manuscripta mathematica

, Volume 152, Issue 1–2, pp 223–240 | Cite as

On the local Bump–Friedberg L-function II

  • Nadir MatringeEmail author


Let F be a p-adic field with residue field of cardinality q. To each irreducible representation of GL(nF), we attach a local Euler factor \(L^{BF}(q^{-s},q^{-t},\pi )\) via the Rankin–Selberg method, and show that it is equal to the expected factor \(L(s+t+1/2,\phi _\pi )L(2s,\Lambda ^2\circ \phi _\pi )\) of the Langlands’ parameter \(\phi _\pi \) of \(\pi \). The corresponding local integrals were introduced in Bump and Friedberg (The exterior square automorphic L-functions on \(\mathrm{GL}(n)\) 47–65, 1990), and studied in Matringe (J Reine Angew Math doi: 10.1515/crelle-2013-0083). This work is in fact the continuation of Matringe (J Reine Angew Math doi: 10.1515/crelle-2013-0083). The result is a consequence of the fact that if \(\delta \) is a discrete series representation of GL(2mF), and \(\chi \) is a character of Levi subgroup \(L=GL(m,F)\times GL(m,F)\) which is trivial on GL(mF) embedded diagonally, then \(\delta \) is \((L,\chi )\)-distinguished if an only if it admits a Shalika model. This result was only established for \(\chi =\mathbf {1}\) before.

Mathematics Subject Classification

11F66 11F70 22E50 


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I thank the referee for his careful reading and comments. This work was supported by the research project ANR-13-BS01 -0012 FERPLAY.


  1. 1.
    Bernstein, J.N., Zelevinsky, A.V.: Induced representations of reductive p-adic groups. Ann. Sc. E.N.S 10(4), 441–472 (1977)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bump, D., Friedberg, S.: The exterior square automorphic \(L\)-functions on \({\rm GL}(n)\). Israel Math. Conference Proceedings, 3 Weizmann Jerusalem 47–65 (1990)Google Scholar
  3. 3.
    Bushnell, C.J., Henniart, G.: The local Langlands conjecture for \({{\rm GL}}(2)\) (Grundlehren der Mathematischen Wissenschaften, 335). Springer-Verlag, Berlin (2006)Google Scholar
  4. 4.
    Delorme, P.: Constant term of smooth \(H_\psi \)-spherical functions on a reductive \(p\)-adic group. Trans. Amer. Math. Soc. 362(2), 933–955 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Friedberg, S., Jacquet, H.: Linear periods. J. Reine Angew. Math. 443, 91–139 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Godement, R., Jacquet, H.: Zeta functions of simple algebras (Lecture Notes in Mathematics, vol. 260). Springer-Verlag, Berlin-New York (1972)Google Scholar
  7. 7.
    Harris, M., Taylor, R.: with an appendix by V.G. Berkovich, The geometry and cohomology of some simple Shimura varieties (Annals of Mathematics Studies 151). Princeton University Press, Princeton (2001)Google Scholar
  8. 8.
    Henniart, G.: Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math. 139(2), 439–455 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jacquet, H.: Principal \(L\)-functions of the linear group (Automorphic forms, representations and \(L\)-functions, Part 2, pp. 63-86, Proceedings Symposium Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I) (1979)Google Scholar
  10. 10.
    Jacquet, H., Rallis, S.: Uniqueness of linear periods. Compos. Math. 102(1), 65–123 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Jacquet, H., Piatetskii-Shapiro, I.I., Shalika, J.A.: Rankin–Selberg convolutions. Amer. J. Math. 105, 367–464 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jacquet, H., Shalika, J.A.: The Whittaker models of induced representations. Pacific J. Math. 109(1), 107–120 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kato, S,I., Takano, K.: Subrepresentation theorem for \(p\)-adic symmetric spaces. Int. Math. Res. Not. 11, rnn028 (2008)Google Scholar
  14. 14.
    Matringe, N.: Derivatives and asymptotics of Whittaker functions. Represent. Theory 15, 646–669 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matringe, N.: Linear and Shalika local periods for the mirabolic group, and some consequences. J. Number Theory 138, 1–19 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Matringe N.: On the local Bump–Friedberg \(L\)-function. J. Reine Angew. Math. doi: 10.1515/crelle-2013-0083
  17. 17.
    Rodier F.: Whittaker models for admissible representations of reductive p-adic split groups. (Harmonic analysis on homogeneous spaces, 425–430. Amer. Math. Soc.) Providence, R.I (1973)Google Scholar
  18. 18.
    Silberger, A.J.: The Langlands quotient theorem for p-adic groups. Math. Ann. 236(2), 95–104 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zelevinsky, A.V.: Induced representations of reductive p-adic groups II. Ann. Sc. E.N.S. 13(2), 165–210 (1977)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et ApplicationsUniversité de PoitiersFuturoscope Chasseneuil CedexFrance

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