova 129 (2013), 277-297. · Zbl 1272.22007 |
| [13] | B. Cahen. Stratonovich-Weyl correspondence for the diamond group. Riv. Mat. Univ. Parma 4 (2013), 197-213. · Zbl 1293.22003 |
| [14] | B. Cahen. Global parametrization of scalar holomorphic coadjoint orbits of a quasi-Hermitian Lie group. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 52 (2013), 35-48. · Zbl 1296.22007 |
| [15] | B. Cahen. Stratonovich-Weyl correspondence for the Jacobi group. Commun. Math. 22 (2014), 31-48. · Zbl 1304.22005 |
| [16] | B. Cahen. Berezin transform and Stratonovich-Weyl correspondence for the multi-dimensional Jacobi group. Rend. Semin. Mat. Univ. Padova 136 (2016), 69-93. · Zbl 1359.22010 |
| [17] | B. Cahen. Weyl calculus on the Fock space and Stratonovich-Weyl correspondence for Heisenberg motion groups. Rend. Semin. Mat. Univ. Politec. Torino 76 (2018), 63-80. · Zbl 1440.22015 |
| [18] | B. Cahen. A note on the harmonic representation of SU(p, q). Preprint. |
| [19] | H. Figueroa, J. M. Gracia-Bondìa and J. C. Vàrilly. Moyal quantization with compact symmetry groups and noncommutative analysis. J. Math. Phys. 31 (1990), 2664-2671. · Zbl 0753.43002 |
| [20] | B. Folland. Harmonic Analysis in Phase Space. Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.43001 |
| [21] | J. M. Gracia-Bondìa. Generalized Moyal quantization on homogeneous symplectic spaces. Deformation Theory and Quantum Groups With Applications to Mathematical Physics (Amherst, MA, 1990) (Contemporary Mathematics, 134). American Mathematical Society, Providence, RI, 1992, pp. 93-114. · Zbl 0788.58024 |
| [22] | L. K. Hua. Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains (Translations of Mathematical Monographs, 6). American Mathematical Society, Providence, RI, 1963. · Zbl 0112.07402 |
| [23] | A. A. Kirillov. Lectures on the Orbit Method (Graduate Studies in Mathematics, 64). American Mathematical Society, Providence, RI, 2004. |
| [24] | S. G. Low. Representations of the canonical group (the semidirect product of the unitary and Weyl-Heisenberg groups), acting as a dynamical group on noncommutative extended phase space. J. Phys. A, Math. Gen. 35 (2002), 5711-5729. · Zbl 1068.81032 |
| [25] | K.-H. Neeb. Holomorphy and Convexity in Lie Theory (de Gruyter Expositions in Mathematics, 28). de Gruyter, Berlin, 2000. |
| [26] | T. Nomura. Berezin transforms and group representations. J. Lie Theory 8 (1998), 433-440. · Zbl 0919.43008 |
| [27] | B. Ørsted and G. Zhang. Weyl quantization and tensor products of Fock and Bergman spaces. Indiana Univ. Math. J. 43(2) (1994), 551-583. · Zbl 0805.46053 |
| [28] | I. Satake. Algebraic Structures of Symmetric Domains. Iwanami Sho-ten, Tokyo; and Princeton University Press, Princeton, NJ, 1971. |
| [29] | R. L. Stratonovich. On distributions in representation space. Sov. Phys. JETP 4 (1957), 891-898. · Zbl 0082.19302 |
| [30] | A. Unterberger and H. Upmeier. Berezin transform and invariant differential operators. Comm. Math. Phys. 164(3) (1994), 563-597. · Zbl 0843.32019 |
| [31] | N. J. Wildberger. On the Fourier transform of a compact semisimple Lie group. J. Aust. Math. Soc. A 56 (1994) 64-116. · Zbl 0842.22015 |
| [32] | J. A. Wolf. Representations of certain semidirect product groups. J. Funct. Anal. 19 (1975), 339-372. · Zbl 0311.22021 |
| [33] | G. Zhang. Berezin transform on compact Hermitian symmetric spaces. Manuscripta Math. 97 (1998), 371-388. · Zbl 0920.22008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
