Advertisement

Indecomposable Harish-Chandra Modules for Jacobi Groups

  • Martin Raum
Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 10)

Abstract

We describe some indecomposable \(({\mathfrak {g}},{\mathrm {K}})\)-modules for Jacobi groups that admit an automorphic realization with possible singularities. A particular tensor product decomposition of universal enveloping algebras of Jacobi Lie algebras, which does not lift to the groups, allows us to study distinguished highest weight modules for the Heisenberg group. We encounter modified theta series as components of vector-valued Jacobi forms, whose arithmetic type is not completely reducible.

Notes

Acknowledgements

The author thanks the referee for comments greatly improving readability of this paper. The author was partially supported by Vetenskapsrøadet Grant 2015-04139.

References

  1. 1.
    Berndt, R., Schmidt, R.: Elements of the Representation Theory of the Jacobi Group, vol. 163. Progress in Mathematics. Birkhäuser, Basel (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bringmann, K., Kudla, S.: A classification of harmonicMaaß forms (2016). arXiv:1609.06999Google Scholar
  3. 3.
    Bringmann, K., Richter, O.K.: Zagier-type dualities and lifting maps for harmonic Maass-Jacobi forms. Adv. Math. 225(4), 2298–2315 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bringmann, K., Raum, M., Richter, O.K.: Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition. Trans. Am. Math. Soc. 367(9), 6647–6670 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 45–90 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Campoamor-Stursberg, R., Low, S.G.: Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators. J. Phys. A 42(6), 065205 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Conley, C.H., Westerholt-Raum, M.: Harmonic Maaß–Jacobi forms of degree 1 with higher rank indices. Int. J. Number Theory 12(7), 1871–1897 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Duke, W., Imamoğlu, Ö., Tóth, Á.: Cycle integrals of the j-function and mock modular forms. Ann. Math. (2) 173(2), 947–981 (2011)Google Scholar
  9. 9.
    Eichler, M., Zagier, D.B.: The Theory of Jacobi Forms, vol. 55. Progress in Mathematics. Birkhäuser, Boston, MA (1985)zbMATHGoogle Scholar
  10. 10.
    Gel’fand, I.M., Ponomarev, V.A.: A classification of the indecomposable infinitesimal representations of the Lorentz group. Dokl. Akad. Nauk SSSR 176, 502–505 (1967)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Goujard, E., Möller, M.: Counting Feynman-like graphs: quasimodularity and Siegel-Veech weight (2016). arXiv:1609.01658Google Scholar
  12. 12.
    Knapp, A.W., Zuckerman, G.: Classification of irreducible tempered representations of semi-simple Lie groups. Proc. Natl. Acad. Sci. USA 73(7), 2178–2180 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Langlands, R.P.: On the classification of irreducible representations of real algebraic groups. In: Representation Theory and Harmonic Analysis on Semisimple Lie Groups, vol. 31. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1989)Google Scholar
  14. 14.
    Ono, K.: Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory. Current Developments in Mathematics, vol. 2008. International Press, Somerville, MA (2009)Google Scholar
  15. 15.
    Pitale, A.: Jacobi Maaß forms. Abh. Math. Semin. Univ. Hambg. 79(1), 87–111 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Quesne, C.: Casimir operators of semidirect sum Lie algebras. J. Phys. A 21(6), L321 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Schulze-Pillot, R.: Weak Maaßforms and \((\mathfrak {g},K)\)-modules. Ramanujan J. 26(3), 437–445 (2011)Google Scholar
  18. 18.
    Skoruppa, N.-P.: Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms. Invent. Math. 102(3), 501–520 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Skoruppa, N.-P., Zagier, D.B.: Jacobi forms and a certain space of modular forms. Invent. Math. 94(1), 113–146 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vogan Jr., D.A., Zuckerman, G.J.: Unitary representations with nonzero cohomology. Compos. Math. 53(1), 51–90 (1984)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Wallach, N.: Real Reductive Groups. I, vol. 132. Pure and Applied Mathematics. Academic, Boston, MA (1988)Google Scholar
  22. 22.
    Westerholt-Raum, M.: Harmonic weak Siegel Maaß forms I: Preimages of non-holomorphic Saito-Kurokawa lift. Int. Math. Res. Not., rnw288 (2016).  https://doi.org/10.1093/imrn/rnw288
  23. 23.
    Westerholt-Raum, M.: H-harmonic Maaß-Jacobi forms of degree 1. Res. Math. Sci. 2, 30pp. (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Westerholt-Raum, M.: Indefinite theta series on tetrahedral cones (2016). arXiv:1608.08874Google Scholar
  25. 25.
    Ziegler, C.D.: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg 59, 191–224 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zwegers, S.: Mock theta functions. Ph.D. thesis. Universiteit Utrecht (2002)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institutionen för Matematiska vetenskaperChalmers tekniska högskola och Göteborgs UniversitetGöteborgSweden

Personalised recommendations