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The essentially tame local Langlands correspondence, I

Authors: Colin J. Bushnell and Guy Henniart
Journal: J. Amer. Math. Soc. 18 (2005), 685-710
MSC (2000): Primary 22E50
Published electronically: April 25, 2005
MathSciNet review: 2138141
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Abstract: Let $F$ be a non-Archimedean local field (of characteristic $0$ or $p$) with finite residue field of characteristic $p$. An irreducible smooth representation of the Weil group of $F$ is called essentially tame if its restriction to wild inertia is a sum of characters. The set of isomorphism classes of irreducible, essentially tame representations of dimension $n$ is denoted $\boldsymbol{\mathscr{G}}^{et}_n(F)$. The Langlands correspondence induces a bijection of $\boldsymbol{\mathscr{G}}^{et}_n(F)$ with a certain set $\boldsymbol{\mathscr{A}}^{et}_n(F)$ of irreducible supercuspidal representations of ${GL}_n(F)$. We consider the set $P_n(F)$ of isomorphism classes of certain pairs $(E/F,\xi)$, called ``admissible'', consisting of a tamely ramified field extension $E/F$ of degree $n$ and a quasicharacter $\xi$ of $E^\times$. There is an obvious bijection of $P_n(F)$ with $\boldsymbol{\mathscr{G}}^{et}_n(F)$. Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of $P_n(F)$ with $\boldsymbol{\mathscr{A}}^{et}_n(F)$, generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of $\boldsymbol{\mathscr{G}}^{et}_n(F)$ with $\boldsymbol{\mathscr{A}}^{et}_n(F)$. We show that one obtains the Langlands correspondence by composing the map $P_n(F) \to \boldsymbol{\mathscr{A}}^{et}_n(F)$ with a permutation of $P_n(F)$ of the form $(E/F,\xi)\mapsto (E/F,\mu_\xi\xi)$, where $\mu_\xi$ is a tamely ramified character of $E^\times$ depending on $\xi$. This answers a question of Moy (1986). We calculate the character $\mu_\xi$in the case where $E/F$ is totally ramified of odd degree.

References [Enhancements On Off] (What's this?)

  • 1. J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Math. Studies, vol. 120, Princeton University Press, 1989. MR 1007299 (90m:22041)
  • 2. C.J. Bushnell and A. Fröhlich, Gauss sums and $p$-adic division algebras, Lecture Notes in Math., vol. 987, Springer, Berlin, Heidelberg, New York, 1983. MR 0701540 (84m:12017)
  • 3. C.J. Bushnell and G. Henniart, Local tame lifting for $GL(N)$I: simple characters, Publ. Math. IHES 83 (1996), 105-233. MR 1423022 (98m:11129)
  • 4. -, Local tame lifting for $\mathrm{GL}(n)$ II: wildly ramified supercuspidals, Astérisque 254 (1999). MR 1685898 (2000d:11147)
  • 5. -, Davenport-Hasse relations and an explicit Langlands correspondence, J. reine angew. Math. 519 (2000), 171-199. MR 1739725 (2002e:11171)
  • 6. -, Local tame lifting for $\mathrm{GL}(n)$ IV: simple characters and base change, Proc. London Math. Soc. (3) 87 (2003), 337-362.MR 1990931 (2004f:22017)
  • 7. -, The essentially tame local Langlands correspondence, II: totally ramified representations, Compositio Mathematica, to appear.
  • 8. C.J. Bushnell, G. Henniart and P.C. Kutzko, Correspondance de Langlands locale pour $\mathrm{GL}_{n}$ et conducteurs de paires, Ann. Scient. École Norm. Sup. (4) 31 (1998), 537-560. MR 1634095 (99h:22011)
  • 9. C.J. Bushnell and P.C. Kutzko, The admissible dual of $GL(N)$via compact open subgroups, Annals of Math. Studies, vol. 129, Princeton University Press, 1993.MR 1204652 (94h:22007)
  • 10. -, The admissible dual of $SL(N)$ II, Proc. London Math. Soc. (3) 68 (1994), 317-379.MR 1253507 (94k:22035)
  • 11. L.J. Corwin and R.E. Howe, Computing characters of tamely ramified $p$-adic division algebras, Pacific J. Math. 73 (1977), 461-477.MR 0492084 (58:11238)
  • 12. G. Glauberman, Correspondences of characters for relatively prime operator groups, Canad. J. Math. 20 (1968), 1465-1488. MR 0232866 (38:1189)
  • 13. J.A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402-447. MR 0072878 (17:345e)
  • 14. M. Harris and R. Taylor, On the geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies, vol. 151, Princeton University Press, 2001. MR 1876802 (2002m:11050)
  • 15. G. Henniart, Une preuve simple des conjectures de Langlands pour $\mathrm{GL}(n)$ sur un corps $p$-adique, Invent. Math. 139 (2000), 439-455.MR 1738446 (2001e:11052)
  • 16. G. Henniart and R. Herb, Automorphic induction for $GL(n)$(over local non-archimedean fields), Duke Math. J. 78 (1995), 131-192.MR 1328755 (96i:22038)
  • 17. G. Henniart and B. Lemaire, Work in progress.
  • 18. R.E. Howe, Tamely ramified supercuspidal representations of $GL_{n}$, Pacific J. Math. 73 (1977), 437-460. MR 0492087 (58:11241)
  • 19. H. Jacquet, I. Piatetski-Shapiro and J. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), 367-483. MR 0701565 (85g:11044)
  • 20. H. Koch and E.-W. Zink, Zur Korrespondenz von Darstellungen der Galoisgruppen und der zentralen Divisionsalgebren über lokalen Körpern (der zahme Fall), Math. Nachr. 98 (1980), 83-119. MR 0623696 (83h:12025)
  • 21. G. Laumon, M. Rapoport and U. Stuhler, $\mathscr{D}$-elliptic sheaves and the Langlands correspondence, Invent. Math. 113 (1993), 217-338.MR 1228127 (96e:11077)
  • 22. C. M\oeglin, Sur la correspondance de Langlands-Kazhdan, J. Math. Pures et Appl. (9) 69 (1990), 175-226. MR 1067450 (91g:11141)
  • 23. A. Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 108 (1986), 863-930. MR 0853218 (88b:11081)
  • 24. H. Reimann, Representations of tamely ramified $p$-adic division and matrix algebras, J. Number Theory 38 (1991), 58-105. MR 1105671 (92h:11103)
  • 25. J. Rogawski, Representations of $\mathrm{GL}(n)$ and division algebras over a local field, Duke Math. J. 50 (1983), 161-196. MR 0700135 (84j:12018)
  • 26. J.-P. Serre, Corps locaux, Hermann, Paris, 1965. MR 0150130 (27:133)
  • 27. F. Shahidi, Fourier transforms of intertwining operators and Plancherel measures for $\mathrm{GL}(n)$, Amer. J. Math. 106 (1984), 67-111. MR 0729755 (86b:22031)

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Additional Information

Colin J. Bushnell
Affiliation: Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom

Guy Henniart
Affiliation: Département de Mathématiques & UMR 8628 du CNRS, Bâtiment 425, Université de Paris-Sud, 91405 Orsay cedex, France

Keywords: Explicit local Langlands correspondence, base change, automorphic induction, tame lifting, admissible pair
Received by editor(s): March 29, 2004
Published electronically: April 25, 2005
Additional Notes: Much of the work in this programme was carried out while the first-named author was visiting, and partly supported by, l’Université de Paris-Sud. Both authors were also partially supported by the EU network “Arithmetical Algebraic Geometry”.
Dedicated: To the memory of Albrecht Fröhlich
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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