Advertisement

R-groups for metaplectic groups

  • Marcela HanzerEmail author
Article
  • 7 Downloads

Abstract

In this short note, we completely describe a parabolically induced representation \(Ind_P^{Sp\widetilde{(2n,F)}}(\sigma)\), in particular, its length and multiplicities. Here, \(Sp\widetilde{(2n,F)}\) is a p-adic metaplectic group, and σ is a discrete series representation of a Levi subgroup of P. A multiplicity one result follows.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Arthur, The Endoscopic Classification of Representations, American Mathematical Society Colloquium Publications, Vol. 61, American Mathematical Society, Providence, RI, 2013.Google Scholar
  2. [2]
    H. Atobe and W. T. Gan, Local theta correspondence of tempered representations and Langlands parameters, Inventiones Mathematicae 210 (2017), 341–415.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    D. Ban and C. Jantzen, The Langlands quotient theorem for finite central extensions of p-adic groups, Glasnik Matematički. Serija III 48(68) (2013), 313–334.Google Scholar
  4. [4]
  5. [5]
    W. Casselman, Introduction to the theory of admissible representations of p-adic reductive groups, https://doi.org/www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf.
  6. [6]
    W. T. Gan and G. Savin, Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence, Compositio Mathematica 148 (2012), 1655–1694.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    W. T. Gan and S. Takeda, A proof of the Howe duality conjecture, Journal of the American Mathematical Society 29 (2016), 473–493.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S. S. Gelbart, Weil’s Representation and the Spectrum of the Metaplectic Group, Lecture Notes in Mathematics, Vol. 530, Springer-Verlag Berlin–New York, 1976.Google Scholar
  9. [9]
    D. Goldberg, Reducibility of induced representations for Sp(2n) and SO(n), American Journal of Mathematics 116 (1994), 1101–1151.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M. Hanzer, R groups for quaternionic Hermitian groups, Glasnik Matematički, Serija III 39(59) (2004), 31–48.Google Scholar
  11. [11]
    M. Hanzer, The generalized injectivity conjecture for classical p-adic groups, International Mathematics Research Notices (2010), 195–237.Google Scholar
  12. [12]
    M. Hanzer and G. Muić, Parabolic induction and Jacquet functors for metaplectic groups, Journal of Algebra 323 (2010), 241–260.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    R. A. Herb, Elliptic representations for Sp(2n) and SO(n), Pacific Journal of Mathematics 161 (1993), 347–358.MathSciNetCrossRefGoogle Scholar
  14. [14]
    R. Howe, θ-series and invariant theory, in Automorphic Forms, Representations and Lfunctions (Proceedings of Symposium in Pure Mathematics, Oregon State University, Corvallis, Ore., 1977), Part 1, Proceedings of Symposia in Pure Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 1979, pp. 275–285.Google Scholar
  15. [15]
    R. Howe, Transcending classical invariant theory, Journal of the American Mathematical Society 2 (1989), 535–552.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    C. D. Keys, L-indistinguishability and R-groups for quasisplit groups: unitary groups in even dimension, Annales Scientifiques de l’École Normale Supérieure 20 (1987), 31–64.MathSciNetGoogle Scholar
  17. [17]
    C. D. Keys, On the decomposition of reducible principal series representations of p-adic Chevalley groups, Pacific Journal of Mathematics 101 (1982), 351–388.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    S. S. Kudla, On the local theta-correspondence, Inventiones Mathematicae 83 (1986), 229–255.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S. S. Kudla, Notes on the local theta correspondence, (1996).Google Scholar
  20. [20]
    S. S. Kudla and S. Rallis, On first occurrence in the local theta correspondence, in Automorphic Representations, L-functions and Applications: Progress and Prospects, Ohio State University Mathematical Research Institute Publications, Vol. 11, de Gruyter, Berlin, 2005, pp. 273–308.Google Scholar
  21. [21]
    C. Moeglin, M.-F. Vignéras and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics, Vol. 1291, Springer-Verlag, Berlin, 1987.Google Scholar
  22. [22]
    G. Muić, A proof of Casselman–Shahidi’s conjecture for quasi-split classical groups, Canadian Mathematical Bulletin 44 (2001), 298–312.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    G. Muić, On the structure of theta lifts of discrete series for dual pairs (Sp(n), O(V )), Israel Journal of Mathematics 164 (2008), 87–124.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    R. Ranga Rao, On some explicit formulas in the theory of Weil representation, Pacific Journal of Mathematics 157 (1993), 335–371.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    B. Sun and C.-B. Zhu, Conservation relations for local theta correspondence, Journal of the American Mathematical Society 28 (2015), 939–983.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    D. Szpruch, The Langlands–Shahidi method quotient for the metaplectic group and applications, PhD Thesis, Tel Aviv University, 2009.Google Scholar
  27. [27]
    M. Tadić, Representations of p-adic symplectic groups, Compositio Mathematica 90 (1994), 123–181.MathSciNetzbMATHGoogle Scholar
  28. [28]
    M. Tadić, Structure arising from induction and Jacquet modules of representations of classical p-adic groups, Journal of Algebra 177 (1995), 1–33.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    J.-L. Waldspurger, Démonstration d’une conjecture de dualité de Howe dans le cas padique, p = 2, in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday, Part I (Ramat Aviv, 1989), Israel Mathematical Conference Proceedings, Vol. 2, Weizmann Press, Jerusalem, 1990, pp. 267–324.Google Scholar
  30. [30]
    N. Winarsky, Reducibility of principal series representations of p-adic Chevalley groups, American Journal of Mathematics 100 (1978), 941–956.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    A. V. Zelevinsky, Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n), Annales Scientifiques de l’École Normale Supérieure 13 (1980), 165–210.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia

Personalised recommendations