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manuscripta mathematica

, Volume 155, Issue 1–2, pp 229–302 | Cite as

Non-Siegel Eisenstein series for symplectic groups

  • Marcela HanzerEmail author
Article

Abstract

We explicitly (in terms of Langlands parameters) describe the image of the degenerate Eisenstein series in the case of a symplectic group. We study these series for any maximal parabolic subgroup and any Grossencharacter. We do that by an explicit analysis of the constant term of the Eisenstein series. Using already available information on the constituents of the local degenerate principal series, we further analyze the action of the local intertwining operators and determine their images. The local components of the images are given explicitly, in terms of their Langlands parameters. In this way, we have a complete description of the images of Eisenstein series in the right half-plane, for any Grossencharacter of the adeles over \(\mathbb {Q}.\) Thus, we complete previously known results obtained by different techniques for the trivial character and extend them to the case of the non-trivial characters.

Mathematics Subject Classification

11F70 22E55 22E50 

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References

  1. 1.
    Arthur, J.: The endoscopic classification of representations: orthogonal and symplectic groups. http://www.claymath.org/cw/arthur/pdf/Book.pdf
  2. 2.
    Arthur, J.: Intertwining operators and residues. I. Weighted characters. J. Funct. Anal. 84, 19–84 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arthur, J.: An introduction to the trace formula, in Harmonic analysis, the trace formula, and Shimura varieties, vol. 4 of Clay Math. Proc., Amer. Math. Soc., Providence, pp. 1–263 (2005)Google Scholar
  4. 4.
    Aubert, A.-M.: Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif \(p\)-adique. Trans. Am. Math. Soc. 347, 2179–2189 (1995)zbMATHGoogle Scholar
  5. 5.
    Ban, D.: The Aubert involution and R-groups. Ann. Sci. École Norm. Sup. 4(35), 673–693 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borel, A.: Introduction to automorphic forms. In: Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), pp. 199–210. Amer. Math. Soc., Providence (1966)Google Scholar
  7. 7.
    Casselman, W.: Introduction to the theory of admissible representations of p-adic reductive groups, preprintGoogle Scholar
  8. 8.
    Gelbart, S., Piatetski-Shapiro, I., Rallis, S.: Explicit constructions of automorphic \(L\)-functions. Lecture Notes in Mathematics, vol. 1254. Springer, Berlin (1987)Google Scholar
  9. 9.
    Gindikin, S., Karpelevich, F.: On an integral connected with symmetric Riemann spaces of nonpositive curvature. Transl. Ser. 2. Am. Math. Soc 85, 249–258 (1969)zbMATHGoogle Scholar
  10. 10.
    Ginzburg, D., Rallis, S., Soudry, D.: The Descent Map from Automorphic Representations of \({\rm GL}(n)\) to Classical Groups. World Scientific Publishing Co Pte. Ltd., Hackensack (2011)CrossRefzbMATHGoogle Scholar
  11. 11.
    Goldberg, D.: Reducibility of induced representations for \({\rm Sp}(2n)\) and \({\rm SO}(n)\). Am. J. Math. 116, 1101–1151 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hanzer, M.: The generalized injectivity conjecture for classical \(p\)-adic groups. Int. Math. Res. Not. IMRN 195–237 (2010)Google Scholar
  13. 13.
    Hanzer, M.: Degenerate eisenstein series for symplectic groups. Glas. Mat. Ser. II I(50), 289–332 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hanzer, M., Muić, G.: On the images and poles of degenerate Eisenstein series for \({ GL}(n, { \mathbb{A}}) \) and \({ GL}(n, { \mathbb{R}})\). Am. J. Math 137, 907–951 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Howe, R., Lee, S.T.: Degenerate principal series representations of \( \text{ GL }_n({ C})\) and \( \text{ GL }_n({ R})\). J. Funct. Anal. 166, 244–309 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jantzen, C., Kim, H.H.: Parametrization of the image of normalized intertwining operators. Pac. J. Math. 199, 367–415 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jiang, D.: The first term identities for Eisenstein series. J. Number Theory 70, 67–98 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kim, H.H.: The residual spectrum of \( \text{ Sp }_4\). Compos. Math. 99, 129–151 (1995)MathSciNetGoogle Scholar
