Inventiones mathematicae

, Volume 217, Issue 3, pp 985–1068 | Cite as

Doubling constructions and tensor product L-functions: the linear case

  • Yuanqing Cai
  • Solomon Friedberg
  • David Ginzburg
  • Eyal KaplanEmail author


We present an integral representation for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. Our construction is uniform over all classical groups, and is applicable to all cuspidal representations; it does not require genericity. The main new ideas of the construction are the use of generalized Speh representations as inducing data for the Eisenstein series and the introduction of a new (global and local) model, which generalizes the Whittaker model. Here we consider linear groups, but our construction also extends to arbitrary degree metaplectic coverings; this will be the topic of an upcoming work.

Mathematics Subject Classification

Primary 11F70 Secondary 11F55 11F66 22E50 22E55 



Part of this work was done while the fourth named author was a Zassenhaus Assistant Professor at The Ohio State University, under the supervision of Jim Cogdell. Eyal wishes to express his gratitude to Jim for his kind encouragement and support. We thank the referee for a very careful reading of the manuscript and for many helpful suggestions. Eyal dedicates his part of the work to his beloved Sophie Kaplan who passed away unexpectedly a few weeks before the submission of the first version of this work.


  1. 1.
    Aizenbud, A., Gourevitch, D., Sahi, S.: Derivatives for smooth representations of \(GL(n,\mathbb{R})\) and \(GL(n,\mathbb{C})\). Isr. J. Math. 206(1), 1–38 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aizenbud, A., Gourevitch, D., Sahi, S.: Twisted homology for the mirabolic nilradical. Isr. J. Math. 206(1), 39–88 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Banks, W.D.: A corollary to Bernstein’s theorem and Whittaker functionals on the metaplectic group. Math. Res. Lett. 5(6), 781–790 (1998)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beineke, J., Bump, D.: A summation formula for divisor functions associated to lattices. Forum Math. 18(6), 869–906 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bernstein, I.N., Zelevinsky, A.V.: Representations of the group \({GL(n, F)}\) where \({F}\) is a local non-Archimedean field. Russ. Math. Surv. 31(3), 1–68 (1976)MathSciNetGoogle Scholar
  6. 6.
    Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \({p}\)-adic groups I. Ann. Sci. Éc. Norm. Sup. 10(4), 441–472 (1977)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Böcherer, S., Schmidt, C.-G.: \(p\)-adic measures attached to Siegel modular forms. Ann. Inst. Fourier (Grenoble) 50(5), 1375–1443 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Borel, A.: Automorphic \(L\)-functions. In: Automorphic Forms, Representations, and \(L\)-functions, vol. 33 Part II, pp. 27–61 (1979)Google Scholar
  9. 9.
    Bump, D.: Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  10. 10.
    Bump, D., Friedberg, S.: The exterior square automorphic \(L\)-functions on \({\rm GL}(n)\). In: Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989), vol. 3 of Israel Mathematical Conference Proceedings, pp. 47–65. Weizmann, Jerusalem (1990)Google Scholar
  11. 11.
    Bump, D., Ginzburg, D.: Symmetric square \(L\)-functions on \({\rm GL}(r)\). Ann. Math. (2) 136(1), 137–205 (1992)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cai, Y.: Fourier coefficients for degenerate Eisenstein series and the descending decomposition. Manuscripta Math. 156(3–4), 469–501 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cai, Y., Friedberg, S., Kaplan, E.: Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representations. Preprint 2018. arxiv:1802.02637
  14. 14.
    Carter, R.W.: Finite Groups of Lie Type. Wiley Classics Library. Wiley, Chichester (1993). Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience PublicationGoogle Scholar
  15. 15.
    Casselman, W.: The unramified principal series of \({p}\)-adic groups I: the spherical function. Compositio Math. 40(3), 387–406 (1980)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Casselman, W., Shalika, J.A.: The unramified principal series of \({p}\)-adic groups II: the Whittaker function. Compositio Math. 41(2), 207–231 (1980)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Cogdell, J.W., Kim, H.H., Piatetski-Shapiro, I.I., Shahidi, F.: Functoriality for the classical groups. Publ. Math. Inst. Hautes Études Sci. 99(1), 163–233 (2004)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Cogdell, J.W., Piatetski-Shapiro, I.I.: Converse theorems for \({\rm GL}_n\). Publ. Math. Inst. Hautes Études Sci. 79(1), 157–214 (1994)zbMATHGoogle Scholar
