mathematicsВестник российских университетов. МатематикаRussian Universities Reports. Mathematics2686-96672782-3342Тамбовский государственный университет имени Г.Р. Державина10.20310/2686-9667-2019-24-128-345-353mathematics-246Research ArticleСТАТЬИARTICLESГруппа Якоби и ее представления голоморфной дискретной серии на областях Зигеля-ЯкобиThe Jacobi group and its holomorphic discrete series representations on Siegel-Jacobi domainsБерчануСтефанBerceanuStefanBerceanu@theory.nipne.roГеоргеАлександруGheorgheAlexandrunoemail@neicon.ruНациональный институт исследований и разработок в области физики и ядерной инженерии им. Хории ХулубеяРоссияHoria Hulubei National Institute for R&D in Physics and Nuclear EngineeringRussian Federation20190101202024128345353Copyright © Берчану С., Георге А., 20202020Берчану С., Георге А.Berceanu S., Gheorghe A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathematics.elpub.ru/jour/article/view/246

Эта статья - краткое изложение части лекции, прочитанной на конференции в Тамбовском университете в октябре 2012, излагающее некоторые результаты о группах Якоби и их голоморфных представлениях, полученные авторами.

This is the summary of a part of the talk delivered at the workshop held at the Tambov University in September 2012, reporting several results on Jacobi groups and its holomorphic representations published by the authors.

