Nous construisons un complexe de représentations localement analytiques de , associé à certaines représentations semi-stables de dimension du groupe de Galois absolu de . Nous montrons ensuite que l’on peut retrouver le -module filtré de la représentation galoisienne en considérant les morphismes, dans la catégorie dérivée des -modules, de ce complexe dans le complexe de de Rham de l’espace de Drinfel’d de dimension . La preuve requiert le calcul de certains espaces de cohomologie localement analytiques de sous-groupes unipotents à coefficients dans des séries principales localement analytiques.
We construct a complex of locally analytic representations of , which is associated to some semi-stable -dimensional representations of the absolute Galois group of . Then we show that we can retrieve the -filtered module of the Galois representation in the space of morphisms, in the derived category of -modules, of this complex in the de Rham-complex of the -dimensional Drinfel’d’s space. For the proof, we compute some spaces of locally analytic cohomology of unipotent subgroups with coefficients in some locally analytic principal series.
Mot clés : correspondance de Langlands $p$-adique, espaces de Drinfel’d, représentations localement analytiques $p$-adiques
Keywords: $p$-adic Langlands correspondence, Drinfel’d’s spaces, $p$-adic locally analytic representations
@article{ASENS_2011_4_44_1_43_0, author = {Schraen, Benjamin}, title = {Repr\'esentations localement analytiques de $\mathrm {GL}_3(\mathbb {Q}_{p})$}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {43--145}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {4e s{\'e}rie, 44}, number = {1}, year = {2011}, doi = {10.24033/asens.2140}, mrnumber = {2760195}, zbl = {1235.11108}, language = {fr}, url = {http://archive.numdam.org/articles/10.24033/asens.2140/} }
TY - JOUR AU - Schraen, Benjamin TI - Représentations localement analytiques de $\mathrm {GL}_3(\mathbb {Q}_{p})$ JO - Annales scientifiques de l'École Normale Supérieure PY - 2011 SP - 43 EP - 145 VL - 44 IS - 1 PB - Société mathématique de France UR - http://archive.numdam.org/articles/10.24033/asens.2140/ DO - 10.24033/asens.2140 LA - fr ID - ASENS_2011_4_44_1_43_0 ER -
%0 Journal Article %A Schraen, Benjamin %T Représentations localement analytiques de $\mathrm {GL}_3(\mathbb {Q}_{p})$ %J Annales scientifiques de l'École Normale Supérieure %D 2011 %P 43-145 %V 44 %N 1 %I Société mathématique de France %U http://archive.numdam.org/articles/10.24033/asens.2140/ %R 10.24033/asens.2140 %G fr %F ASENS_2011_4_44_1_43_0
Schraen, Benjamin. Représentations localement analytiques de $\mathrm {GL}_3(\mathbb {Q}_{p})$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 1, pp. 43-145. doi : 10.24033/asens.2140. http://archive.numdam.org/articles/10.24033/asens.2140/
[1] Induced representations of reductive -adic groups. I, Ann. Sci. École Norm. Sup. 10 (1977), 441-472. | Numdam | Zbl
& ,[2] Higher regulators, algebraic -theory, and zeta functions of elliptic curves, CRM Monograph Series 11, Amer. Math. Soc., 2000. | Zbl
,[3] Cohomologie de et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 613-636. | Numdam | Zbl
,[4] Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Math. Studies 94, Princeton Univ. Press, 1980. | Zbl
& ,[5] Topologie générale. Chapitre 9, Hermann, 1974.
,[6] Sur quelques représentations modulaires et -adiques de . II, J. Inst. Math. Jussieu 2 (2003), 23-58. | Zbl
,[7] Invariant et série spéciale -adique, Ann. Sci. École Norm. Sup. 37 (2004), 559-610. | Zbl
,[8] Série spéciale -adique et cohomologie étale complétée, Astérisque 331 (2010), 65-115. | Numdam | Zbl
,[9] Représentations semi-stables de , demi-plan -adique et réduction modulo , Astérisque 331 (2010), 117-178. | Numdam | MR | Zbl
& ,[10] First steps towards -adic Langlands functoriality, J. reine angew. Math. 610 (2007), 149-180. | Zbl
& ,[11] The local Langlands conjecture for , Grund. Math. Wiss. 335, Springer, 2006. | Zbl
& ,[12] Continuous cohomology and a conjecture of Serre's, Invent. Math. 25 (1974), 199-211. | Zbl
& ,[13] Dilogarithms, regulators and -adic -functions, Invent. Math. 69 (1982), 171-208. | Zbl
,[14] Hidden structures on semistable curves, Astérisque 331 (2010), 179-254. | Numdam | Zbl
& ,[15] Une correspondance de Langlands locale -adique pour les représentations semi-stables de dimension , preprint, 2004.