  19. 19.
    Cogdell, J.W., Piatetski-Shapiro, I.I.: Converse theorems for \({\rm GL}_n\). II. J. Reine Angew. Math. 1999(507), 165–188 (1999)zbMATHGoogle Scholar
  20. 20.
    Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York (1993)Google Scholar
  21. 21.
    Dixmier, J., Malliavin, P.: Factorisations de fonctions et de vecteurs indéfiniment différentiables. Bull. Sci. Math. (2) 102(4), 307–330 (1978)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Eischen, E., Harris, M., Li, J.-S., Skinner, C.M.: \(p\)-adic \(L\)-functions for unitary groups. Preprint 2016. arxiv:1602.01776
  23. 23.
    Frahm, J., Kaplan, E.: A Godement–Jacquet type integral and the metaplectic Shalika model. Am. J. Math. 141(1), 219–282 (2019)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Friedberg, S., Ginzburg, D.: Criteria for the existence of cuspidal theta representations. Res. Number Theory 2 2, 16 (2016)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Gan, W.T.: Doubling zeta integrals and local factors for metaplectic groups. Nagoya Math. J. 208, 67–95 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Gan, W.T., Savin, G.: Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence. Compositio Math. 148, 1655–1694 (2012)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Gao, F.: The Langlands–Shahidi \(L\)-functions for Brylinski–Deligne extensions. Am. J. Math. 140(1), 83–137 (2018)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Garrett, P.B.: Pullbacks of Eisenstein series; applications. In: Automorphic Forms of Several Variables (Katata, 1983), vol. 46 of Progress in Mathematics, pp. 114–137. Birkhäuser Boston, Boston (1984)Google Scholar
  29. 29.
    Gelbart, S., Piatetski-Shapiro, I., Rallis, S.: \({L}\)-Functions for \({G\times GL(n)}\), Lecture Notes in Math, vol. 1254. Springer, New York (1987)Google Scholar
  30. 30.
    Ginzburg, D.: \({L}\)-functions for \({{\rm SO}_{n}\times {\rm GL}_{k}}\). J. Reine Angew. Math. 1990(405), 156–180 (1990)Google Scholar
  31. 31.
    Ginzburg, D.: Certain conjectures relating unipotent orbits to automorphic representations. Isr. J. Math. 151(1), 323–355 (2006)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Ginzburg, D., Jiang, D., Rallis, S., Soudry, D.: \(L\)-functions for symplectic groups using Fourier-Jacobi models. In: Arithmetic Geometry and Automorphic Forms, vol. 19 of Advanced Lectures in Mathematics (ALM), pp. 183–207. International Press, Somerville (2011)Google Scholar
  33. 33.
    Ginzburg, D., Piatetski-Shapiro, I., Rallis, S.: \(L\) functions for the orthogonal group. Mem. Am. Math. Soc. 128(611), viii+218 (1997)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Ginzburg, D., Rallis, S., Soudry, D.: \({L}\)-functions for symplectic groups. Bull. Soc. Math. Fr. 126, 181–244 (1998)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Ginzburg, D., Rallis, S., Soudry, D.: On a correspondence between cuspidal representations of \({\rm GL}_{2n}\) and \(\widetilde{\rm Sp}_{2n}\). J. Am. Math. Soc. 12(3), 849–907 (1999)zbMATHGoogle Scholar
  36. 36.
    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)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Ginzburg, D., Rallis, S., Soudry, D.: The Descent Map from Automorphic Representations of \({\rm GL}(n)\) to Classical Groups. World Scientific Publishing, Singapore (2011)zbMATHGoogle Scholar
  38. 38.
    Godement, R., Jacquet, H.: Zeta Functions of Simple Algebras, Lecture Notes in Math, vol. 260. Springer, Berlin (1972)zbMATHGoogle Scholar
  39. 39.
    Gomez, R., Gourevitch, D., Sahi, S.: Whittaker supports for representations of reductive groups. Preprint 2016. arxiv:1610.00284
  40. 40.
    Gomez, R., Gourevitch, D., Sahi, S.: Generalized and degenerate Whittaker models. Compos. Math. 153(2), 223–256 (2017)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Harris, M., Kudla, S.S., Sweet, W.J.: Theta dichotomy for unitary groups. J. Am. Math. Soc. 9(4), 941–1004 (1996)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Harris, M., Li, J.-S., Skinner, C.M.: The Rallis inner product formula and \(p\)-adic \(L\)-functions. In: Automorphic representations, \(L\)-functions and applications: progress and prospects, vol. 11 of Ohio State University Mathematical Research Institute Publications, pp. 225–255. de Gruyter, Berlin (2005)Google Scholar
