Группа Якоби и ее представления голоморфной дискретной серии на областях Зигеля-Якоби

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.

группа Якоби, область Зигеля-Якоби, канонический фактор автоморфности, каноническое ядро (функция), скалярная голоморфная дискретная серия

1. S. Berceanu, “A holomorphic representation of the Jacobi algebra”, Reviews in Mathematical Physics, 18:2 (2006), 163-199, arXiv:math/0408219v3.

2. 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.

3. 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.

4. S. Berceanu, A. Gheorghe, “Applications of the Jacobi group to Quantum Mechanics”, Romanian Journal of Physics, 53:9-10 (2008), 1013-1021.

5. S. Berceanu and A. Gheorghe, “On the geometry of Siegel-Jacobi domains”, International Journal of Geometric Methods in Modern Physics, 8 (2011), 1783-1798.

6. S. Berceanu, “A convenient coordinatization of Siegel-Jacobi domains”, Reviews in Mathematical Physics, 24:10 (2012), 1-38 c r os s r ef.

7. 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.

8. F.A. Berezin, “Quantization in complex symmetric spaces”, Math. USSR-Izv., 9:2 (1975), 341-379 c r os s r ef.

9. R. Berndt and R. Schmidt, Progress in Mathematics: Elements of the Representation Theory of the Jacobi Group, 163, Birkh¨auser, Basel, 1998, 216 pp.

10. M. Eichler and D. Zagier, Progress in Mathematics: The Theory of Jacobi Forms, 55, Birkh¨auser, Boston, 1985.

11. 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.

12. 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.

13. A.M. Perelomov, Generalized Coherent States and their Applications, Springer, Berlin, 1986.

14. 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.

15. I. Satake, “Unitary representations of a semi-direct products of Lie groups on ∂ -cohomology spaces”, Mathematische Annalen, 190:3 (1971), 177-202.

16. I. Satake, Algebraic Structures of Symmetric Domains, Princeton University Press, Princeton, 1980.

17. 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.

18. K. Takase, “A note on automorphic forms”, J. Reine Angew. Math., 409 (1990), 138-171.

19. K. Takase, “On unitary representations of Jacobi groups”, J. Reine Angew. Math., 430 (1992), 130-149.

20. K. Takase, “On Siegel modular forms of half-integral weights and Jacobi forms”, Trans. Amer. Math. Soc., 351:2 (1999), 735-780.

21. J.-H. Yang, “The method of orbits for real Lie groups”, Kyungpook Mathematical Journal, 42:2 (2002), 199-272.

22. J.-H. Yang, “A partial Cayley transform for Siegel-Jacobi disk”, Journal of the Korean Mathematical Society, 45 (2008), 781-794.

23. C. Ziegler, “Jacobi forms of higher degree”, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 59:1 (1989), 191-224.