Représentations localement analytiques de GL 3 ( p )
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 1, pp. 43-145.

Nous construisons un complexe de représentations localement analytiques de GL 3 ( p ), associé à certaines représentations semi-stables de dimension 3 du groupe de Galois absolu de p . Nous montrons ensuite que l’on peut retrouver le (ϕ,N)-module filtré de la représentation galoisienne en considérant les morphismes, dans la catégorie dérivée des D( GL 3 ( p ))-modules, de ce complexe dans le complexe de de Rham de l’espace de Drinfel’d de dimension 2. 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 GL 3 ( p ), which is associated to some semi-stable 3-dimensional representations of the absolute Galois group of p . Then we show that we can retrieve the (ϕ,N)-filtered module of the Galois representation in the space of morphisms, in the derived category of D( GL 3 ( p ))-modules, of this complex in the de Rham-complex of the 2-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.

DOI : 10.24033/asens.2140
Classification : 11F70, 11S20, 11S37, 11S80, 14G22, 22E50
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://www.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://www.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://www.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://www.numdam.org/articles/10.24033/asens.2140/

[1] I. N. Bernstein & A. V. Zelevinsky, Induced representations of reductive 𝔭-adic groups. I, Ann. Sci. École Norm. Sup. 10 (1977), 441-472. | Numdam | Zbl

[2] S. J. Bloch, Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, CRM Monograph Series 11, Amer. Math. Soc., 2000. | Zbl

[3] A. Borel, Cohomologie de SL n et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 613-636. | Numdam | Zbl

[4] A. Borel & N. R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Math. Studies 94, Princeton Univ. Press, 1980. | Zbl

[5] N. Bourbaki, Topologie générale. Chapitre 9, Hermann, 1974.

[6] C. Breuil, Sur quelques représentations modulaires et p-adiques de GL 2 (𝐐 p ). II, J. Inst. Math. Jussieu 2 (2003), 23-58. | Zbl

[7] C. Breuil, Invariant et série spéciale p-adique, Ann. Sci. École Norm. Sup. 37 (2004), 559-610. | Zbl

[8] C. Breuil, Série spéciale p-adique et cohomologie étale complétée, Astérisque 331 (2010), 65-115. | Numdam | Zbl

[9] C. Breuil & A. Mézard, Représentations semi-stables de GL 2 ( p ), demi-plan p-adique et réduction modulo p, Astérisque 331 (2010), 117-178. | Numdam | MR | Zbl

[10] C. Breuil & P. Schneider, First steps towards p-adic Langlands functoriality, J. reine angew. Math. 610 (2007), 149-180. | Zbl

[11] C. J. Bushnell & G. Henniart, The local Langlands conjecture for GL (2), Grund. Math. Wiss. 335, Springer, 2006. | Zbl

[12] W. Casselman & D. Wigner, Continuous cohomology and a conjecture of Serre's, Invent. Math. 25 (1974), 199-211. | Zbl

[13] R. F. Coleman, Dilogarithms, regulators and p-adic L-functions, Invent. Math. 69 (1982), 171-208. | Zbl

[14] R. F. Coleman & A. Iovita, Hidden structures on semistable curves, Astérisque 331 (2010), 179-254. | Numdam | Zbl

[15] P. Colmez, Une correspondance de Langlands locale p-adique pour les représentations semi-stables de dimension 2, preprint, 2004.

[16] P. Colmez, La série principale unitaire de GL 2 (𝐐 p ), Astérisque 330 (2010), 213-262. | Numdam | Zbl

[17] P. Colmez, Représentations de GL 2 (𝐐 p ) et (φ,Γ)-modules, Astérisque 330 (2010), 281-509. | Numdam | Zbl

[18] P. Colmez & J.-M. Fontaine, Construction des représentations p-adiques semi-stables, Invent. Math. 140 (2000), 1-43. | MR | Zbl

[19] J. F. Dat, Espaces symétriques de Drinfeld et correspondance de Langlands locale, Ann. Sci. École Norm. Sup. 39 (2006), 1-74. | Zbl

[20] J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, 1974, Cahiers scientifiques, fasc. XXXVII. | Zbl

[21] M. Emerton, p-adic L-functions and unitary completions of representations of p-adic reductive groups, Duke Math. J. 130 (2005), 353-392. | MR | Zbl

[22] M. Emerton, Jacquet modules of locally analytic representations of p-adic reductive groups II. The relation to parabolic induction, à paraître dans J. Inst. Math. de Jussieu.

