Locally analytic representations of GL 3 ( ℚ p )

Benjamin Schraen

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 1, page 43-145
  • ISSN: 0012-9593

Abstract

top
We construct a complex of locally analytic representations of GL 3 ( ℚ p ) , which is associated to some semi-stable 3 -dimensional representations of the absolute Galois group of ℚ 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 ( ℚ 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.

How to cite

top

Schraen, Benjamin. "Représentations localement analytiques de $\mathrm {GL}_3(\mathbb {Q}_{p})$." Annales scientifiques de l'École Normale Supérieure 44.1 (2011): 43-145. <http://eudml.org/doc/272235>.

@article{Schraen2011,
abstract = {Nous construisons un complexe de représentations localement analytiques de $\mathrm \{GL\}_3(\mathbb \{Q\}_\{p\})$, associé à certaines représentations semi-stables de dimension $3$ du groupe de Galois absolu de $\mathbb \{Q\}_\{p\}$. Nous montrons ensuite que l’on peut retrouver le $(\varphi ,N)$-module filtré de la représentation galoisienne en considérant les morphismes, dans la catégorie dérivée des $D(\mathrm \{GL\}_3(\mathbb \{Q\}_\{p\}))$-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.},
author = {Schraen, Benjamin},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {$p$-adic Langlands correspondence; Drinfel’d’s spaces; $p$-adic locally analytic representations},
language = {fre},
number = {1},
pages = {43-145},
publisher = {Société mathématique de France},
title = {Représentations localement analytiques de $\mathrm \{GL\}_3(\mathbb \{Q\}_\{p\})$},
url = {http://eudml.org/doc/272235},
volume = {44},
year = {2011},
}

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
PB - Société mathématique de France
VL - 44
IS - 1
SP - 43
EP - 145
AB - Nous construisons un complexe de représentations localement analytiques de $\mathrm {GL}_3(\mathbb {Q}_{p})$, associé à certaines représentations semi-stables de dimension $3$ du groupe de Galois absolu de $\mathbb {Q}_{p}$. Nous montrons ensuite que l’on peut retrouver le $(\varphi ,N)$-module filtré de la représentation galoisienne en considérant les morphismes, dans la catégorie dérivée des $D(\mathrm {GL}_3(\mathbb {Q}_{p}))$-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.
LA - fre
KW - $p$-adic Langlands correspondence; Drinfel’d’s spaces; $p$-adic locally analytic representations
UR - http://eudml.org/doc/272235
ER -

