Skip to main content
SpringerLink
Log in

Theta distinguished representations, inflation and the symmetric square L-function

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

A theta distinguished representation is a quotient of a tensor of exceptional representations, where “exceptional” is in the sense of Kazhdan and Patterson. We study relations between theta distinguished representations of \(\mathrm {GL}_n\) and \(\mathrm {GSpin}_{2n+1}\). In the case of \(\mathrm {GSpin}_{2n+1}\) (or \(\mathrm {SO}_{2n+1}\)) exceptional (or small) representations were constructed by Bump, Friedberg and Ginzburg. We prove a Rodier-type hereditary property: a tempered representation \(\tau \) is distinguished if and only if the representation \({\mathrm {I}}(\tau )\) induced to \(\mathrm {GSpin}_{2n+1}\) is distinguished, and the multiplicity of both quotients is at most one. If \(\tau \) is supercuspidal and distinguished, then so is the Langlands quotient of \({\mathrm {I}}(\tau )\). As a corollary, we characterize supercuspidal distinguished representations, in terms of the pole of the symmetric square L-function at \(s=0\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In [40, p. 25] it was incorrectly stated they are equal.

References

  1. Asgari, M.: On holomorphy of local Langlands L-functions. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)-Purdue University (2000)

  2. Asgari, M.: Local \(L\)-functions for split spinor groups. Can. J. Math. 54(4), 673–693 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Asgari, M., Shahidi, F.: Generic transfer for general spin groups. Duke Math. J. 132(1), 137–190 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Asgari, M., Shahidi, F.: Image of functoriality for general spin groups. Manuscr. Math. 144(3–4), 609–638 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernstein, J.N.: \(P\)-invariant distributions on \({{\rm GL}}(N)\) and the classification of unitary representations of \({{\rm GL}}(N)\) (non-Archimedean case). In: Lie group representations, II (College Park, Md., 1982/1983), Lecture Notes in Mathematics, vol. 1041, pp. 50–102. Springer, Berlin (1984)

  6. Bump, D., Friedberg, S., Ginzburg, D.: A Rankin–Selberg integral using the automorphic minimal representation of \({{\rm SO}}(7)\). J. Ramanujan Math. Soc. 15(2), 81–124 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Bump, D., Friedberg, S., Ginzburg, D.: Small representations for odd orthogonal groups. Int. Math. Res. Not. 2003(25), 1363–1393 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bump, D., Friedberg, S., Ginzburg, D.: Lifting automorphic representations on the double covers of orthogonal groups. Duke Math. J. 131(2), 363–396 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bump, D., Ginzburg, D.: Symmetric square \(L\)-functions on \({{\rm GL}}(r)\). Ann. Math. (2) 136(1), 137–205 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Banks, W.D., Levy, J., Sepanski, M.R.: Block-compatible metaplectic cocycles. J. Reine Angew. Math. 507, 131–163 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. Bernstein, I.N., Zelevinsky, A.V.: Representations of the group \({GL(n, F)}\) where \({F}\) is a local non-Archimedean field. Russian Math. Surv. 31(3), 1–68 (1976)

    Article  MathSciNet  Google Scholar 

  13. Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \({p}\)-adic groups I. Ann. Scient. Éc. Norm. Sup. 10(4), 441–472 (1977)

    MathSciNet  MATH  Google Scholar 

  14. Carter, R.W.: Finite groups of Lie type. Wiley Classics Library. John Wiley & Sons Ltd., Chichester, 1993. Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience Publication

  15. Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York (1993)

    MATH  Google Scholar 

  16. Casselman, W., Shahidi, F.: On irreducibility of standard modules for generic representations. Ann. Scient. Éc. Norm. Sup. 31(4), 561–589 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Flicker, Y.Z., Kazhdan, D.A.: Metaplectic correspondence. Inst. Hautes Études Sci. Publ. Math. 64(1), 53–110 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ginzburg, D.: \({L}\)-functions for \({SO_n\times GL_k}\). J. Reine Angew. Math. 405, 156–180 (1990)

    MathSciNet  Google Scholar 

  19. Ginzburg, D.: Certain conjectures relating unipotent orbits to automorphic representations. Israel J. Math. 151(1), 323–355 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ginzburg, D., Jiang, D., Soudry, D.: Periods of automorphic forms, poles of \(L\)-functions and functorial lifting. Sci. China Math. 53(9), 2215–2238 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ginzburg, D., Jiang, D., Soudry, D.: On certain automorphic descents to \({{\rm GL}}_2\). Int. Math. Res. Not. 2011(21), 4779–4820 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Ginzburg, D., Rallis, S., Soudry, D.: On the automorphic theta representation for simply laced groups. Israel J. Math. 100(1), 61–116 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ginzburg, D., Rallis, S., Soudry, D.: Self-dual automorphic \({GL_n}\) modules and construction of a backward lifting from \({GL_n}\). Int. Math. Res. Not. 1997(14), 687–701 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ginzburg, D., Rallis, S., Soudry, D.: On a correspondence between cuspidal representations of \({GL_{2n}}\) and \(\widetilde{Sp}_{2n}\). J. Am. Math. Soc. 12(3), 849–907 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ginzburg, D., Rallis, S., Soudry, D.: On explicit lifts of cusp forms from \({{\rm GL}}_m\) to classical groups. Ann. Math. (2) 150(3), 807–866 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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. 2001(14), 729–764 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ginzburg, D., Rallis, S., Soudry, D.: On Fourier coefficients of automorphic forms of symplectic groups. Manuscr. Math. 111(1), 1–16 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ginzburg, D., Rallis, S., Soudry, D.: The Descent Map from Automorphic Representations of GL(n) to Classical Groups. World Scientific Publishing, Singapore (2011)

