Glasnik Matematicki, Vol. 52, No. 2 (2017), 257-274.

REDUCIBILITY OF SOME GENERALIZED PRINCIPAL SERIES OF THE METAPLECTIC GROUP

Igor Ciganović

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: igor.ciganovic@math.hr

Abstract.   We determine reducibility of the representation of the metaplectic group induced from the tensor product of an essentially square integrable representation attached to the Zelevinsky segment and a genuine cuspidal representation of the metaplectic group.

2010 Mathematics Subject Classification.   22D12, 22E50, 22D30, 11F85.

Key words and phrases.   Metaplectic group, p-adic field, parabolic induction, Jacquet module, generalized principal series.


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DOI: 10.3336/gm.52.2.06


References:

  1. I. N. Bernštein and A. V. Zelevinskii, Representations of the group GL(n,F), where F is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), 5-70.
    MathSciNet    

  2. I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive p-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), 441-472.
    MathSciNet     CrossRef

  3. I. Ciganović and N. Grbac, The Zelevinsky classification of unramified representations of the metaplectic group, J. Algebra 454 (2016), 357-399.
    MathSciNet     CrossRef

  4. W. T. Gan and G. Savin, Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence, Compos. Math. 148 (2012), 1655-1694.
    MathSciNet     CrossRef

  5. W. T. Gan and S. Takeda, A proof of the Howe duality conjecture, J. Amer. Math. Soc. 29 (2016), 473-493.
    MathSciNet     CrossRef

  6. D. Goldberg, Reducibility of induced representation for Sp(2n) and SO(n), Amer. J. Math. 116 (1994), 1101-1151.
    MathSciNet     CrossRef

  7. M. Hanzer and I. Matić, The unitary dual of p-adic , Pacific J. Math. 248 (2010), 107-137.
    MathSciNet     CrossRef

  8. M. Hanzer and G. Muić, Parabolic induction and Jacquet functors for metaplectic groups, J. Algebra 323 (2010), 241-260.
    MathSciNet     CrossRef

  9. M. Hanzer and G. Muić, Rank one reducibility for metaplectic groups via theta correspondence, Canad. J. Math. 63 (2011), 591-615.
    MathSciNet     CrossRef

  10. S. S. Kudla, On the local theta-correspondence, Invent. Math. 83 (1986), 229-255.
    MathSciNet     CrossRef

  11. I. Matić, First occurrence indices of tempered representations of metaplectic groups, Proc. Amer. Math. Soc. 144 (2016), 3157-3172.
    MathSciNet     CrossRef

  12. I. Matić, Strongly positive representations of metaplectic groups, J. Algebra 334 (2011), 255-274.
    MathSciNet     CrossRef

  13. I. Matić, Jacquet modules of strongly positive representations of the metaplectic group , Trans. Amer. Math. Soc. 365 (2013), 2755-2778.
    MathSciNet     CrossRef

  14. I. Matić and M. Tadić, On Jacquet modules of representations of segment type, Manuscripta Math. 147 (2015), 437-476.
    MathSciNet     CrossRef

  15. C. Mœglin, Paquetes stables des séries discrètes accessibles par endoscopie tordue; leur paramètre de Langlands, in Automorphic forms and related geometry: assessing the Legacy of I. I. Piatetski-Shapiro, Contemp. Math. 614, Amer. Math. Soc., Providence, 2014, 295-336.
    MathSciNet     CrossRef

  16. C. Mœglin and M. Tadić, Construction of discrete series for classical p-adic groups, J. Amer. Math. Soc. 15 (2002), 715-786.
    MathSciNet     CrossRef

  17. C. Mœglin, M.-F. Vignéras and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics 1291, Springer-Verlag, Berlin, 1987.
    MathSciNet     CrossRef

  18. R. Ranga Rao, On some explicit formulas in the theory of Weil representation, Pacific J. Math. 157 (1993), 335-371.
    MathSciNet     CrossRef

  19. M. Tadić, On reducibility of parabolic induction, Israel J. Math. 107 (1998), 29-91.
    MathSciNet     CrossRef

  20. M. Tadić, Structure arising from induction and Jacquet modules of representations of classical p-adic groups, J. Algebra 177 (1995), 1-33.
    MathSciNet     CrossRef

  21. A. V. Zelevinsky, Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n), Ann. Sci. École Norm. Sup. (4) 13 (1980), 165-210.
    MathSciNet     CrossRef

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