The Conjectural Relation Between Generalized Shalika Models on \(\mathop{\mathrm{SO}}\nolimits _{4n}(F)\) and Symplectic Linear Models on \(\mathop{\mathrm{Sp}}\nolimits _{4n}(F)\): A Toy Example

Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 2)

Abstract

We show that if an irreducible admissible representation of \(\mathop{\mathrm{SO}}\nolimits _{4}(F)\) has a generalized Shalika model, its theta lift to \(\mathop{\mathrm{Sp}}\nolimits _{4}(F)\) is non-zero and has a symplectic linear model.

References

  1. Arthur, J.: The Endoscopic Classification of Representations. American Mathematical Society Colloquium Publications, vol. 61. American Mathematical Society, Providence (2013). Orthogonal and symplectic groupsGoogle Scholar
  2. Ban, D.: Parabolic induction and Jacquet modules of representations of O(2n, F). Glas. Mat. Ser. III 34(54), 147–185 (1999)MathSciNetMATHGoogle Scholar
  3. Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive p-adic groups I. Ann. Sci. École Norm. Sup. 10(4), 441–472 (1977)MathSciNetMATHGoogle Scholar
  4. J. Bernstein, Draft of: Representations of p-adic groups, preprint, 1992, available at http://www.math.tau.ac.il/~bernstei Google Scholar
  5. Cogdell, J.W., Piatetski-Shapiro, I.I., Shahidi, F.: Functoriality for the quasisplit classical groups. In: On Certain L-Functions. Clay Mathematics Institute, vol. 13, pp. 117–140. American Mathematical Society, Providence (2011)Google Scholar
  6. Gan, W.T., Ichino, A.: Formal degrees and local theta correspondence. Invent. Math. 195, 509–672 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. Gan, W.T., Savin, G.: Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence. Compos. Math. 148, 1655–1694 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. Gan, W.T., Takeda, S.: The local Langlands conjecture for Sp(4). Int. Math. Res. Not. IMRN 15, 2987–3038 (2010a)MathSciNetGoogle Scholar
  9. Gan, W.T., Takeda, S.: On Shalika periods and a theorem of Jacquet-Martin. Am. J. Math. 132, 475–528 (2010b)MathSciNetCrossRefMATHGoogle Scholar
  10. Gan, W.T., Takeda, S.: The local Langlands conjecture for GSp(4). Ann. Math. 173(2), 1841–1882 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. Ginzburg, D., Rallis, S., Soudry, D.: On a correspondence between cuspidal representations of \(\mathrm{GL}_{2n}\) and \(\widetilde{\mathrm{Sp}}_{2n}\). J. Am. Math. Soc. 12, 849–907 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. Ginzburg, D., Rallis, S., Soudry, D.: Generic automorphic forms on SO(2n + 1): functorial lift to GL(2n), endoscopy, and base change. Int. Math. Res. Not. 14, 729–764 (2001)MathSciNetCrossRefGoogle Scholar
  13. Jacquet, H., Rallis, S.: Uniqueness of linear periods. Compos. Math. 102, 65–123 (1996)MathSciNetMATHGoogle Scholar
  14. Jiang, D., Qin, Y.: Residues of Eisenstein series and generalized Shalika models for \(\mathrm{SO}_{4n}\). J. Ramanujan Math. Soc. 22, 101–133 (2007)MathSciNetMATHGoogle Scholar
  15. Jiang, D., Soudry, D.: Appendix: On the local descent from GL(n) to classical groups [appendix to mr2931222]. Am. J. Math. 134, 767–772 (2012)MathSciNetCrossRefGoogle Scholar
  16. Jiang, D., Nien, C., Qin, Y.: Symplectic supercuspidal representations and related problems. Sci. China Math. 53, 533–546 (2010a)MathSciNetCrossRefMATHGoogle Scholar
  17. Jiang, D., Nien, C., Qin, Y.: Symplectic supercuspidal representations of GL(2n) over p-adic fields. Pac. J. Math. 245, 273–313 (2010b)MathSciNetCrossRefMATHGoogle Scholar
  18. Jiang, D., Nien, C., Qin, Y.: Generalized Shalika models of p-adic SO4n and functoriality. Israel J. Math. 195, 135–169 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. Kudla, S.S.: Notes on the local theta correspondence (lectures at the European School in Group Theory), preprint, 1996, available at http://www.math.utoronto.ca/~skudla/castle.pdf Google Scholar
  20. Kudla, S.: Notes on the local theta correspondence (lectures at the European School in Group Theory) (1996)Google Scholar
  21. Labesse, J.-P., Langlands, R.P.: L-indistinguishability for SL(2). Can. J. Math. 31, 726–785 (1979)MathSciNetCrossRefMATHGoogle Scholar
  22. Mœglin, C., Vignéras, M.-F., Waldspurger, J.-L.: Correspondances de Howe sur un corps p-adique. Lecture Notes in Mathematics, vol. 1291. Springer, Berlin (1987)Google Scholar
  23. Sally, Jr., P.J., Tadić, M.: Induced representations and classifications for GSp(2, F) and Sp(2, F). Mém. Soc. Math. France (N.S.) 52, 75–133 (1993)Google Scholar
  24. Sun, B., Zhu, C.-B.: Conservation relations for local theta correspondence. http://arxiv.org/pdf/1204.2969v2.pdf (2012)
  25. Tadić, M.: On reducibility of parabolic induction. Israel J. Math. 107, 29–91 (1998)MathSciNetCrossRefMATHGoogle Scholar
  26. Zassenhaus, H.: On the spinor norm. Arch. Math. 13, 434–451 (1962)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Mathematical Research UnitUniversity of LuxembourgLuxembourgLuxembourg
  2. 2.Department of MathematicsUniversity of ZagrebZagrebCroatia
  3. 3.Mathematical InstituteUniversity of BonnBonnGermany

Personalised recommendations