Manuscripta Mathematica

, Volume 142, Issue 3–4, pp 307–346 | Cite as

Multiplicativity of the gamma factors of Rankin–Selberg integrals for SO 2l × GL n

  • Eyal KaplanEmail author


Let SO 2l be the special even orthogonal group, split or quasi–split, defined over a local non–Archimedian field. The Rankin–Selberg method for a pair of generic representations of SO 2l × GL n constructs a family of integrals, which are used to define γ and L-factors. Here we prove full multiplicative properties for the γ-factor, namely that it is multiplicative in each variable. As a corollary, the γ-factor is identical with Shahidi’s standard γ-factor.

Mathematics Subject Classifications (2010)

Primary: 11S40 Secondary: 11F70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aizenbud A., Gourevitch D., Rallis S., Schiffmann G.: Multiplicity one theorems. Ann. Math. 172(2), 1407–1434 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Banks W.: A corollary to Bernstein’s theorem and Whittaker functionals on the metaplectic group. Math. Res. Lett. 5(6), 781–790 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bernstein I.N., Zelevinsky A.V.: Representations of the group GL(n,F) where F is a local non-Archimedean field. Russ. Math. Surveys 31(3), 1–68 (1976)CrossRefzbMATHGoogle Scholar
  4. 4.
    Casselman W., Shalika J.A.: The unramified principal series of p-adic groups II: the Whittaker function. Compos. Math. 41, 207–231 (1980)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cogdell J.W., Kim H.H., Piatetski-Shapiro I., Shahidi F.: Functoriality for the classical groups. Publ. Math. IHES 99, 163–233 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gan, W.T., Gross, B.H., Prasad, D.: Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Asterisque 346 (2012).
  7. 7.
    Gelbart, S., Piatetski-Shapiro, I., Rallis, S.: L-functions for G × GL(n). Lecture Notes in Mathemetics, vol. 1254, Springer, New York (1987)Google Scholar
  8. 8.
    Ginzburg D.: L-functions for SO n × GL k. J. Reine Angew. Math. 405, 156–180 (1990)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ginzburg, D., Piatetski-Shapiro, I., Rallis, S.: L-functions for the orthogonal group. Mem. Am. Math. Soc. 128(611), (1997)Google Scholar
  10. 10.
    Ginzburg D., Rallis S., Soudry D.: Generic automorphic forms of SO 2n+1: functorial lift to GL 2, endoscopy, and base change. Int. Math. Res. Notices 729(14), 729–764 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jacquet H., Piatetski-Shapiro I., Shalika J.A.: Rankin–Selberg convolutions. Am. J. Math. 105(2), 367–464 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jacquet, H., Shalika, J.A.: Rankin–Selberg convolutions: Archimedean theory, Festschrift in Honor of I. Piatetskiv-Shapiro, Part I, pp. 125–207. Weizmann Science Press, Jerusalem (1990)Google Scholar
  13. 13.
    Jiang D., Soudry D.: The local converse theorem for SO(2n + 1) and applications. Ann. Math. 157(3), 743–806 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kaplan E.: An invariant theory approach for the unramified computation of Rankinv-Selberg integrals for quasi-split SO 2n × GL n. J. Number Theory 130, 1801–1817 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaplan E.: The unramified computation of Rankinv–Selberg integrals for SO 2l × GL n. Israel J. Math. 191(1), 137–184 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moeglin, C., Waldspurger, J.-L.: La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général (2010).
  17. 17.
    Muić G.: A geometric construction of intertwining operators for reductive p-vadic groups. Manuscripta Math. 125, 241–272 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shahidi F.: Functional equation satisfied by certain L-functions. Compos. Math. 37, 171–208 (1978)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Shahidi F.: On certain L-functions. Am. J. Math. 103(2), 297–355 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shahidi F.: A proof of Langlands’ conjecture on Plancherel measures; complementary series of \({\mathfrak{p}}\) -adic groups. Ann. Math. 132(2), 273–330 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Soudry, D.: Rankin–Selberg convolutions for SO 2l+1 × GL n: local theory. Mem. Am. Math. Soc., 105(500) (1993)Google Scholar
  22. 22.
    Soudry D.: On the Archimedean theory of Rankin–Selberg convolutions for SO 2l+1 × GL n. Ann. Sci. Éc. Norm. Sup. 28(2), 161–224 (1995)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Soudry D.: Full multiplicativity of gamma factors for SO 2l+1 × GL n. Israel J. Math. 120(1), 511–561 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Soudry, D.: Rankin–Selberg integrals, the descent method, and Langlands functoriality. In: Proceedings of the International Congress of Mathematicians, pp. 1311–1325. EMS, Madrid (2006)Google Scholar
  25. 25.
    Waldspurger J.-L.: La formule de Plancherel d’après Harish–Chandra. J. Inst. Math. Jussieu 2(2), 235–333 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Sackler Faculty of Exact SciencesTel-Aviv UniversityTel-AvivIsrael

Personalised recommendations