  43. 43.
    Harris, M., Li, J.-S., Skinner, C.M.: \(p\)-adic \(L\)-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure. Doc. Math. Extra Volume: John H. Coates’ Sixtieth Birthday 393–464 (2006)Google Scholar
  44. 44.
    Ikeda, T.: On the location of poles of the triple L-functions. Compositio Math. 83(2), 187–237 (1992)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Ikeda, T.: On the gamma factor of the triple \({L}\)-function. I. Duke Math. J. 97(2), 301–318 (1999)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Jacquet, H.: On the residual spectrum of \({\rm GL}(n)\). In: Lie Group Representations, II (College Park, Md., 1982/1983), vol. 1041 of Lecture Notes in Mathematics, pp. 185–208. Springer, Berlin (1984)Google Scholar
  47. 47.
    Jacquet, H., Langlands, R.: Automorphic Forms on \({\rm GL}(2)\), Lecture Notes in Mathematics, vol. 114. Springer, Berlin (1970)zbMATHGoogle Scholar
  48. 48.
    Jacquet, H., Piatetski-Shapiro, I.I., Shalika, J.A.: Rankin–Selberg convolutions. Am. J. Math. 105(2), 367–464 (1983)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Jacquet, H., Rallis, S.: Symplectic periods. J. Reine Angew. Math. 1992(423), 175–197 (1992)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Jacquet, H., Shalika, J.: Exterior square \(L\)-functions. In: Automorphic forms, Shimura varieties, and \(L\)-functions, vol. II (Ann Arbor, MI, 1988), volume 11 of Perspect. Math., pp. 143–226. Academic Press, Boston (1990)Google Scholar
  51. 51.
    Jacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic representations I. Am. J. Math. 103(3), 499–558 (1981)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Jiang, D., Liu, B.: On Fourier coefficients of automorphic forms of \({\rm GL}(n)\). Int. Math. Res. Not. IMRN 2013(17), 4029–4071 (2013)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Jiang, D., Zhang, L.: A product of tensor product \(L\)-functions of quasi-split classical groups of Hermitian type. Geom. Funct. Anal. 24(2), 552–609 (2014)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Kaplan, E.: Doubling constructions and tensor product \({L}\)-functions: coverings of the symplectic group. Preprint 2019. arxiv:1902.00880
  55. 55.
    Kaplan, E.: Multiplicativity of the gamma factors of Rankin–Selberg integrals for \(SO_{2l}\times GL_n\). Manuscripta Math. 142(3–4), 307–346 (2013)MathSciNetGoogle Scholar
  56. 56.
    Kaplan, E.: On the gcd of local Rankin–Selberg integrals for even orthogonal groups. Compositio Math. 149, 587–636 (2013)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Kaplan, E.: On the local theory of Rankin–Selberg convolutions for \({{\rm SO}_{2l}\times {\rm GL}_{n}}\). Thesis, Tel Aviv University, Israel (2013)Google Scholar
  58. 58.
    Kim, J.: Gamma factors of certain supercuspidal representations. Math. Ann. 317(4), 751–781 (2000)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Kudla, S.S., Rallis, S.: A regularized Siegel–Weil formula: the first term identity. Ann. Math. (2) 140(1), 1–80 (1994)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Langlands, R.P.: Euler Products. Yale University Press, New Haven (1967)zbMATHGoogle Scholar
  61. 61.
    Langlands, R.P.: On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, vol. 544. Springer, Berlin (1976)zbMATHGoogle Scholar
  62. 62.
    Lapid, E.M., Rallis, S.: On the local factors of representations of classical groups. In: Cogdell, J.W., Jiang, D., Kudla, S.S., Soudry, D., Stanton, R. (eds.) Automorphic Representations, \({L}\)-Functions and Applications: Progress and Prospects, pp. 309–359. Ohio State Univ. Math. Res. Inst. Publ., 11, de Gruyter, Berlin (2005)Google Scholar
  63. 63.
    Mœglin, C., Waldspurger, J.-L.: Modèles de Whittaker dégénérés pour des groupes \(p\)-adiques. Math. Z. 196(3), 427–452 (1987)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Mœglin, C., Waldspurger, J.-L.: Le spectre résiduel de \({\rm GL}(n)\). Ann. Sci. École Norm. Sup. (4) 22(4), 605–674 (1989)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Mœglin, C., Waldspurger, J.-L.: Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics, vol. 113. Cambridge University Press, Cambridge (1995). Une paraphrase de l’Écriture [A paraphrase of Scripture]zbMATHGoogle Scholar
  66. 66.
