The essentially tame local Langlands correspondence, I

Bushnell, Colin¹ ; Henniart, Guy²

¹ Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom
² Département de Mathématiques & UMR 8628 du CNRS, Bâtiment 425, Université de Paris-Sud, 91405 Orsay cedex, France

Journal of the American Mathematical Society, Tome 18 (2005) no. 3, pp. 685-710

Résumé

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 $\mathcal {G}^\mathrm {et}_n(F)$. The Langlands correspondence induces a bijection of $\mathcal {G}^\mathrm {et}_n(F)$ with a certain set $\mathcal {A}^\mathrm {et}_n(F)$ of irreducible supercuspidal representations of $\mathrm {GL}_n(F)$. 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 $\mathcal {G}^\mathrm {et}_n(F)$. Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of $P_n(F)$ with $\mathcal {A}^\mathrm {et}_n(F)$, generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of $\mathcal {G}^\mathrm {et}_n(F)$ with $\mathcal {A}^\mathrm {et}_n(F)$. We show that one obtains the Langlands correspondence by composing the map $P_n(F) \to \mathcal {A}^\mathrm {et}_n(F)$ 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.

Détail

DOI : 10.1090/S0894-0347-05-00487-X

Bibliographie

[1] Arthur, James, Clozel, Laurent Simple algebras, base change, and the advanced theory of the trace formula 1989

[2] Bushnell, Colin J., Frã¶Hlich, Albrecht Gauss sums and 𝑝-adic division algebras 1983

[3] Bushnell, Colin J., Henniart, Guy Local tame lifting for 𝐺𝐿(𝑁). I. Simple characters Inst. Hautes Études Sci. Publ. Math. 1996 105 233

[4] Bushnell, Colin J., Henniart, Guy Local tame lifting for 𝐺𝐿(𝑛). II. Wildly ramified supercuspidals Astérisque 1999

[5] Bushnell, Colin J., Henniart, Guy Davenport-Hasse relations and an explicit Langlands correspondence J. Reine Angew. Math. 2000 171 199

[6] Bushnell, Colin J., Henniart, Guy Local tame lifting for 𝐺𝐿(𝑛). IV. Simple characters and base change Proc. London Math. Soc. (3) 2003 337 362

[7] Bushnell, Colin J., Henniart, Guy, Kutzko, Philip C. Correspondance de Langlands locale pour 𝐺𝐿_{𝑛} et conducteurs de paires Ann. Sci. École Norm. Sup. (4) 1998 537 560

[8] Bushnell, Colin J., Kutzko, Philip C. The admissible dual of 𝐺𝐿(𝑁) via compact open subgroups 1993

[9] Bushnell, Colin J., Kutzko, Philip C. The admissible dual of 𝑆𝐿(𝑁). II Proc. London Math. Soc. (3) 1994 317 379

[10] Corwin, Lawrence, Howe, Roger E. Computing characters of tamely ramified 𝑝-adic division algebras Pacific J. Math. 1977 461 477

[11] Glauberman, George Correspondences of characters for relatively prime operator groups Canadian J. Math. 1968 1465 1488

[12] Green, J. A. The characters of the finite general linear groups Trans. Amer. Math. Soc. 1955 402 447

[13] Harris, Michael, Taylor, Richard The geometry and cohomology of some simple Shimura varieties 2001

[14] Henniart, Guy Une preuve simple des conjectures de Langlands pour 𝐺𝐿(𝑛) sur un corps 𝑝-adique Invent. Math. 2000 439 455

[15] Henniart, Guy, Herb, Rebecca Automorphic induction for 𝐺𝐿(𝑛) (over local non-Archimedean fields) Duke Math. J. 1995 131 192

[16] Howe, Roger E. Tamely ramified supercuspidal representations of 𝐺𝑙_{𝑛} Pacific J. Math. 1977 437 460

[17] Jacquet, H., Piatetskii-Shapiro, I. I., Shalika, J. A. Rankin-Selberg convolutions Amer. J. Math. 1983 367 464

[18] Koch, Helmut, Zink, Ernst-Wilhelm Zur Korrespondenz von Darstellungen der Galoisgruppen und der zentralen Divisionsalgebren über lokalen Körpern (der zahme Fall) Math. Nachr. 1980 83 119

[19] Laumon, G., Rapoport, M., Stuhler, U. 𝒟-elliptic sheaves and the Langlands correspondence Invent. Math. 1993 217 338

[20] Må“Glin, C. Sur la correspondance de Langlands-Kazhdan J. Math. Pures Appl. (9) 1990 175 226

[21] Moy, Allen Local constants and the tame Langlands correspondence Amer. J. Math. 1986 863 930

[22] Reimann, Harry Representations of tamely ramified 𝑝-adic division and matrix algebras J. Number Theory 1991 58 105

[23] Rogawski, Jonathan D. Representations of 𝐺𝐿(𝑛) and division algebras over a 𝑝-adic field Duke Math. J. 1983 161 196

[24] Serre, Jean-Pierre Corps locaux 1962 243

[25] Shahidi, Freydoon Fourier transforms of intertwining operators and Plancherel measures for 𝐺𝐿(𝑛) Amer. J. Math. 1984 67 111