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On the cuspidal support of discrete series for p-adic quasisplit \(\textit{Sp}(N)\) and \(\textit{SO}(N)\)

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Abstract

Zelevinsky’s classification theory of discrete series of p-adic general linear groups has been well known. Mœglin and Tadić gave the same kind of theory for p-adic classical groups, which is more complicated due to the occurrence of nontrivial structure of L-packets. Nonetheless, their work is independent of the endoscopic classification theory of Arthur (also Mok in the unitary case), which concerns the structure of L-packets in these cases. So our goal in this paper is to make more explicit the connection between these two very different types of theories. To do so, we reprove the results of Mœglin and Tadić in the case of quasisplit symplectic groups and orthogonal groups by using Arthur’s theory.

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References

  1. Arthur, J.: On local character relations. Sel. Math. 2(4), 501–579 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arthur, J.: The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, vol. 61. Colloquium Publications, American Mathematical Society, Providence, RI (2013)

  3. 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(6), 2179–2189 (1995)

    MATH  Google Scholar 

  4. Ban, D.: Parabolic induction and Jacquet modules of representations of \({\rm O} (2n, F)\). Glas. Mat. Ser. 34(2), 147–185 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Bernstein, J., Deligne, P., Kazhdan, D.: Trace Paley–Wiener theorem for reductive \(p\)-adic groups. J. Anal. Math. 47, 180–192 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Mathematical Surveys and Monographs, vol. 67, 2nd edn. American Mathematical Society, Providence (2000)

    Book  MATH  Google Scholar 

  7. Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \(p\)-adic groups I. Ann. Sci. École Norm. Sup. (4) 10(4), 441–472 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. Casselman, W.: Characters and Jacquet modules. Math. Ann. 230(2), 101–105 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  9. Clozel, L.: Characters of nonconnected, reductive \(p\)-adic groups. Can. J. Math. 39(1), 149–167 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Harish-Chandra, Admissible invariant distributions on reductive \(p\)-adic groups, University Lecture Series, vol. 16, American Mathematical Society, Providence, RI: Preface and notes by Stephen DeBacker and Paul J. Sally, Jr (1999)

  11. Henniart, G.: Une preuve simple des conjectures de Langlands pour \({\rm GL}(n)\) sur un corps \(p\)-adique. Invent. Math. 139(2), 439–455 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hiraga, K.: On functoriality of Zelevinski involutions. Compos. Math. 140, 1625–1656 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, With an appendix by Vladimir G. Berkovich (2001)

  14. Jantzen, C.: Jacquet modules of induced representations for \(p\)-adic special orthogonal groups. J. Algebra 305(2), 802–819 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jacquet, H., Piatetskii-Shapiro, I.I., Shalika, J.A.: Rankin–Selberg convolutions. Am. J. Math. 105(2), 367–464 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kazhdan, D.: Cuspidal geometry of \(p\)-adic groups. J. Anal. Math. 47, 1–36 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kottwitz, R.E.: Rational conjugacy classes in reductive groups. Duke Math. J. 49(4), 785–806 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kottwitz, R.E., Shelstad, D.: Foundations of twisted endoscopy. Astérisque 55(255), vi+190 (1999)

    MathSciNet  MATH  Google Scholar 

  19. Mœglin, C.: Sur la classification des séries discrètes des groupes classiques \(p\)-adiques: paramètres de Langlands et exhaustivité. J. Eur. Math. Soc.: JEMS 4(2), 143–200 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mœglin, C.: Multiplicité 1 dans les paquets d’Arthur aux places \(p\)-adiques. In: On Certain \(L\)-Functions, Clay Mathematics Proceedings, vol. 13, pp. 333–374. American Mathematical Society, Providence (2011)

  21. Mœglin, C.: Paquets stables des séries discrètes accessibles par endoscopie tordue; leur paramètre de Langlands. Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro, Contemporary Mathematics, vol. 614, pp. 295–336. American Mathematical Society, Providence (2014)

  22. Mœglin, C., Tadić, M.: Construction of discrete series for classical \(p\)-adic groups. J. Am. Math. Soc. 15(3), 715–786 (2002). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  23. Muić, G., Tadić, M.: Unramified unitary duals for split classical \(p\)-adic groups; the topology and isolated representations. In: On certain \(L\)-functions, Clay Mathematics Proceedings, vol. 13, pp. 375–438. American Mathematical Society, Providence (2011)

  24. Mœglin, C., Waldspurger, J.-L.: Le spectre résiduel de \({\rm GL}(n)\). Ann. Sci. École Norm. Sup. (4) 22(4), 605–674 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mœglin, C., Waldspurger, J.-L.: Sur le transfert des traces d’un groupe classique \(p\)-adique à un groupe linéaire tordu. Sel. Math. 12(3–4), 433–515 (2006)

    MATH  Google Scholar 

  26. Mœglin, C., Waldspurger, J.-L.: Stabilisation de la formule des traces tordue. Progress in Mathematics, vol. 316/317. Birkhäuser, Basel (2016)

    MATH  Google Scholar 

  27. Ngô, B.C.: Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. 111, 1–169 (2010)

    Article  MATH  Google Scholar 

  28. Rogawski, J.D.: Trace Paley–Wiener theorem in the twisted case. Trans. Am. Math. Soc. 309(1), 215–229 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Scholze, P.: The local Langlands correspondence for \({GL}_n\) over \(p\)-adic fields. Invent. Math. 192(3), 663–715 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shahidi, F.: On certain \(L\)-functions. Am. J. Math. 103(2), 297–355 (1981)

    Article  MATH  Google Scholar 

  31. 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 

  32. Silberger, A.J.: Introduction to harmonic analysis on reductive \(p\)-adic groups. Mathematical Notes, vol. 23, Princeton University Press, Princeton (1979). Based on lectures by Harish-Chandra at the Institute for Advanced Study (1971–1973)

  33. Tadić, M.: Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case). Ann. Sci. École Norm. Sup. (4) 19(3), 335–382 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tadić, M.: On reducibility and unitarizability for classical \(p\)-adic groups, some general results. Can. J. Math. 61(2), 427–450 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Waldspurger, J.-L.: Une formule des traces locale pour les algèbres de Lie \(p\)-adiques. J. Reine Angew. Math. 465, 41–99 (1995)

    MathSciNet  MATH  Google Scholar 

  36. Waldspurger, J.-L.: Le lemme fondamental implique le transfert. Compos. Math. 105(2), 153–236 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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)

    Article  MathSciNet  MATH  Google Scholar 

  38. Waldspurger, J.-L.: Endoscopie et changement de caractéristique. J. Inst. Math. Jussieu 5(3), 423–525 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Waldspurger, J.-L.: L’endoscopie tordue n’est pas si tordue. Mem. Am. Math. Soc. 194(908), x+261 (2008)

    MathSciNet  MATH  Google Scholar 

  40. Waldspurger, J.-L.: La formule des traces locale tordue, à paraître aux Memoirs of AMS (2017)

  41. Zelevinsky, A.V.: Induced representations of reductive \(p\)-adic groups. II. On irreducible representations of \(\text{ GL }(n)\). Ann. Sci. École Norm. Sup. (4) 13(2), 165–210 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics and Statistics, University of Calgary, Calgary, Canada

    Bin Xu

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  1. Bin Xu
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Xu, B. On the cuspidal support of discrete series for p-adic quasisplit \(\textit{Sp}(N)\) and \(\textit{SO}(N)\) . manuscripta math. 154, 441–502 (2017). https://doi.org/10.1007/s00229-017-0923-x

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  • DOI: https://doi.org/10.1007/s00229-017-0923-x

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