    Piatetski-Shapiro, I., Rallis, S.: \(\epsilon \) factor of representations of classical groups. Proc. Natl. Acad. Sci. USA 83(13), 4589–4593 (1986)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Piatetski-Shapiro, I., Rallis, S.: \({L}\)-Functions for the Classical Groups, Lecture Notes in Math, vol. 1254. Springer, New York (1987)zbMATHGoogle Scholar
  68. 68.
    Piatetski-Shapiro, I., Rallis, S.: Rankin triple \(L\) functions. Compositio Math. 64(1), 31–115 (1987)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Piatetski-Shapiro, I.I.: Euler subgroups. In: Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), pp. 597–620. Halsted, New York (1975)Google Scholar
  70. 70.
    Rallis, S., Soudry, D.: Stability of the local gamma factor arising from the doubling method. Math. Ann. 333(2), 291–313 (2005)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Satake, I.: Theory of spherical functions on reductive algebraic groups over \({p}\)-adic fields. Publ. Math. Inst. Hautes Études Sci. 18(1), 5–69 (1963)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Shahidi, F.: On certain \({L}\)-functions. Am. J. Math. 103(2), 297–355 (1981)zbMATHGoogle Scholar
  73. 73.
    Shahidi, F.: A proof of Langlands’ conjecture on Plancherel measures; complementary series for \(p\)-adic groups. Ann. Math. (2) 132(2), 273–330 (1990)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Shahidi, F.: Twisted endoscopy and reducibility of induced representations for \({p}\)-adic groups. Duke Math. J. 66(1), 1–41 (1992)MathSciNetzbMATHGoogle Scholar
  75. 75.
    Shalika, J.A.: The multiplicity one theorem for \({GL_n}\). Ann. Math. 100, 171–193 (1974)MathSciNetzbMATHGoogle Scholar
  76. 76.
    Silberger, A.J.: Introduction to Harmonic Analysis on Reductive p-adic Groups. Princeton University Press and University of Tokyo Press, Princeton, New Jersey (1979)zbMATHGoogle Scholar
  77. 77.
    Soudry, D.: Rankin–Selberg convolutions for \({\rm SO}_{2l+1}\times {\rm GL}_n\): local theory. Mem. Am. Math. Soc. 105(500), vi+100 (1993)Google Scholar
  78. 78.
    Soudry, D.: On the Archimedean theory of Rankin–Selberg convolutions for \({\rm SO}_{2l+1}\times {\rm GL}_n\). Ann. Sci. École Norm. Sup. (4) 28(2), 161–224 (1995)MathSciNetGoogle Scholar
  79. 79.
    Soudry, D.: Full multiplicativity of gamma factors for \({\rm SO}_{2l+1}\times {\rm GL}_n\). Isr. J. Math. 120(1), 511–561 (2000)MathSciNetzbMATHGoogle Scholar
  80. 80.
    Soudry, D.: On Langlands functoriality from classical groups to \({\rm GL}_n\). Astérisque 298, 335–390 (2005)zbMATHGoogle Scholar
  81. 81.
    Soudry, D.: The unramified computation of Rankin–Selberg integrals expressed in terms of Bessel models for split orthogonal groups: part I. Isr. J. Math. 222(2), 711–786 (2017)MathSciNetzbMATHGoogle Scholar
  82. 82.
    Soudry, D.: The unramified computation of Rankin–Selberg integrals expressed in terms of Bessel models for split orthogonal groups: part II. J. Number Theory 186, 62–102 (2018)MathSciNetzbMATHGoogle Scholar
  83. 83.
    Spaltenstein, N.: Classes Unipotentes et sous-Groupes de Borel, Lecture Notes in Mathematics, vol. 946. Springer, Berlin (1982)zbMATHGoogle Scholar
  84. 84.
    Takano, K.: On standard \(L\)-functions for unitary groups. Proc. Jpn. Acad. Ser. A Math. Sci. 73(1), 5–9 (1997)MathSciNetzbMATHGoogle Scholar
  85. 85.
    Takeda, S.: The twisted symmetric square L-function of GL(r). Duke Math. J. 163(1), 175–266 (2014)MathSciNetzbMATHGoogle Scholar
  86. 86.
    Waldspurger, J.-L.: La formule de Plancherel pour les groupes p-adiques, d’après Harish-Chandra. J. Inst. Math. Jussieu 2(2), 235–333 (2003)MathSciNetzbMATHGoogle Scholar
  87. 87.
    Yamana, S.: L-functions and theta correspondence for classical groups. Invent. Math. 196(3), 651–732 (2014)MathSciNetzbMATHGoogle Scholar
  88. 88.
    Zelevinsky, A.V.: Induced representations of reductive \({p}\)-adic groups. II. On irreducible representations of \({\rm GL}(n)\). Ann. Sci. École Norm. Sup. (4) 13(2), 165–210 (1980)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Yuanqing Cai
    • 1
    • 2
  • Solomon Friedberg
    • 1
  • David Ginzburg
    • 3
  • Eyal Kaplan
    • 4
    Email author
  1. 1.Department of MathematicsBoston CollegeChestnut HillUSA
  2. 2.Department of MathematicsWeizmann Institute of ScienceRehovotIsrael
  3. 3.School of Mathematical SciencesTel Aviv UniversityRamat Aviv, Tel AvivIsrael
  4. 4.Department of MathematicsBar Ilan UniversityRamat GanIsrael

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