A Note on Maass-Jacobi Forms II

Yang, Jae-Hyun

  • Received : 2011.08.23
  • Accepted : 2012.09.17
  • Published : 2013.03.23


This article is a continuation of the paper [21]. In this paper we deal with Maass-Jacobi forms on the Siegel-Jacobi space $\mathbb{H}{\times}\mathbb{C}^m$, where H denotes the Poincar$\acute{e}$ upper half plane and $m$ is any positive integer.


Maass-Jacobi forms;invariant differential operators;fundamental domains;Casimir operators;skew-holomorphic Jacobi forms;covariant differential operators


  1. R. Campoamor-Stursburg and S. G. Low, Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators, J. Phys. A: Math. Theor., 42(2009), 065205.
  2. S. Helgason, Groups and geometric analysis, Academic Press, New York (1984).
  3. R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proceedings, vol. 8 (1995), 1-182.
  4. M. Itoh, H. Ochiai and J.-H. Yang, Invariant differential operators on Siegel-Jacobi space, preprint, 2013.
  5. N. N. Lebedev, Special Functions and their Applications, Dover, New York (1972).
  6. H. Maass, Die Differentialgleichungen in der Theorie der Siegelschen Modulfunktionen, Math. Ann., 126(1953), 44-68.
  7. H. Maass, Siegel modular forms and Dirichlet series, Lecture Notes in Math., vol. 216, Springer-Verlag, Berlin-Heidelberg-New York (1971).
  8. H. Ochiai, A remark on the generators of invariant differential operators on Siegel- Jacobi space of the smallest size, preprint, 2011.
  9. A. Pitale, Jacobi Maass forms, Abh. Math. Sem. Univ. Hamburg, 79(2009), 87-111.
  10. C. Quesne, Casimir operators of semidirect sum Lie algebras, J. Phys. A: Math. Gen., 21(1988), L321-L324.
  11. J.-P. Serre, A Course in Arithmetic, Springer-Verlag, Berlin-Heidelberg-New York (1973).
  12. C. L. Siegel, Symplectic Geometry, Amer. J. Math., 65(1943), 1-86
  13. N.-P. Skoruppa, Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms, Invent. Math., 102(1990), 501-520.
  14. G. N. Watson, The Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, London (1962).
  15. J.-H. Yang, The Method of Orbits for Real Lie Groups, Kyungpook Math. J., 42(2)(2002), 199-272 or arXiv:math.RT/0602056.
  16. J.-H. Yang, A note on Maass-Jacobi forms, Kyungpook Math. J. 43(4)(2003), 547- 566 or arXiv:math.NT/0612387.
  17. J.-H. Yang, A note on a fundamental domain for Siegel-Jacobi space, Houston J. Math., 32(3)(2006), 701-712 or arXiv:math.NT/0507218.
  18. J.-H. Yang, Invariant metrics and Laplacians on Siegel-Jacobi space, Journal of Number Theory, 127(2007), 83-102 or arXiv:math.NT/0507215.
  19. J.-H. Yang, A partial Cayley transform for Siegel-Jacobi disk, J. Korean Math. Soc., 45(2008), 781-794 or arXiv:math.NT/0507216.
  20. J.-H. Yang, Invariant metrics and Laplacians on Siegel-Jacobi disk, Chinese Annals of Mathematics, 31B(1)(2010), 85-100 or arXiv:math.NT/0507217.
  21. J.-H. Yang, Invariant Differential Operators on the Siegel-Jacobi Space, arXiv: 1107.0509v1 [math.NT], 4 July 2011.
  22. R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Progress in Mathematics, 163, Birkhauser, Basel, 1998.
  23. W. Borho, Primitive und vollprimitive Ideale in Einhullenden von so(5;C), J. Algebra, 43(1976), 619-654.
  24. K. Bringmann, C. Conley and O. K. Richter, Jacobi forms over complex quadratic fields via the cubic Casimier operators, preprint.
  25. K. Bringmann and O. K. Richter, Zagier-type dualities and lifting maps for harmonic Maass-Jacobi forms, Advances in Math., 225(2010), 2298-2315.
  26. C. Conley and M. Raum, Harmonic Maass-Jacobi forms of degree 1 with higher rank indices, arXiv:1012.289/v1 [math.NT], 13 Dec 2010.
  27. C. L. Siegel, Symplectic Geometry, Academic Press, New York and London (1964)
  28. C. L. Siegel, Symplectic Geometry, Gesammelte Abhandlungen, no. 41, vol. II, Springer-Verlag (1966), 274-359

Cited by

  1. THETA SUMS OF HIGHER INDEX vol.53, pp.6, 2016,


Supported by : National Research Foundation of Korea(NRF)