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To an effective local Langlands Correspondence
About this Title
Colin J. Bushnell, King’s College London, Department of Mathematics, Strand, London WC2R 2LS, UK. and Guy Henniart, Université de Paris-Sud, Laboratoire de Mathématiques d’Orsay, Orsay Cedex, F-91405; CNRS, Orsay cedex, F-91405.
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 231, Number 1087
ISBNs: 978-0-8218-9417-0 (print); 978-1-4704-1723-9 (online)
DOI: https://doi.org/10.1090/memo/1087
Published electronically: February 10, 2014
Keywords: Explicit local Langlands correspondence,
automorphic induction,
simple type
MSC: Primary 22E50
Table of Contents
Chapters
- Introduction
- 1. Representations of Weil groups
- 2. Simple characters and tame parameters
- 3. Action of tame characters
- 4. Cuspidal representations
- 5. Algebraic induction maps
- 6. Some properties of the Langlands correspondence
- 7. A naïve correspondence and the Langlands correspondence
- 8. Totally ramified representations
- 9. Unramified automorphic induction
- 10. Discrepancy at a prime element
- 11. Symplectic signs
- 12. Main Theorem and examples
Abstract
Let $F$ be a non-Archimedean local field. Let $\mathcal {W}_{F}$ be the Weil group of $F$ and $\mathcal {P}_{F}$ the wild inertia subgroup of $\mathcal {W}_{F}$. Let $\widehat {\mathcal {W}}_{F}$ be the set of equivalence classes of irreducible smooth representations of $\mathcal {W}_{F}$. Let $\mathcal {A}^{0}_{n}(F)$ denote the set of equivalence classes of irreducible cuspidal representations of $\mathrm {GL}_{n}(F)$ and set $\widehat {\mathrm {GL}}_{F} = \bigcup _{n\ge 1} \mathcal {A}^{0}_{n}(F)$. If $\sigma \in \widehat {\mathcal {W}}_{F}$, let $^{L}{\sigma }\in \widehat {\mathrm {GL}}_{F}$ be the cuspidal representation matched with $\sigma$ by the Langlands Correspondence. If $\sigma$ is totally wildly ramified, in that its restriction to $\mathcal {P}_{F}$ is irreducible, we treat $^{L}{\sigma }$ as known. From that starting point, we construct an explicit bijection $\mathbb {N}:\widehat {\mathcal {W}}_{F} \to \widehat {\mathrm {GL}}_{F}$, sending $\sigma$ to $^{N}{\sigma }$. We compare this “naïve correspondence” with the Langlands correspondence and so achieve an effective description of the latter, modulo the totally wildly ramified case. A key tool is a novel operation of “internal twisting” of a suitable representation $\pi$ (of $\mathcal {W}_{F}$ or $\mathrm {GL}_{n}(F)$) by tame characters of a tamely ramified field extension of $F$, canonically associated to $\pi$. We show this operation is preserved by the Langlands correspondence.- James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299
- Colin J. Bushnell and Albrecht Fröhlich, Gauss sums and $p$-adic division algebras, Lecture Notes in Mathematics, vol. 987, Springer-Verlag, Berlin-New York, 1983. MR 701540
- Colin J. Bushnell and Guy Henniart, Local tame lifting for $\textrm {GL}(N)$. I. Simple characters, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 105–233. MR 1423022
- Colin J. Bushnell and Guy Henniart , Local tame lifting for $\mathrm {GL}(n)$ II: wildly ramified supercuspidals, Astérisque 254 (1999).
- Colin J. Bushnell and Guy Henniart, Davenport-Hasse relations and an explicit Langlands correspondence, J. Reine Angew. Math. 519 (2000), 171–199. MR 1739725, DOI 10.1515/crll.2000.011
- Colin J. Bushnell and Guy Henniart, Davenport-Hasse relations and an explicit Langlands correspondence. II. Twisting conjectures, J. Théor. Nombres Bordeaux 12 (2000), no. 2, 309–347 (English, with English and French summaries). Colloque International de Théorie des Nombres (Talence, 1999). MR 1823188
- Colin J. Bushnell and Guy Henniart, The local Rankin-Selberg convolution for $\textrm {GL}(n)$: divisibility of the conductor, Math. Ann. 321 (2001), no. 2, 455–461. MR 1866496, DOI 10.1007/s002080100237
- Colin J. Bushnell and Guy Henniart, Local tame lifting for $\textrm {GL}(n)$. IV. Simple characters and base change, Proc. London Math. Soc. (3) 87 (2003), no. 2, 337–362. MR 1990931, DOI 10.1112/S0024611503014114
- Colin J. Bushnell and Guy Henniart, Local tame lifting for $\textrm {GL}(n)$. III. Explicit base change and Jacquet-Langlands correspondence, J. Reine Angew. Math. 580 (2005), 39–100. MR 2130587, DOI 10.1515/crll.2005.2005.580.39
- Colin J. Bushnell and Guy Henniart, The essentially tame local Langlands correspondence. I, J. Amer. Math. Soc. 18 (2005), no. 3, 685–710. MR 2138141, DOI 10.1090/S0894-0347-05-00487-X
- Colin J. Bushnell and Guy Henniart, The essentially tame local Langlands correspondence. II. Totally ramified representations, Compos. Math. 141 (2005), no. 4, 979–1011. MR 2148193, DOI 10.1112/S0010437X05001363
- Colin J. Bushnell and Guy Henniart, The local Langlands conjecture for $\rm GL(2)$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Springer-Verlag, Berlin, 2006. MR 2234120
- Colin J. Bushnell and Guy Henniart, The essentially tame local Langlands correspondence, III: the general case, Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 497–553. MR 2679700, DOI 10.1112/plms/pdp053
- Colin J. Bushnell and Guy Henniart, The essentially tame Jacquet-Langlands correspondence for inner forms of $\textrm {GL}(n)$, Pure Appl. Math. Q. 7 (2011), no. 3, Special Issue: In honor of Jacques Tits, 469–538. MR 2848585, DOI 10.4310/PAMQ.2011.v7.n3.a2
- Colin J. Bushnell and Guy Henniart, Intertwining of simple characters in $\mathrm {GL}(n)$, Internat. Math. Res. Notices (2012), doi:10.1093/imrn/rns162.
