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The local Langlands correspondence for \(\text{GL}_n\) and conductors of pairs. (Correspondance de Langlands locale pour \(\text{GL}_n\) et conducteurs de paires.) (French) Zbl 0915.11055

Abstract reworded: Fix a finite extension \(F\) of \(\mathbb Q_p\), and \(n\geq 1\). Denote by \(\pi_n\) an admissible irreducible supercuspidal representation of \(GL_n(F)\). Denote by \(\sigma_n\) a semisimple continuous \(n\)-dimensional representation of the Weil group of \(F\). M. Harris attached to each \(\pi_n\) a \(\sigma_n\), denoted \(\sigma(\pi_n)\). We show that the map \(L_n: \pi_n\mapsto\sigma(\pi_n)\) induces a bijection from the set of equivalence classes of such \(\pi_n\) to the set of equivalence classes of irreducible such \(\sigma_n\). In addition, we show that the maps \(L_n\) preserve conductors of pairs.

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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[1] J. ARTHUR et 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 90m:22041 | Zbl 0682.10022 · Zbl 0682.10022
[2] C. J. BUSHNELL et A. FRÖHLICH , Gauss sums and p-adic division algebras (Lecture Notes in Math., vol. 987, Springer, Berlin, 1983 ). MR 84m:12017 | Zbl 0507.12008 · Zbl 0507.12008
[3] C. J. BUSHNELL et G. HENNIART , Local tame lifting for GLn I : simple characters (Publ. Math. IHES, vol. 83, 1996 , p. 105-233). Numdam | MR 98m:11129 | Zbl 0878.11042 · Zbl 0878.11042 · doi:10.1007/BF02698646
[4] C. J. BUSHNELL et G. HENNIART , Local tame lifting for GLn II : wildly ramified supercuspidals (Manuscrit, Mai 1997 ). · Zbl 0920.11079
[5] C. J. BUSHNELL , G. HENNIART et P. C. KUTZKO , Local Rankin-Selberg convolutions for GLn : Explicit conductor formula (J. Amer. Math. Soc., à paraître). Zbl 0899.22017 · Zbl 0899.22017 · doi:10.1090/S0894-0347-98-00270-7
[6] C. J. BUSHNELL et P. C. KUTZKO , Simple types in GL(N) : computing conjugacy classes . Representation theory and analysis on homogeneous spaces (S. Gindikin et al, eds.) (Contemp. Math, vol. 177, Amer. Math. Soc., Providence, 1995 , p. 107-135). MR 96c:22027 | Zbl 0835.22009 · Zbl 0835.22009
[7] R. GODEMENT et H. JACQUET , Zeta functions of simple algebras (Lecture Notes in Math., vol. 260, Springer, Berlin 1972 ). MR 49 #7241 | Zbl 0244.12011 · Zbl 0244.12011 · doi:10.1007/BFb0070263
[8] M. HARRIS , Supercuspidal representations in the cohomology of Drinfel’d upper half-spaces ; elaboration of Carayol’s program (Invent. Math., vol. 129, 1997 , p. 75-120). MR 98i:11100 | Zbl 0886.11029 · Zbl 0886.11029 · doi:10.1007/s002220050159
[9] M. HARRIS , p-adic uniformization and Galois properties of automorphic forms (Manuscrit, 1997 ).
[10] G. HENNIART , Représentations du groupe de Weil d’un corps local (L’Ens. Math., vol. 26, 1980 , p. 155-172). MR 81j:12012 | Zbl 0452.12006 · Zbl 0452.12006
[11] G. HENNIART , La conjecture de Langlands locale pour GL(n) . Journées Arithmétiques de Metz (Astérisque, vol. 94, 1982 , p. 67-85). MR 84g:12021 | Zbl 0504.12017 · Zbl 0504.12017
[12] G. HENNIART , La conjecture de Langlands locale pour GL(3) (Mém. Soc. Math. France, nouvelle série, vol. 11/12, 1984 ). Numdam | Zbl 0577.12011 · Zbl 0577.12011
[13] G. HENNIART , On the local Langlands conjecture for GL(n) : the cyclic case (Ann. Math., vol. 123, 1986 , p. 145-203). MR 87k:11132 | Zbl 0588.12010 · Zbl 0588.12010 · doi:10.2307/1971354
[14] G. HENNIART , La conjecture de Langlands locale numérique pour GL(n) (Ann. Scient. Éc. Norm. Sup., (4), vol. 21, 1988 , p. 497-544). Numdam | MR 90f:11094 | Zbl 0666.12013 · Zbl 0666.12013
[15] G. HENNIART , Une conséquence de la théorie du changement de base pour GL(n) (Analytic Number Theory, Tokyo, Lecture Notes in Math. 1988 , vol. 1434, Springer, Berlin, 1990 , p. 138-142). MR 91i:22021 | Zbl 0703.11069 · Zbl 0703.11069
[16] G. HENNIART , Caractérisation de la correspondance de Langlands locale par les facteurs \epsilon de paires (Invent. Math., vol. 113, 1993 , p. 339-350). MR 96e:11078 | Zbl 0810.11069 · Zbl 0810.11069 · doi:10.1007/BF01244309
[17] G. HENNIART , lettre à Michael Harris, janvier 1994 .
