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Représentations galoisiennes associées aux représentations automorphes autoduales de \(GL(n)\). (Galois representations associated with self-dual automorphic representations of \(GL(n)\)). (French) Zbl 0739.11020

To certain automorphic representations of \(GL(n,A_ F)\), \(F\) a totally real number field, \(\ell\)-adic Galois representations are attached relating eigenvalues of Hecke operators to eigenvalues of Frobenius. This is done by first transporting the representation of \(GL(n,A_ F)\) to an appropriate unitary group \(U\). This involves base change from \(GL(n)\) over \(F\) to \(GL(n)\) over a totally imaginary quadratic extension \(F_ c\) of \(F\), correspondence between the latter and the multiplicative group \(D^*\) of a division algebra over \(F\) with involution of the second kind and descent from \(D^*\) to the unitary group \(U\). Then Kottwitz’ results are applied. Essential is the hypothesis that the original representation \(\pi\) of \(GL(n,A_ F)\) be self-dual.
The result implies the Ramanujan conjecture at almost all places for \(\pi\). There is an analogous result for a \(CM\)-field.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
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References:

[1] Automorphic Forms, Representations, and L-functions, Proceedings of the Corvallis Conference,A. Borel etW. Casselman eds,Proc. Symp. Pure Math.,33, A.M.S., Providence, 1979, I, II.
[2] Automorphic Forms, Shimura Varieties, and L-functions, Proceedings of the Ann Arbor Conference,L. Clozel etJ. S. Milne eds, Academic Press, 1990, I, II.
[3] J. Adams, D. Vogan,Lifting of characters and Harish-Chandra’s method of descent, preprint. · Zbl 0808.22001
[4] J. Arthur, A Paley-Wiener theorem for real reductive groups,Acta Math.,150 (1983), 1–89. · Zbl 0514.22006 · doi:10.1007/BF02392967
[5] J. Arthur, L. Clozel,Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Math. Studies, Princeton U. Press, 1989. · Zbl 0682.10022
[6] D. Blasius, Automorphic forms and Galois representations : some examples,in [AA], II, 1–13.
[7] A. Borel, Automorphic L-functions,in [C], II, 27–62.
[8] A. Borel, N. Wallach,Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Annals of Math. Studies, Princeton U. Press, 1980. · Zbl 0443.22010
[9] A. Bouaziz, Relèvement des caractères d’un groupe endoscopique pour le changement de baseC/R,Astérisque,171–172 (1989), 163–194.
[10] L. Clozel, Changement de base pour les représentations tempérées des groupes réductifs réels,Ann. Sc. E.N.S. (4e sér.),15 (1982), 45–115.
[11] L. Clozel, The fundamental lemma for stable base change,Duke Math. J.,61 (1990), 255–302. · Zbl 0731.22011 · doi:10.1215/S0012-7094-90-06112-5
[12] L. Clozel, On the cuspidal cohomology of arithmetic subgroups of GL(2n)...,Duke Math. J.,55 (1987), 475–486. · Zbl 0648.22007 · doi:10.1215/S0012-7094-87-05525-6
[13] L. Clozel, Motifs et formes automorphes : applications du principe de fonctorialité,in [AA], I, 77–159.