группа Якобиобласть Зигеля-Якобиканонический фактор автоморфностиканоническое ядро (функция)скалярная голоморфная дискретная серияJacobi groupSiegel-Jacobi domaincanonical automorphy factorcanonical kernel functionscalar holomorphic discrete seriesReferencesS. Berceanu, “A holomorphic representation of the Jacobi algebra”, Reviews in Mathematical Physics, 18:2 (2006), 163-199, arXiv:math/0408219v3.S. Berceanu, “A holomorphic representation of the Jacobi algebra”, Reviews in Mathematical Physics, 18:2 (2006), 163-199, arXiv:math/0408219v3.S. Berceanu, “Coherent states associated to the Jacobi group - a variation on a theme by Erich K¨ahler”, Journal of Geometry and Symmetry in Physics, 9 (2007), 1-8 c r os s r ef.S. Berceanu, “Coherent states associated to the Jacobi group - a variation on a theme by Erich K¨ahler”, Journal of Geometry and Symmetry in Physics, 9 (2007), 1-8 c r os s r ef.S. Berceanu, “A holomorphic representation of Jacobi algebra in several dimensions”, The Theta Foundation, Perspectives in Operator Algebras and Mathematical Physics (Bucharest, August 10-17, 2005), Theta Series in Advanced Mathematics, 11, eds. F.P. Boca, R. Purice, S. Strˇatilˇa, Conference Proceedings, 2008, 1-25.S. Berceanu, “A holomorphic representation of Jacobi algebra in several dimensions”, The Theta Foundation, Perspectives in Operator Algebras and Mathematical Physics (Bucharest, August 10-17, 2005), Theta Series in Advanced Mathematics, 11, eds. F.P. Boca, R. Purice, S. Strˇatilˇa, Conference Proceedings, 2008, 1-25.S. Berceanu, A. Gheorghe, “Applications of the Jacobi group to Quantum Mechanics”, Romanian Journal of Physics, 53:9-10 (2008), 1013-1021.S. Berceanu, A. Gheorghe, “Applications of the Jacobi group to Quantum Mechanics”, Romanian Journal of Physics, 53:9-10 (2008), 1013-1021.S. Berceanu and A. Gheorghe, “On the geometry of Siegel-Jacobi domains”, International Journal of Geometric Methods in Modern Physics, 8 (2011), 1783-1798.S. Berceanu and A. Gheorghe, “On the geometry of Siegel-Jacobi domains”, International Journal of Geometric Methods in Modern Physics, 8 (2011), 1783-1798.S. Berceanu, “A convenient coordinatization of Siegel-Jacobi domains”, Reviews in Mathematical Physics, 24:10 (2012), 1-38 c r os s r ef.S. Berceanu, “A convenient coordinatization of Siegel-Jacobi domains”, Reviews in Mathematical Physics, 24:10 (2012), 1-38 c r os s r ef.S. Berceanu, “Consequences of the fundamental conjecture for the motion on the Siegel-Jacobi disk”, International Journal of Geometric Methods in Modern Physics, 10:01 (2013), 1-18 c r os s r ef.S. Berceanu, “Consequences of the fundamental conjecture for the motion on the Siegel-Jacobi disk”, International Journal of Geometric Methods in Modern Physics, 10:01 (2013), 1-18 c r os s r ef.F.A. Berezin, “Quantization in complex symmetric spaces”, Math. USSR-Izv., 9:2 (1975), 341-379 c r os s r ef.F.A. Berezin, “Quantization in complex symmetric spaces”, Math. USSR-Izv., 9:2 (1975), 341-379 c r os s r ef.R. Berndt and R. Schmidt, Progress in Mathematics: Elements of the Representation Theory of the Jacobi Group, 163, Birkh¨auser, Basel, 1998, 216 pp.R. Berndt and R. Schmidt, Progress in Mathematics: Elements of the Representation Theory of the Jacobi Group, 163, Birkh¨auser, Basel, 1998, 216 pp.M. Eichler and D. Zagier, Progress in Mathematics: The Theory of Jacobi Forms, 55, Birkh¨auser, Boston, 1985.M. Eichler and D. Zagier, Progress in Mathematics: The Theory of Jacobi Forms, 55, Birkh¨auser, Boston, 1985.P. Kramer and M. Saraceno, “Semicoherent states and the group ISp(2,R)”, Physica A: Statistical Mechanics and its Applications, 114:1-3 (1982), 448-453 c r os s r ef.P. Kramer and M. Saraceno, “Semicoherent states and the group ISp(2,R)”, Physica A: Statistical Mechanics and its Applications, 114:1-3 (1982), 448-453 c r os s r ef.M.H. Lee, “Theta functions on hermitian symmetric domains and Fock representations”, Journal of the Australian Mathematical Society, 74:2 (2003), 201-234 c r os s r ef.M.H. Lee, “Theta functions on hermitian symmetric domains and Fock representations”, Journal of the Australian Mathematical Society, 74:2 (2003), 201-234 c r os s r ef.A.M. Perelomov, Generalized Coherent States and their Applications, Springer, Berlin, 1986.A.M. Perelomov, Generalized Coherent States and their Applications, Springer, Berlin, 1986.C. Quesne, “Vector coherent state theory of the semidirect sum Lie algebras wsp (2N,R)”, Journal of Physics A: Mathematical and General, 23:6 (1990), 847-862.C. Quesne, “Vector coherent state theory of the semidirect sum Lie algebras wsp (2N,R)”, Journal of Physics A: Mathematical and General, 23:6 (1990), 847-862.I. Satake, “Unitary representations of a semi-direct products of Lie groups on ∂ -cohomology spaces”, Mathematische Annalen, 190:3 (1971), 177-202.I. Satake, “Unitary representations of a semi-direct products of Lie groups on ∂ -cohomology spaces”, Mathematische Annalen, 190:3 (1971), 177-202.I. Satake, Algebraic Structures of Symmetric Domains, Princeton University Press, Princeton, 1980.I. Satake, Algebraic Structures of Symmetric Domains, Princeton University Press, Princeton, 1980.K. Shuman, “Complete signal processing bases and the Jacobi group”, Journal of Mathematical Analysis and Applications, 278:1 (2003), 203-213 c r os s r ef.K. Shuman, “Complete signal processing bases and the Jacobi group”, Journal of Mathematical Analysis and Applications, 278:1 (2003), 203-213 c r os s r ef.K. Takase, “A note on automorphic forms”, J. Reine Angew. Math., 409 (1990), 138-171.K. Takase, “A note on automorphic forms”, J. Reine Angew. Math., 409 (1990), 138-171.K. Takase, “On unitary representations of Jacobi groups”, J. Reine Angew. Math., 430 (1992), 130-149.K. Takase, “On unitary representations of Jacobi groups”, J. Reine Angew. Math., 430 (1992), 130-149.K. Takase, “On Siegel modular forms of half-integral weights and Jacobi forms”, Trans. Amer. Math. Soc., 351:2 (1999), 735-780.K. Takase, “On Siegel modular forms of half-integral weights and Jacobi forms”, Trans. Amer. Math. Soc., 351:2 (1999), 735-780.J.-H. Yang, “The method of orbits for real Lie groups”, Kyungpook Mathematical Journal, 42:2 (2002), 199-272.J.-H. Yang, “The method of orbits for real Lie groups”, Kyungpook Mathematical Journal, 42:2 (2002), 199-272.J.-H. Yang, “A partial Cayley transform for Siegel-Jacobi disk”, Journal of the Korean Mathematical Society, 45 (2008), 781-794.J.-H. Yang, “A partial Cayley transform for Siegel-Jacobi disk”, Journal of the Korean Mathematical Society, 45 (2008), 781-794.C. Ziegler, “Jacobi forms of higher degree”, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 59:1 (1989), 191-224.C. Ziegler, “Jacobi forms of higher degree”, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 59:1 (1989), 191-224.

The authors declare that there are no conflicts of interest present.