,[16] La série principale unitaire de , Astérisque 330 (2010), 213-262. | Numdam | Zbl
,[17] Représentations de et -modules, Astérisque 330 (2010), 281-509. | Numdam | Zbl
,[18] Construction des représentations -adiques semi-stables, Invent. Math. 140 (2000), 1-43. | MR | Zbl
& ,[19] Espaces symétriques de Drinfeld et correspondance de Langlands locale, Ann. Sci. École Norm. Sup. 39 (2006), 1-74. | Zbl
,[20] Algèbres enveloppantes, Gauthier-Villars, 1974, Cahiers scientifiques, fasc. XXXVII. | Zbl
,[21] -adic -functions and unitary completions of representations of -adic reductive groups, Duke Math. J. 130 (2005), 353-392. | MR | Zbl
,[22] Jacquet modules of locally analytic representations of -adic reductive groups II. The relation to parabolic induction, à paraître dans J. Inst. Math. de Jussieu.
,[23] Représentations -adiques semi-stables, Astérisque 223 (1994), 113-184. | Numdam | MR | Zbl
,[24] The locally analytic principal series of split reductive groups, preprint SFB 478/265 http://wwwmath.uni-muenster.de/sfb/about/publ/heft265.ps, 2003.
,[25] Frobenius and monodromy operators in rigid analysis, and Drinfelʼd's symmetric space, J. Algebraic Geom. 14 (2005), 391-437. | MR | Zbl
,[26] On the -adic cohomology of some -adically uniformized varieties, J. Algebraic Geom. (2010). | MR | Zbl
,[27] Introduction to Lie algebras and representation theory, Graduate Texts in Math. 9, Springer, 1978. | MR | Zbl
,[28] Representations of semisimple Lie algebras in the BGG category , Graduate Studies in Math. 94, Amer. Math. Soc., 2008. | MR | Zbl
,[29] Logarithmic differential forms on -adic symmetric spaces, Duke Math. J. 110 (2001), 253-278. | MR | Zbl
& ,[30] Weight-monodromy conjecture for -adically uniformized varieties, Invent. Math. 159 (2005), 607-656. | MR | Zbl
,[31] Derived categories and their uses, in Handbook of algebra, Vol. 1, North-Holland, 1996, 671-701. | MR | Zbl
,[32] Lie groups, Lie algebras, and cohomology, Mathematical Notes 34, Princeton Univ. Press, 1988. | MR | Zbl
,[33] Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton Univ. Press, 2001. | MR | Zbl
,[34] Invariant distributions on -adic analytic groups, Duke Math. J. 137 (2007), 19-62. | MR | Zbl
,[35] The cohomology of locally analytic representations, preprint SFB 478/491 http://wwwmath.uni-muenster.de/sfb/about/publ/heft491.pdf, à paraître dans J. reine angew. Math. | MR | Zbl
,[36] Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), 65-127. | Numdam | MR | Zbl
,[37] Einige Resultate über die topologischen Darstellungen -adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem -adischen Körper, Schriftenreihe Math. Inst. Univ. Münster 23 (1999), 1-111. | MR | Zbl
,[38] Group extensions and cohomology for locally compact groups. III, Trans. Amer. Math. Soc. 221 (1976), 1-33. | MR | Zbl
,[39] On extensions of generalized Steinberg representations, J. Algebra 293 (2005), 611-630. | MR | Zbl
,[40] Equivariant vector bundles on Drinfeld's upper half space, Invent. Math. 172 (2008), 585-656. | MR | Zbl
,[41] On Jordan-Hölder Series of some Locally Analytic Representations, preprint arXiv :1001.0323.
& ,[42] On the irreducibility of locally analytic principal series representations, preprint arXiv :math/0612809, à paraître dans Representation Theory. | MR | Zbl
& ,[43] Locally algebraic representations of -adic groups, Representation Theory 5 (2001), 111-128. | MR | Zbl
,[44] The cohomology of local systems on -adically uniformized varieties, Math. Ann. 293 (1992), 623-650. | MR | Zbl
,[45] Nonarchimedean functional analysis, Springer Monographs in Math., Springer, 2002. | MR | Zbl
,[46] The cohomology of -adic symmetric spaces, Invent. Math. 105 (1991), 47-122. | MR | Zbl
& ,[47] -finite locally analytic representations, Represent. Theory 5 (2001), 111-128. | MR | Zbl
& ,[48] Locally analytic distributions and -adic representation theory, with applications to , J. Amer. Math. Soc. 15 (2002), 443-468. | MR | Zbl
& ,[49] -adic boundary values, Astérisque 278 (2002), 51-125. | Numdam | MR | Zbl
& ,[50] Algebras of -adic distributions and admissible representations, Invent. Math. 153 (2003), 145-196. | MR | Zbl
& ,[51] Duality for admissible locally analytic representations, Represent. Theory 9 (2005), 297-326. | MR | Zbl
& ,[52] Représentations -adiques de et catégories dérivées, Israel J. Math. 176 (2010), 307-361. | MR | Zbl
,[53] The -adic monodromy-weight conjecture for -adically uniformized varieties, Compos. Math. 141 (2005), 101-120. | MR | Zbl
,[54] An introduction to homological algebra, Cambridge Studies in Advanced Math. 38, Cambridge Univ. Press, 1994. | MR | Zbl
,[55] On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 507-576. | MR | Zbl
,Cité par Sources :