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Indecomposable Harish-Chandra Modules for Jacobi Groups

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L-Functions and Automorphic Forms

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 10))

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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.

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References

  1. Berndt, R., Schmidt, R.: Elements of the Representation Theory of the Jacobi Group, vol. 163. Progress in Mathematics. Birkhäuser, Basel (1998)

    Book  MATH  Google Scholar 

  2. Bringmann, K., Kudla, S.: A classification of harmonicMaaß forms (2016). arXiv:1609.06999

    Google Scholar 

  3. Bringmann, K., Richter, O.K.: Zagier-type dualities and lifting maps for harmonic Maass-Jacobi forms. Adv. Math. 225(4), 2298–2315 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 45–90 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Eichler, M., Zagier, D.B.: The Theory of Jacobi Forms, vol. 55. Progress in Mathematics. Birkhäuser, Boston, MA (1985)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  11. Goujard, E., Möller, M.: Counting Feynman-like graphs: quasimodularity and Siegel-Veech weight (2016). arXiv:1609.01658

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Pitale, A.: Jacobi Maaß forms. Abh. Math. Semin. Univ. Hambg. 79(1), 87–111 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Quesne, C.: Casimir operators of semidirect sum Lie algebras. J. Phys. A 21(6), L321 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Schulze-Pillot, R.: Weak Maaßforms and \((\mathfrak {g},K)\)-modules. Ramanujan J. 26(3), 437–445 (2011)

    Google Scholar 

  18. Skoruppa, N.-P.: Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms. Invent. Math. 102(3), 501–520 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Skoruppa, N.-P., Zagier, D.B.: Jacobi forms and a certain space of modular forms. Invent. Math. 94(1), 113–146 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Vogan Jr., D.A., Zuckerman, G.J.: Unitary representations with nonzero cohomology. Compos. Math. 53(1), 51–90 (1984)

    MathSciNet  MATH  Google Scholar 

  21. Wallach, N.: Real Reductive Groups. I, vol. 132. Pure and Applied Mathematics. Academic, Boston, MA (1988)

    Google Scholar 

  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. Westerholt-Raum, M.: H-harmonic Maaß-Jacobi forms of degree 1. Res. Math. Sci. 2, 30pp. (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Westerholt-Raum, M.: Indefinite theta series on tetrahedral cones (2016). arXiv:1608.08874

    Google Scholar 

  25. Ziegler, C.D.: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg 59, 191–224 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zwegers, S.: Mock theta functions. Ph.D. thesis. Universiteit Utrecht (2002)

    Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Institutionen för Matematiska vetenskaper, Chalmers tekniska högskola och Göteborgs Universitet, SE-412 96, Göteborg, Sweden

    Martin Raum

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  1. Martin Raum
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Correspondence to Martin Raum .

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Editors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Jan Hendrik Bruinier

  2. Mathematisches Institut, Universität Heidelberg, Heidelberg, Germany

    Winfried Kohnen

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Raum, M. (2017). Indecomposable Harish-Chandra Modules for Jacobi Groups. In: Bruinier, J., Kohnen, W. (eds) L-Functions and Automorphic Forms. Contributions in Mathematical and Computational Sciences, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-69712-3_13

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