    Book  MATH  Google Scholar 

  29. Gan, W.T., Savin, G.: On minimal representations definitions and properties. Represent. Theory 9, 46–93 (2005). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ginzburg, D., Rallis, S., Soudry, D.: Endoscopic representations of \({\widetilde{\rm Sp}}_{2n}\). J. Inst. Math. Jussieu 1(1), 77–123 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hundley, J., Sayag, E.: Descent construction for GSpin groups. Mem. Am. Math. Soc. (2012, preprint), (to appear). arXiv:1110.6788v2

  32. Ikeda, T.: On the theory of Jacobi forms and Fourier–Jacobi coefficients of Eisenstein series. J. Math. Kyoto Univ. 34(3), 615–636 (1994)

    MathSciNet  MATH  Google Scholar 

  33. Jacquet, H., Rallis, S.: Symplectic periods. J. Reine Angew. Math. 423, 175–197 (1992)

    MathSciNet  MATH  Google Scholar 

  34. Jacquet, H., Rallis, S.: Uniqueness of linear periods. Compos. Math. 102(1), 65–123 (1996)

    MathSciNet  MATH  Google Scholar 

  35. Jiang, D., Soudry, D.: The local converse theorem for \({SO(2n+1)}\) and applications. Ann. Math. (2) 157(3), 743–806 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Jiang, D., Soudry, D.: Generic representations and local Langlands reciprocity law for \({p}\)-adic \({SO(2n+1)}\). In: Hida, H., Ramakrishnan, D., Shahidi, F. (eds.) Contributions to Automorphic Forms, Geometry, and Number Theory: A Volume in Honor of Joseph Shalika, pp. 457–519. Johns Hopkins University Press, Baltimore (2004)

    Google Scholar 

  37. Kable, A.C.: The tensor product of exceptional representations on the general linear group. Ann. Sci. École Norm. Sup. (4) 34(5), 741–769 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Kable, A.C.: On a conjecture of Savin. Int. Math. Res. Not. 2002(30), 1601–1627 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kaplan, E.: The characterization of theta-distinguished representations of \({GL_n}\). (2015) arXiv:1502.06643. http://arxiv.org/pdf/1502.06643v1

  40. Kaplan, E.: The theta period of a cuspidal automorphic representation of \({GL(n)}\). Int. Math. Res. Not. 2015(8), 2168–2209 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Kaplan, E.: The double cover of odd general spin groups, small representations and applications. J. Inst. Math. Jussieu (2016, to appear). doi:10.1017/S1474748015000250

  42. Kaplan, E.: Representations distinguished by pairs of exceptional representations and a conjecture of Savin. Int. Math. Res. Not. (to appear). doi:10.1093/imrn/rnv150

  43. Klyachko, A.A.: Models for complex representations of groups \({{\rm GL}}(n,\, q)\). Mat. Sb. (N.S.) 120(162)(3), 371–386 (1983)

    MathSciNet  MATH  Google Scholar 

  44. Kazhdan, D.A., Patterson, S.J.: Metaplectic forms. Inst. Hautes Études Sci. Publ. Math. 59(1), 35–142 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kazhdan, D., Savin, G.: The smallest representation of simply laced groups. 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, pp. 209–223. Weizmann, Jerusalem (1990)

  46. Lapid, E., Mao, Z.: Whittaker-Fourier coefficients of cusp forms on \(\widetilde{Sp}_n\): reduction to a local statement (2013). arXiv:1401.0198. http://arxiv.org/pdf/1401.0198v3

  47. Lapid, E., Mao, Z.: On an analogue of the Ichino-Ikeda conjecture for Whittaker coefficients on the metaplectic group (2014). arXiv:1404.2905. http://arxiv.org/pdf/1404.2905v2

  48. Lapid, E., Mao, Z.: A conjecture on Whittaker–Fourier coefficients of cusp forms. J. Number Theory 146, 448–505 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lapid, E., Mao, Z.: Model transition for representations of metaplectic type. Int. Math. Res. Not. 2015(19), 9486–9568 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Loke, H.Y., Savin, G.: Modular forms on non-linear double covers of algebraic groups. Trans. Am. Math. Soc. 362(9), 4901–4920 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. Matsumoto, H.: Sur les sous-groupes arithmétiques des groupes semi-simples déployés. Ann. Sci. École Norm. Sup. (4) 2(1), 1–62 (1969)

