Berceanu, S.; Gheorghe, A. On the geometry of Siegel-Jacobi domains. (English) Zbl 1250.22010 Int. J. Geom. Methods Mod. Phys. 8, No. 8, 1783-1798 (2011). The Jacobi group \(G^J\) is the semidirect product of a Zariski connected semisimple real algebraic group of Hermitian type \(G^s\) and the Heisenberg group \(H(V)\) associated with the symplectic \(\mathbb{R}\)-space \(V\). The Hermitian symmetric domain associated to \(G^s\) is \(\mathcal{D}=G^s/K^s\), where \(K^s\) is a maximal compact subgroup. The Jacobi-Siegel domain associated to the Jacobi group \(G^J\) is \(\mathcal{D}^J=\mathcal{D}\times \mathbb{C}^N\), where dim \(V=2N\). The authors study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. The Siegel disk \(\mathcal{D}_n\) of degree \(n\) consists of all the symmetric matrices \(W\in M_n(\mathbb{C})\) with \(I_n-W\overline{W}>0\). Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk \(\mathcal{D}_n^J=\mathcal{D}_n \times \mathbb{C}^n\) are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given. Reviewer: Nicolae Cotfas (Bucureşti) Cited in 9 Documents MSC: 22E30 Analysis on real and complex Lie groups 20G05 Representation theory for linear algebraic groups 11F50 Jacobi forms 12E10 Special polynomials in general fields 81R30 Coherent states Keywords:Jacobi group; Siegel-Jacobi domain; canonical automorphy factor; canonical kernel function; Fock representation; scalar holomorphic discrete series PDF BibTeX XML Cite \textit{S. Berceanu} and \textit{A. Gheorghe}, Int. J. Geom. Methods Mod. Phys. 8, No. 8, 1783--1798 (2011; Zbl 1250.22010) Full Text: DOI arXiv References: [1] DOI: 10.1002/cpa.3160140303 · Zbl 0107.09102 [2] DOI: 10.1142/S0129055X06002619 · Zbl 1099.81036 [3] Berceanu S., J. Geom. Symmetry Phys. 5 pp 5– [4] Berceanu S., J. Geom. Symmetry Phys. 9 pp 1– [5] S. Berceanu, Perspectives in Operator Algebra and Mathematical Physics (The Theta Foundation, Bucharest, 2008) pp. 1–25. [6] Berceanu S., Romanian J. Phys. 53 pp 1013– [7] DOI: 10.1070/IM1975v009n02ABEH001480 · Zbl 0324.53049 [8] DOI: 10.1007/BF02570858 · Zbl 0695.10024 [9] DOI: 10.1007/978-3-0348-0283-3 · Zbl 1235.11046 [10] Cirelli R., J. Phys. A 15 pp 3829– [11] DOI: 10.1007/BF01645755 · Zbl 0152.23204 [12] DOI: 10.1016/0378-4371(82)90330-2 [13] DOI: 10.1017/S1446788700003256 · Zbl 1041.22008 [14] DOI: 10.1090/S0273-0979-1991-16019-2 · Zbl 0736.22008 [15] Moscovici H., Ann. Inst. H. Poincaré Sect. A 29 pp 139– [16] DOI: 10.2140/pjm.1996.174.497 · Zbl 0894.22008 [17] DOI: 10.1007/BF01077706 · Zbl 0738.47035 [18] DOI: 10.1007/978-3-642-61629-7 [19] DOI: 10.1088/0305-4470/23/6/011 · Zbl 0715.22026 [20] Satake I., Ann. of Math. Studies 66 pp 393– [21] DOI: 10.1007/BF01433209 [22] DOI: 10.1016/0001-8708(71)90043-0 · Zbl 0219.22016 [23] Satake I., Publications of the Mathematical Society of Japan 14, in: Algebraic structures of symmetric domains (1980) · Zbl 0483.32017 [24] DOI: 10.1016/S0022-247X(02)00653-4 · Zbl 1021.94503 [25] Takase K., J. Reine Angew. Math. 409 pp 138– [26] Takase K., J. Reine Angew. Math. 430 pp 130– [27] DOI: 10.1090/S0002-9947-99-02168-6 · Zbl 0930.11030 [28] Yang J.-H., Kyungpook Math. J. 42 pp 199– [29] DOI: 10.1016/j.jnt.2006.12.014 · Zbl 1133.32011 [30] DOI: 10.4134/JKMS.2008.45.3.781 · Zbl 1223.32015 [31] DOI: 10.1007/BF02942329 · Zbl 0707.11035 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.