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Types et contragrédientes

  • Guy Henniart (a1) and Vincent Sécherre (a2)

Résumé

Soit $\text{G}$ un groupe réductif $p$ -adique, et soit $\text{R}$ un corps algébriquement clos. Soit $\text{ }\!\!\pi\!\!\text{ }$ une représentation lisse de $\text{G}$ dans un espace vectoriel $\text{V}$ sur $\text{R}$ . Fixons un sous-groupe ouvert et compact $\text{K}$ de $\text{G}$ et une représentation lisse irréductible $\varrho$ de $\text{K}$ dans un espace vectoriel $\text{W}$ de dimension finie sur $\text{R}$ . Sur l'espace $\text{Ho}{{\text{m}}_{\text{K}}}(\text{W,}\,\text{V)}$ agit l'algèbre d'entrelacement $\mathcal{H}(\text{G,}\,\text{K,}\,\text{W)}$ . Nous examinons la compatibilité de ces constructions avec le passage aux représentations contragrédientes ${{\text{V}}^{\vee }}$ et ${{\text{W}}^{\vee }}$ , et donnons en particulier des conditions sur $\text{W}$ ou sur la caractéristique de $\text{R}$ pour que le comportement soit semblable au cas des représentations complexes. Nous prenons un point de vue abstrait, n'utilisant que des propriétés générales de $\text{G}$ . Nous terminons par une application á la théorie des types pour le groupe $\text{G}{{\text{L}}_{n}}$ et ses formes intérieures sur un corps local non archimédien.

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References

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[1] Bourbaki, N., Algébre. Chapitre 8, seconde édition, Springer-Verlag, Berlin, 2012.
[2] Bourbaki, N., Théorie des ensembles. Springer-Verlag, Berlin, 2006.
[3] Bushnell, C. J. et Kutzko, P. C. , The admissible dual of GL(N) via compact open subgroups. Princeton University Press, Princeton, NJ, 1993.
[4] Bushnell, C. J., Smooth representations of reductive p-adic groups: structure theory via types. Proc. London Math. Soc. (3) 77 (1998), 582634.http://dx.doi.org/10.1112/S0024611598000574
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[6] Mínguez, A et Sécherre, V., Types modulo l pour les formes intérieures de GLn sur un corps local non archimédien. Prépublication, voir http://lmv.math.cnrs.fr/annuaire/vincent-secherre/.
[7] Mínguez, A et Sécherre, V. , Représentations lisses modulo l de GLm(D). Duke Math. J., á paraître.
[8] Tadić, M., Induced representations of GL(n,A) for p-adic division algebras A. J. Reine Angew. Math. 405(1990), 4877.
[9] Vignéras, M.-F., Représentations l-modulaires d’un groupe réductif p-adique avec l≠ p. Progr. Math. 137, Birkhäuser Boston Inc., Boston, MA, 1996.
[10] Vignéras, M.-F., Induced R-representations of p-adic reductive groups. Selecta Math. (N.S.) 4(1998),549623.With an appendix by Alberto Arabia. http://dx.doi.org/10.1007/s000290050040
[11] Vignéras, M.-F., Irreducible modular representations of a reductive p-adic group and simple modules for Hecke algebras. Progr. Math. 201, Birkhäuser, Basel, 2001, 117133.
[12] Zelevinsky, A. V., Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. (4) 13(1980), 165210.
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Types et contragrédientes

  • Guy Henniart (a1) and Vincent Sécherre (a2)

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