[23] J.-M. Fontaine, Représentations p-adiques semi-stables, Astérisque 223 (1994), 113-184. | MR | Zbl

[24] H. Frommer, 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] E. Grosse-Klönne, Frobenius and monodromy operators in rigid analysis, and Drinfelʼd's symmetric space, J. Algebraic Geom. 14 (2005), 391-437. | MR | Zbl

[26] E. Grosse-Klönne, On the p-adic cohomology of some p-adically uniformized varieties, J. Algebraic Geom. (2010). | MR | Zbl

[27] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Math. 9, Springer, 1978. | MR | Zbl

[28] J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category 𝒪, Graduate Studies in Math. 94, Amer. Math. Soc., 2008. | MR | Zbl

[29] A. Iovita & M. Spiess, Logarithmic differential forms on p-adic symmetric spaces, Duke Math. J. 110 (2001), 253-278. | MR | Zbl

[30] T. Ito, Weight-monodromy conjecture for p-adically uniformized varieties, Invent. Math. 159 (2005), 607-656. | MR | Zbl

[31] B. Keller, Derived categories and their uses, in Handbook of algebra, Vol. 1, North-Holland, 1996, 671-701. | MR | Zbl

[32] A. W. Knapp, Lie groups, Lie algebras, and cohomology, Mathematical Notes 34, Princeton Univ. Press, 1988. | MR | Zbl

[33] A. W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton Univ. Press, 2001. | MR | Zbl

[34] J. Kohlhaase, Invariant distributions on p-adic analytic groups, Duke Math. J. 137 (2007), 19-62. | MR | Zbl

[35] J. Kohlhaase, 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] J.-L. Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), 65-127. | Numdam | MR | Zbl

[37] C. T. Féaux De Lacroix, Einige Resultate über die topologischen Darstellungen p-adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem p-adischen Körper, Schriftenreihe Math. Inst. Univ. Münster 23 (1999), 1-111. | MR | Zbl

[38] C. C. Moore, Group extensions and cohomology for locally compact groups. III, Trans. Amer. Math. Soc. 221 (1976), 1-33. | MR | Zbl

[39] S. Orlik, On extensions of generalized Steinberg representations, J. Algebra 293 (2005), 611-630. | MR | Zbl

[40] S. Orlik, Equivariant vector bundles on Drinfeld's upper half space, Invent. Math. 172 (2008), 585-656. | MR | Zbl

[41] S. Orlik & M. Strauch, On Jordan-Hölder Series of some Locally Analytic Representations, preprint arXiv :1001.0323.

[42] S. Orlik & M. Strauch, On the irreducibility of locally analytic principal series representations, preprint arXiv :math/0612809, à paraître dans Representation Theory. | MR | Zbl

[43] D. Prasad, Locally algebraic representations of p-adic groups, Representation Theory 5 (2001), 111-128. | MR | Zbl

[44] P. Schneider, The cohomology of local systems on p-adically uniformized varieties, Math. Ann. 293 (1992), 623-650. | MR | Zbl

[45] P. Schneider, Nonarchimedean functional analysis, Springer Monographs in Math., Springer, 2002. | MR | Zbl

[46] P. Schneider & U. Stuhler, The cohomology of p-adic symmetric spaces, Invent. Math. 105 (1991), 47-122. | MR | Zbl

[47] P. Schneider & J. Teitelbaum, U(𝔤)-finite locally analytic representations, Represent. Theory 5 (2001), 111-128. | MR | Zbl

[48] P. Schneider & J. Teitelbaum, Locally analytic distributions and p-adic representation theory, with applications to GL 2 , J. Amer. Math. Soc. 15 (2002), 443-468. | MR | Zbl

[49] P. Schneider & J. Teitelbaum, p-adic boundary values, Astérisque 278 (2002), 51-125. | Numdam | MR | Zbl

[50] P. Schneider & J. Teitelbaum, Algebras of p-adic distributions and admissible representations, Invent. Math. 153 (2003), 145-196. | MR | Zbl

[51] P. Schneider & J. Teitelbaum, Duality for admissible locally analytic representations, Represent. Theory 9 (2005), 297-326. | MR | Zbl

[52] B. Schraen, Représentations p-adiques de GL 2 (L) et catégories dérivées, Israel J. Math. 176 (2010), 307-361. | MR | Zbl

[53] E. De Shalit, The p-adic monodromy-weight conjecture for p-adically uniformized varieties, Compos. Math. 141 (2005), 101-120. | MR | Zbl

[54] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Math. 38, Cambridge Univ. Press, 1994. | MR | Zbl

[55] N. Yoneda, On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 507-576. | MR | Zbl

Cité par Sources :