    MATH  Google Scholar 

  52. Matić, I.: Levi subgroups of \(p\)-adic \({{\rm Spin}}(2n+1)\). Math. Commun. 14(2), 223–233 (2009)

    MathSciNet  MATH  Google Scholar 

  53. Mezo, P.: Metaplectic tensor products for irreducible representations. Pac. J. Math. 215(1), 85–96 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  54. 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)

  55. Offen, O., Sayag, E.: Uniqueness and disjointness of Klyachko models. J. Funct. Anal. 254(11), 2846–2865 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. Perrin, P.: Représentations de Schrödinger, indice de Maslov et groupe metaplectique. In: Non Commutative Harmonic Analysis and Lie Groups (Proceedings Marseille-Luminy 1980), Lecture Notes in Mathematics, vol. 880, pp. 370–407. Springer, Berlin (1981)

  57. Patterson, S.J., Piatetski-Shapiro, I.I.: The symmetric-square \(L\)-function attached to a cuspidal automorphic representation of \({{\rm GL}}_3\). Math. Ann. 283(4), 551–572 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  58. Prasad, D., Ramakrishnan, D.: Self-dual representations of division algebras and Weil groups: a contrast. Am. J. Math. 134(3), 749–767 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  59. Prasad, D.: Weil representation, Howe duality, and the theta correspondence. In: Theta Functions: From the Classical to the Modern, CRM Proceedings Lecture Notes, vol. 1, pp. 105–127. American Mathematical Society, Providence, RI (1993)

  60. Rao, R.: On some explicit formulas in the theory of Weil representations. Pac. J. Math. 157(2), 335–371 (1993)

    Article  MATH  Google Scholar 

  61. Roskies, J.R.: The Minimal Representation of SO(4,3) Over a P-Adic Field. Stanford University, California (1996)

    Google Scholar 

  62. Sabourin, H.: Une représentation unipotente associée à l’orbite minimale: le cas de \({{\rm SO}}(4,3)\). J. Funct. Anal. 137(2), 394–465 (1996)

    Article  MathSciNet  Google Scholar 

  63. Savin, G.: On the tensor product of theta representations of \({{\rm GL}}_3\). Pac. J. Math. 154(2), 369–380 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  64. Savin, G.: An analogue of the Weil representation for \(G_2\). J. Reine Angew. Math. 434, 115–126 (1993)

    MathSciNet  MATH  Google Scholar 

  65. Shahidi, F.: Twisted endoscopy and reducibility of induced representations for \({p}\)-adic groups. Duke Math. J. 66(1), 1–41 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  66. Soudry, D.: On Langlands functoriality from classical groups to \({GL_{n}}\). In: Tilouine, J., Carayol, H., Harris, M., Vignéras, M.-F. (eds.), Formes Automorphes (I); Actes du Semestre du CEB, Printemps 2000, Astérisque, vol. 298 pp. 335–390 (2005)

  67. Soudry, D.: Rankin-Selberg integrals, the descent method, and Langlands functoriality. In: ICM, pp. 1311–1325, Madrid, (2006). EMS (Zürich, Switzerland)

  68. Sun, H.: Examples of metaplectic tensor products (1997, preprint). http://www.geocities.ws/mathtester/pubs/tensor. Last accessed (December 2013)

  69. Takeda, S.: Metaplectic tensor products for automorphic representations of GL(r) (2013). arXiv:1303.2785. http://arxiv.org/abs/1303.2785

  70. Takeda, S.: The twisted symmetric square L-function of GL(r). Duke Math. J. 163(1), 175–266 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  71. Torasso, P.: Méthode des orbites de Kirillov-Duflo et représentations minimales des groupes simples sur un corps local de caractéristique nulle. Duke Math. J. 90(2), 261–377 (1997)

    Article  MathSciNet  Google Scholar 

  72. van Dijk, G.: Smooth and admissible representations of \(p\)-adic unipotent groups. Compos. Math. 37(1), 77–101 (1978)

    MathSciNet  MATH  Google Scholar 

  73. Vogan, Jr. D.A.: Singular unitary representations. In: Noncommutative harmonic analysis and Lie groups (Marseille, 1980), Lecture Notes in Mathematics, vol. 880, pp. 506–535. Springer, Berlin (1981)

  74. Weil, A.: Sur certains groupes d’opérateurs unitaires. Acta Math. 111(1), 143–211 (1964)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I wish to express my gratitude to Erez Lapid for suggesting this project to me. I would like to thank Eitan Sayag, for explaining to me how to use his results with Offen [55, Proposition 1]. I am grateful to Jim Cogdell for numerous encouraging and useful conversations throughout this work. Lastly, I thank the referee for his/her careful reading of the manuscript and helpful remarks, which helped improve the presentation.

Author information

Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Eyal Kaplan

Authors
  1. Eyal Kaplan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eyal Kaplan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaplan, E. Theta distinguished representations, inflation and the symmetric square L-function. Math. Z. 283, 909–936 (2016). https://doi.org/10.1007/s00209-016-1627-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1627-8

Keywords

Mathematics Subject Classification

Search

Navigation