[14] L. Clozel, P. Delorme, Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs II,Ann. Sc. E.N.S. (4e sér.),23 (1990), 193–228. · Zbl 0724.22012
[15] P. Delorme, Théorème de Paley-Wiener invariant tordu pour le changement de baseC/R, preprint. · Zbl 0765.22007
[16] T. Enright, Relative Lie algebra cohomology and unitary representations of complex Lie groups,Duke Math. J.,46 (1979), 513–525. · Zbl 0427.22010 · doi:10.1215/S0012-7094-79-04626-X
[17] M. Harris, Automorphic forms of {\(\sigma\)}-cohomology type as coherent cohomology classes,J. Diff. Geom.,32 (1990), 1–63. · Zbl 0711.14012
[18] G. Harder, Ueber die Galoiskohomologie halbeinfacher Matrizengruppen II,Math. Zeitschrift,92 (1966), 396–415. · Zbl 0152.01001 · doi:10.1007/BF01112219
[19] J. Johnson, Stable base changeC/R of certain derived functor modules,Math. Ann.,287 (1990), 467–493. · Zbl 0672.22016 · doi:10.1007/BF01446906
[20] R. Kottwitz, Rational conjugacy classes in reductive groups,Duke Math. J.,49 (1982), 785–806. · Zbl 0506.20017 · doi:10.1215/S0012-7094-82-04939-0
[21] R. Kottwitz, Shimura varieties and twisted orbital integrals,Math. Ann.,269 (1984), 287–300. · Zbl 0547.14013 · doi:10.1007/BF01450697
[22] R. Kottwitz, Stable trace formula: cuspidal tempered terms,Duke Math. J.,51 (1984), 611–650. · Zbl 0576.22020 · doi:10.1215/S0012-7094-84-05129-9
[23] R. Kottwitz, Stable trace formula: elliptic singular terms,Math. Ann.,275 (1986), 365–399. · Zbl 0591.10020 · doi:10.1007/BF01458611
[24] R. Kottwitz, Shimura varieties and {\(\lambda\)}-adic representations,in [AA], I, 161–209. · Zbl 0743.14019
[25] R. Kottwitz, en préparation.
[26] R. Kottwitz,On the {\(\lambda\)}-adic representations associated to some simple Shimura varieties, preprint, 1989.
[27] J.-P. Labesse,Pseudo-coefficients très cuspidaux et K-théorie, preprint. · Zbl 0789.22028
[28] J.-P. Labesse, J. Schwermer, On liftings and cusp cohomology of arithmetic groups,Inv. Math.,83 (1983), 383–401. · Zbl 0581.10013 · doi:10.1007/BF01388968
[29] R. P. Langlands,Les débuts d’une formule des traces stable, Publ. Math. Univ. Paris 7, Paris, s.d. · Zbl 0532.22017
[30] R. P. Langlands, Automorphic representations, Shimura varieties, and motives, Ein Märchen,in [C], II, 205–246. · Zbl 0447.12009
[31] R. P. Langlands,On the classification of irreducible representations of real algebraic groups, Institute for Advanced Study, Princeton, 1973.
[32] C. Moeglin, J.-L. Waldspurger, Le spectre résiduel de GL(n),Ann. Sc. E.N.S. (4e sér.),22 (1989), 605–674. · Zbl 0696.10023
[33] J. Rogawski,Automorphic representations of unitary groups of three variables, Annals of Math. Studies, Princeton U. Press, 1989. · Zbl 0724.11031
[34] W. Scharlau,Quadratic and Hermitian Forms, Springer-Verlag, 1985. · Zbl 0584.10010
[35] D. Shelstad, Characters and inner forms of quasi-split groups overR,Compositio Math.,39 (1979), 11–45. · Zbl 0431.22011
[36] D. Shelstad, Base change and a matching theorem for real groups, inNon commutative Harmonic Analysis and Lie Groups, Springer Lecture Notes,880 (1981), 425–482. · doi:10.1007/BFb0090419
[37] D. Shelstad, Endoscopic groups and base changeC/R,Pacific J. Math.,110 (1984), 397–415. · Zbl 0488.22033
[38] D. Shelstad,Twisted endoscopic groups in the abelian case, non publié.
[39] M.-F. Vignéras,On the global correspondence between GL(n)and division algebras, Institute for Advanced Study, Princeton, 1984.
[40] D. Vogan,Representations of real reductive Lie groups, Birkhäuser, 1981. · Zbl 0469.22012
[41] D. Vogan, G. Zuckerman, Unitary representations with non-zero cohomology,Compositio Math.,53 (1984), 51–90. · Zbl 0692.22008
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