- Colin J. Bushnell, Guy M. Henniart, and Philip C. Kutzko, Local Rankin-Selberg convolutions for $\textrm {GL}_n$: explicit conductor formula, J. Amer. Math. Soc. 11 (1998), no. 3, 703–730. MR 1606410, DOI 10.1090/S0894-0347-98-00270-7
- Colin J. Bushnell, Guy Henniart, and Philip C. Kutzko, Correspondance de Langlands locale pour $\textrm {GL}_n$ et conducteurs de paires, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 4, 537–560 (French, with English and French summaries). MR 1634095, DOI 10.1016/S0012-9593(98)80106-7
- Colin J. Bushnell and Philip C. Kutzko, The admissible dual of $\textrm {GL}(N)$ via compact open subgroups, Annals of Mathematics Studies, vol. 129, Princeton University Press, Princeton, NJ, 1993. MR 1204652
- A. M. McEvett, Forms over semisimple algebras with involution, J. Algebra 12 (1969), 105–113. MR 274481, DOI 10.1016/0021-8693(69)90019-2
- A. Fröhlich and A. M. McEvett, The representation of groups by automorphisms of forms, J. Algebra 12 (1969), 114–133. MR 240217, DOI 10.1016/0021-8693(69)90020-9
- George Glauberman, Correspondences of characters for relatively prime operator groups, Canadian J. Math. 20 (1968), 1465–1488. MR 232866, DOI 10.4153/CJM-1968-148-x
- Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
- G. Henniart, La Conjecture de Langlands pour $\mathrm {GL}(3)$, Mém. Soc. Math. France 12 (1984).
- Guy Henniart, Correspondance de Langlands-Kazhdan explicite dans le cas non ramifié, Math. Nachr. 158 (1992), 7–26 (French, with French summary). MR 1235293, DOI 10.1002/mana.19921580102
- Guy Henniart, Caractérisation de la correspondance de Langlands locale par les facteurs $\epsilon$ de paires, Invent. Math. 113 (1993), no. 2, 339–350 (French, with English and French summaries). MR 1228128, DOI 10.1007/BF01244309
- Guy Henniart, Une preuve simple des conjectures de Langlands pour $\textrm {GL}(n)$ sur un corps $p$-adique, Invent. Math. 139 (2000), no. 2, 439–455 (French, with English summary). MR 1738446, DOI 10.1007/s002220050012
- Guy Henniart, Une caractérisation de la correspondance de Langlands locale pour $\textrm {GL}(n)$, Bull. Soc. Math. France 130 (2002), no. 4, 587–602 (French, with English and French summaries). MR 1947454, DOI 10.24033/bsmf.2431
- Guy Henniart and Rebecca Herb, Automorphic induction for $\textrm {GL}(n)$ (over local non-Archimedean fields), Duke Math. J. 78 (1995), no. 1, 131–192. MR 1328755, DOI 10.1215/S0012-7094-95-07807-7
- Guy Henniart and Bertrand Lemaire, Formules de caractères pour l’induction automorphe, J. Reine Angew. Math. 645 (2010), 41–84 (French, with English summary). MR 2673422, DOI 10.1515/CRELLE.2010.059
- Guy Henniart and Bertrand Lemaire , Changement de base et induction automorphe pour $\mathrm {GL}_{n}$ en caractéristique non nulle, Mém. Soc. Math. France 108 (2010).
- H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, DOI 10.2307/2374264
- Philip Kutzko, The Langlands conjecture for $\textrm {Gl}_{2}$ of a local field, Ann. of Math. (2) 112 (1980), no. 2, 381–412. MR 592296, DOI 10.2307/1971151
- Philip Kutzko and Allen Moy, On the local Langlands conjecture in prime dimension, Ann. of Math. (2) 121 (1985), no. 3, 495–517. MR 794371, DOI 10.2307/1971207
- G. Laumon, M. Rapoport, and U. Stuhler, ${\scr D}$-elliptic sheaves and the Langlands correspondence, Invent. Math. 113 (1993), no. 2, 217–338. MR 1228127, DOI 10.1007/BF01244308
- C. Mœglin, Sur la correspondance de Langlands-Kazhdan, J. Math. Pures Appl. (9) 69 (1990), no. 2, 175–226 (French). MR 1067450
- Freydoon Shahidi, Fourier transforms of intertwining operators and Plancherel measures for $\textrm {GL}(n)$, Amer. J. Math. 106 (1984), no. 1, 67–111. MR 729755, DOI 10.2307/2374430