[18] G. HENNIART et R. HERB , Automorphic induction for GL(n) (over local non-archimedean fields) (Duke Math. J., vol. 78, 1995 , p. 131-192). Article | MR 96i:22038 | Zbl 0849.11092 · Zbl 0849.11092 · doi:10.1215/S0012-7094-95-07807-7
[19] H. JACQUET , I. I. PIATETSKII-SHAPIRO et J. SHALIKA , Conducteur des représentations du groupe linéaire (Math. Ann., vol. 236, 1981 , p. 199-214). MR 83c:22025 | Zbl 0443.22013 · Zbl 0443.22013 · doi:10.1007/BF01450798
[20] H. JACQUET , I. I. PIATETSKII-SHAPIRO et J. A. SHALIKA , Rankin-Selberg convolutions (Amer. J. Math., vol. 105, 1983 , p. 367-483). MR 85g:11044 | Zbl 0525.22018 · Zbl 0525.22018 · doi:10.2307/2374264
[21] H. JACQUET et J. A. SHALIKA , On Euler products and the classification of automorphic representations II (Amer. J. Math., vol. 103, 1981 , p. 777-815). Zbl 0491.10020 · Zbl 0491.10020 · doi:10.2307/2374050
[22] D. A. KAZHDAN , On lifting. Lie Group Representations II (Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984 , p. 209-249). MR 86h:22029 | Zbl 0538.20014 · Zbl 0538.20014
[23] H. KOCH , Bemerkungen zur numerischen lokalen Langlands Vermutung (Proc. Steklov Inst., vol. 183, 1984 , p. 129-136). Zbl 0557.12009 · Zbl 0557.12009
[24] P. C. KUTZKO , The Langlands conjecture for GL2 of a local field (Ann. Math., vol. 112, 1980 , p. 381-412). MR 82e:12019 | Zbl 0469.22013 · Zbl 0469.22013 · doi:10.2307/1971151
[25] P. C. KUTZKO et A. MOY , On the local Langlands conjecture in prime dimension (Ann. Math., vol. 121, 1985 , p. 495-517). MR 87d:11092 | Zbl 0609.12017 · Zbl 0609.12017 · doi:10.2307/1971207
[26] J.-P. LABESSE , Non-invariant base change identities (Bull. Soc. Math. France, vol. 61, 1995 ). Numdam | MR 97b:11136 | Zbl 0868.11026 · Zbl 0868.11026
[27] R. P. LANGLANDS , Problems in the theory of automorphic forms. Lectures in modern analysis and applications III (Lecture Notes in Math, vol. 170, Springer, Berlin, 1970 , p. 18-86). MR 46 #1758 | Zbl 0225.14022 · Zbl 0225.14022
[28] R. P. LANGLANDS , Base change for GL(2) (Ann. Math. Studies, vol. 96, Princeton University Press, 1980 ). MR 82a:10032 | Zbl 0444.22007 · Zbl 0444.22007
[29] G. LAUMON , M. RAPOPORT et U. STUHLER , D-elliptic sheaves and the Langlands correspondence (Invent. Math., vol. 113, 1993 , p. 217-338). MR 96e:11077 | Zbl 0809.11032 · Zbl 0809.11032 · doi:10.1007/BF01244308
[30] C. MŒGLIN , Sur la correspondance de Langlands-Kazhdan (J. Math. pures et appl., vol. 69, 1990 , p. 175-226). MR 91g:11141 | Zbl 0711.11045 · Zbl 0711.11045
[31] A. MOY , Local constants and the tame Langlands correspondence (Amer. J. Math., vol. 108, 1986 , p. 863-930). MR 88b:11081 | Zbl 0597.12019 · Zbl 0597.12019 · doi:10.2307/2374518
[32] H. REIMANN , Representations of tamely ramified p-adic division and matrix rings (J. Number Theory, vol. 38, 1991 , p. 58-105). MR 92h:11103 | Zbl 0728.11063 · Zbl 0728.11063 · doi:10.1016/0022-314X(91)90093-Q
[33] F. RODIER , Représentations de GL(n, k) où k est un corps p-adique (Séminaire Bourbaki, vol. 587, 1981 / 1982 , Astérisque, vol. 92-93, 1982 , p. 201-218). Numdam | MR 84h:22040 | Zbl 0506.22019 · Zbl 0506.22019
[34] F. SHAHIDI , Fourier transforms of intertwining operators and Plancherel measures for GL(n) (Amer. J. Math., vol. 106, 1984 , p. 67-111). MR 86b:22031 | Zbl 0567.22008 · Zbl 0567.22008 · doi:10.2307/2374430
[35] J. TATE , Number theoretic background. Automorphic forms, representations and L-functions (A. Borel and W. Casselman, eds.) (Proc. Symposia Pure Math, vol. 33, part 2 (Amer. Math. Soc., Providence, 1979 ), p. 3-22). MR 80m:12009 | Zbl 0422.12007 · Zbl 0422.12007
[36] A. V. ZELEVINSKY , Induced representations of reductive p-adic groups II : On irreducible representations of GL(n) (Ann. Scient. Éc. Norm. Sup., (4), vol. 13, 1980 , p. 165-210). Numdam | MR 83g:22012 | Zbl 0441.22014 · Zbl 0441.22014
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