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 : https://doi.org/10.24033/asens.2140
Classification : 11F70,  11S20,  11S37,  11S80,  14G22,  22E50
Mots clés : correspondance de Langlands p-adique, espaces de Drinfel’d, représentations localement analytiques p-adiques
@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},
     zbl = {1235.11108},
     mrnumber = {2760195},
     language = {fr},
     url = {www.numdam.org/item/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/item/ASENS_2011_4_44_1_43_0/

[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 0412.22015

[2] S. J. Bloch, Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, CRM Monograph Series 11, Amer. Math. Soc., 2000. | Zbl 0958.19001

[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 0382.57027

[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 0443.22010

[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 1165.11319

[7] C. Breuil, Invariant et série spéciale p-adique, Ann. Sci. École Norm. Sup. 37 (2004), 559-610. | Zbl 1166.11331

[8] C. Breuil, Série spéciale p-adique et cohomologie étale complétée, Astérisque 331 (2010), 65-115. | Zbl 1246.11106

[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. | Zbl 1271.11106

[10] C. Breuil & P. Schneider, First steps towards p-adic Langlands functoriality, J. reine angew. Math. 610 (2007), 149-180. | Zbl 1180.11036

[11] C. J. Bushnell & G. Henniart, The local Langlands conjecture for GL (2), Grund. Math. Wiss. 335, Springer, 2006. | Zbl 1100.11041

[12] W. Casselman & D. Wigner, Continuous cohomology and a conjecture of Serre's, Invent. Math. 25 (1974), 199-211. | Zbl 0297.20060

[13] R. F. Coleman, Dilogarithms, regulators and p-adic L-functions, Invent. Math. 69 (1982), 171-208. | Zbl 0516.12017

[14] R. F. Coleman & A. Iovita, Hidden structures on semistable curves, Astérisque 331 (2010), 179-254. | Zbl 1251.11047

[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. | Zbl 1242.11095

[17] P. Colmez, Représentations de GL 2 (𝐐 p ) et (φ,Γ)-modules, Astérisque 330 (2010), 281-509. | Zbl 1218.11107

[18] P. Colmez & J.-M. Fontaine, Construction des représentations p-adiques semi-stables, Invent. Math. 140 (2000), 1-43. | MR 1779803 | Zbl 1010.14004

[19] J. F. Dat, Espaces symétriques de Drinfeld et correspondance de Langlands locale, Ann. Sci. École Norm. Sup. 39 (2006), 1-74. | Zbl 1141.22004

[20] J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, 1974, Cahiers scientifiques, fasc. XXXVII. | Zbl 0308.17007

[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 2181093 | Zbl 1092.11024

[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 1293972 | Zbl 0865.14009

[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 2129006 | Zbl 1084.14021

[26] E. Grosse-Klönne, On the p-adic cohomology of some p-adically uniformized varieties, J. Algebraic Geom. (2010). | MR 2729278 | Zbl 1210.14022

[27] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Math. 9, Springer, 1978. | MR 499562 | Zbl 0447.17001

[28] J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category 𝒪, Graduate Studies in Math. 94, Amer. Math. Soc., 2008. | MR 2428237 | Zbl 1177.17001

[29] A. Iovita & M. Spiess, Logarithmic differential forms on p-adic symmetric spaces, Duke Math. J. 110 (2001), 253-278. | MR 1865241 | Zbl 1100.14505

[30] T. Ito, Weight-monodromy conjecture for p-adically uniformized varieties, Invent. Math. 159 (2005), 607-656. | MR 2125735 | Zbl 1154.14014

[31] B. Keller, Derived categories and their uses, in Handbook of algebra, Vol. 1, North-Holland, 1996, 671-701. | MR 1421815 | Zbl 0862.18001

[32] A. W. Knapp, Lie groups, Lie algebras, and cohomology, Mathematical Notes 34, Princeton Univ. Press, 1988. | MR 938524 | Zbl 0648.22010

[33] A. W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton Univ. Press, 2001. | MR 1880691 | Zbl 0993.22001

[34] J. Kohlhaase, Invariant distributions on p-adic analytic groups, Duke Math. J. 137 (2007), 19-62. | MR 2309143 | Zbl 1133.11066

[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 2774315 | Zbl 1226.22020

[36] J.-L. Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), 65-127. | Numdam | MR 36511 | Zbl 0039.02901

[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 1691735 | Zbl 0963.22009

[38] C. C. Moore, Group extensions and cohomology for locally compact groups. III, Trans. Amer. Math. Soc. 221 (1976), 1-33. | MR 414775 | Zbl 0366.22005

[39] S. Orlik, On extensions of generalized Steinberg representations, J. Algebra 293 (2005), 611-630. | MR 2173717 | Zbl 1080.22008

[40] S. Orlik, Equivariant vector bundles on Drinfeld's upper half space, Invent. Math. 172 (2008), 585-656. | MR 2393081 | Zbl 1136.22009

[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 2738585 | Zbl 1247.22018

[43] D. Prasad, Locally algebraic representations of p-adic groups, Representation Theory 5 (2001), 111-128. | MR 1835001 | Zbl 1028.17007

[44] P. Schneider, The cohomology of local systems on p-adically uniformized varieties, Math. Ann. 293 (1992), 623-650. | MR 1176024 | Zbl 0774.14022

[45] P. Schneider, Nonarchimedean functional analysis, Springer Monographs in Math., Springer, 2002. | MR 1869547 | Zbl 0998.46044

[46] P. Schneider & U. Stuhler, The cohomology of p-adic symmetric spaces, Invent. Math. 105 (1991), 47-122. | MR 1109620 | Zbl 0751.14016

[47] P. Schneider & J. Teitelbaum, U(𝔤)-finite locally analytic representations, Represent. Theory 5 (2001), 111-128. | MR 1835001 | Zbl 1028.17007

[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 1887640 | Zbl 1028.11071

[49] P. Schneider & J. Teitelbaum, p-adic boundary values, Astérisque 278 (2002), 51-125. | MR 1922824 | Zbl 1051.14024

[50] P. Schneider & J. Teitelbaum, Algebras of p-adic distributions and admissible representations, Invent. Math. 153 (2003), 145-196. | MR 1990669 | Zbl 1028.11070

[51] P. Schneider & J. Teitelbaum, Duality for admissible locally analytic representations, Represent. Theory 9 (2005), 297-326. | MR 2133762 | Zbl 1146.22301

[52] B. Schraen, Représentations p-adiques de GL 2 (L) et catégories dérivées, Israel J. Math. 176 (2010), 307-361. | MR 2653197 | Zbl 1210.11066

[53] E. De Shalit, The p-adic monodromy-weight conjecture for p-adically uniformized varieties, Compos. Math. 141 (2005), 101-120. | MR 2099771 | Zbl 1087.14019

[54] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Math. 38, Cambridge Univ. Press, 1994. | MR 1269324 | Zbl 0797.18001

[55] N. Yoneda, On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 507-576. | MR 225854 | Zbl 0163.26902