  19. 19.
    Kim, H.H.: Residual spectrum of split classical groups; contribution from Borel subgroups. Pac. J. Math. 199, 417–445 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kim, H.H., Shahidi, F.: Quadratic unipotent Arthur parameters and residual spectrum of symplectic groups. Am. J. Math. 118, 401–425 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kudla, S.S., Rallis, S.: On the Weil–Siegel formula. J. Reine Angew. Math. 387, 1–68 (1988)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kudla, S.S., Rallis, S.: Ramified degenerate principal series representations for \({\rm Sp}(n)\). Israel J. Math. 78, 209–256 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kudla, S.S., Rallis, S.: A regularized Siegel–Weil formula: the first term identity. Ann. Math. 2(140), 1–80 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Langlands, R.P.: Euler Products. Yale University Press, New Haven (1971). A James, K. Whittemore Lecture in Mathematics given at Yale University, 1967, Yale Mathematical Monographs, 1Google Scholar
  25. 25.
    Langlands, R.P.: On the functional equations satisfied by Eisenstein series. Lecture Notes in Mathematics, vol. 544. Springer, Berlin (1976)Google Scholar
  26. 26.
    Lapid, E., Mínguez, A.: On a determinantal formula of Tadić. Am. J. Math. 136, 111–142 (2014)CrossRefzbMATHGoogle Scholar
  27. 27.
    Matić, I.: On discrete series subrepresentations of the generalized principal series. Glas. Mat. Ser. II I(51), 125–152 (2016)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mitra, A., Offen, O., Sayag, E.: Klyachko models for ladder representations. https://arxiv.org/pdf/1605.09197.pdf
  29. 29.
    MŒglin, C.: Représentations unipotentes et formes automorphes de carré intégrable. Forum Math. 6, 651–744 (1994)Google Scholar
  30. 30.
    MŒglin, C.: Sur la classification des séries discrètes des groupes classiques \(p\)-adiques: paramètres de Langlands et exhaustivité. J. Eur. Math. Soc. (JEMS) 4, 143–200 (2002)Google Scholar
  31. 31.
    MŒglin, C., Tadić, M.: Construction of discrete series for classical \(p\)-adic groups. J. Am. Math. Soc. 15, 715–786 (2002). (electronic)Google Scholar
  32. 32.
    MŒglin, C., Waldspurger, J.-L.: Sur l’involution de Zelevinski. J. Reine Angew. Math. 372, 136–177 (1986)Google Scholar
  33. 33.
    MŒglin, C., Waldspurger, J.-L.: Spectral decomposition and Eisenstein series, vol. 113 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, (1995). Une paraphrase de l’Écriture [A paraphrase of Scripture]Google Scholar
  34. 34.
    Muić, G.: Composition series of generalized principal series; the case of strongly positive discrete series. Israel J. Math. 140, 157–202 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Muić, G.: On certain classes of unitary representations for split classical groups. Can. J. Math. 59, 148–185 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Muić, G.: Some applications of degenerate Eisenstein series on \( \text{ Sp }_{2n}\). J. Ramanujan Math. Soc. 23, 223–257 (2008)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Sahi, S.: Jordan algebras and degenerate principal series. J. Reine Angew. Math. 462, 1–18 (1995)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Shahidi, F.: A proof of Langlands’ conjecture on Plancherel measures; complementary series for \(p\)-adic groups. Ann. Math. 2(132), 273–330 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Tadić, M.: Structure arising from induction and Jacquet modules of representations of classical \(p\)-adic groups. J. Algebra 177, 1–33 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Weil, A.: Sur la formule de Siegel dans la théorie des groupes classiques. Acta Math. 113, 1–87 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    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, 165–210 (1980)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.University of ZagrebZagrebCroatia

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