References

top
  1. [1] I. N. Bernstein & A. V. Zelevinsky, Induced representations of reductive 𝔭 -adic groups. I, Ann. Sci. École Norm. Sup. 10 (1977), 441–472. Zbl0412.22015
  2. [2] S. J. Bloch, Higher regulators, algebraic K -theory, and zeta functions of elliptic curves, CRM Monograph Series 11, Amer. Math. Soc., 2000. Zbl0958.19001
  3. [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. Zbl0382.57027
  4. [4] A. Borel & N. R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Math. Studies 94, Princeton Univ. Press, 1980. Zbl0443.22010
  5. [5] N. Bourbaki, Topologie générale. Chapitre 9, Hermann, 1974. 
  6. [6] C. Breuil, Sur quelques représentations modulaires et p -adiques de GL 2 ( 𝐐 p ) . II, J. Inst. Math. Jussieu 2 (2003), 23–58. Zbl1165.11319
  7. [7] C. Breuil, Invariant ℒ et série spéciale p -adique, Ann. Sci. École Norm. Sup.37 (2004), 559–610. Zbl1166.11331
  8. [8] C. Breuil, Série spéciale p -adique et cohomologie étale complétée, Astérisque331 (2010), 65–115. Zbl1246.11106
  9. [9] C. Breuil & A. Mézard, Représentations semi-stables de GL 2 ( ℚ p ) , demi-plan p -adique et réduction modulo p , Astérisque331 (2010), 117–178. Zbl1271.11106
  10. [10] C. Breuil & P. Schneider, First steps towards p -adic Langlands functoriality, J. reine angew. Math. 610 (2007), 149–180. Zbl1180.11036
  11. [11] C. J. Bushnell & G. Henniart, The local Langlands conjecture for GL ( 2 ) , Grund. Math. Wiss. 335, Springer, 2006. Zbl1100.11041
  12. [12] W. Casselman & D. Wigner, Continuous cohomology and a conjecture of Serre’s, Invent. Math.25 (1974), 199–211. Zbl0297.20060
  13. [13] R. F. Coleman, Dilogarithms, regulators and p -adic L -functions, Invent. Math.69 (1982), 171–208. Zbl0516.12017
  14. [14] R. F. Coleman & A. Iovita, Hidden structures on semistable curves, Astérisque331 (2010), 179–254. Zbl1251.11047
  15. [15] P. Colmez, Une correspondance de Langlands locale p -adique pour les représentations semi-stables de dimension 2 , preprint, 2004. 
  16. [16] P. Colmez, La série principale unitaire de GL 2 ( 𝐐 p ) , Astérisque330 (2010), 213–262. Zbl1242.11095
  17. [17] P. Colmez, Représentations de GL 2 ( 𝐐 p ) et ( φ , Γ ) -modules, Astérisque330 (2010), 281–509. Zbl1218.11107
  18. [18] P. Colmez & J.-M. Fontaine, Construction des représentations p -adiques semi-stables, Invent. Math.140 (2000), 1–43. Zbl1010.14004MR1779803
  19. [19] J. F. Dat, Espaces symétriques de Drinfeld et correspondance de Langlands locale, Ann. Sci. École Norm. Sup.39 (2006), 1–74. Zbl1141.22004
  20. [20] J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, 1974, Cahiers scientifiques, fasc. XXXVII. Zbl0308.17007
  21. [21] M. Emerton, p -adic L -functions and unitary completions of representations of p -adic reductive groups, Duke Math. J.130 (2005), 353–392. Zbl1092.11024MR2181093
  22. [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. [23] J.-M. Fontaine, Représentations p -adiques semi-stables, Astérisque223 (1994), 113–184. Zbl0865.14009MR1293972
  24. [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. [25] E. Grosse-Klönne, Frobenius and monodromy operators in rigid analysis, and Drinfelʼd’s symmetric space, J. Algebraic Geom.14 (2005), 391–437. Zbl1084.14021MR2129006
  26. [26] E. Grosse-Klönne, On the p -adic cohomology of some p -adically uniformized varieties, J. Algebraic Geom. (2010). Zbl1210.14022MR2729278
  27. [27] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Math. 9, Springer, 1978. Zbl0447.17001MR499562
  28. [28] J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category 𝒪 , Graduate Studies in Math. 94, Amer. Math. Soc., 2008. Zbl1177.17001MR2428237
  29. [29] A. Iovita & M. Spiess, Logarithmic differential forms on p -adic symmetric spaces, Duke Math. J.110 (2001), 253–278. Zbl1100.14505MR1865241
  30. [30] T. Ito, Weight-monodromy conjecture for p -adically uniformized varieties, Invent. Math.159 (2005), 607–656. Zbl1154.14014MR2125735
  31. [31] B. Keller, Derived categories and their uses, in Handbook of algebra, Vol. 1, North-Holland, 1996, 671–701. Zbl0862.18001MR1421815
  32. [32] A. W. Knapp, Lie groups, Lie algebras, and cohomology, Mathematical Notes 34, Princeton Univ. Press, 1988. Zbl0648.22010MR938524
  33. [33] A. W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton Univ. Press, 2001. Zbl0993.22001MR1880691
  34. [34] J. Kohlhaase, Invariant distributions on p -adic analytic groups, Duke Math. J.137 (2007), 19–62. Zbl1133.11066MR2309143
  35. [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. Zbl1226.22020MR2774315
  36. [36] J.-L. Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France78 (1950), 65–127. Zbl0039.02901MR36511
  37. [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. Münster23 (1999), 1–111. Zbl0963.22009MR1691735
  38. [38] C. C. Moore, Group extensions and cohomology for locally compact groups. III, Trans. Amer. Math. Soc. 221 (1976), 1–33. Zbl0366.22005MR414775
  39. [39] S. Orlik, On extensions of generalized Steinberg representations, J. Algebra293 (2005), 611–630. Zbl1080.22008MR2173717
  40. [40] S. Orlik, Equivariant vector bundles on Drinfeld’s upper half space, Invent. Math.172 (2008), 585–656. Zbl1136.22009MR2393081
  41. [41] S. Orlik & M. Strauch, On Jordan-Hölder Series of some Locally Analytic Representations, preprint arXiv :1001.0323. Zbl1307.22009
  42. [42] S. Orlik & M. Strauch, On the irreducibility of locally analytic principal series representations, preprint arXiv :math/0612809, à paraître dans Representation Theory. Zbl1247.22018MR2738585
  43. [43] D. Prasad, Locally algebraic representations of p -adic groups, Representation Theory5 (2001), 111–128. Zbl1028.17007MR1835001
  44. [44] P. Schneider, The cohomology of local systems on p -adically uniformized varieties, Math. Ann.293 (1992), 623–650. Zbl0774.14022MR1176024
  45. [45] P. Schneider, Nonarchimedean functional analysis, Springer Monographs in Math., Springer, 2002. Zbl0998.46044MR1869547
  46. [46] P. Schneider & U. Stuhler, The cohomology of p -adic symmetric spaces, Invent. Math.105 (1991), 47–122. Zbl0751.14016MR1109620
  47. [47] P. Schneider & J. Teitelbaum, U ( 𝔤 ) -finite locally analytic representations, Represent. Theory5 (2001), 111–128. Zbl1028.17007MR1835001
  48. [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. Zbl1028.11071MR1887640
  49. [49] P. Schneider & J. Teitelbaum, p -adic boundary values, Astérisque278 (2002), 51–125. Zbl1051.14024MR1922824
  50. [50] P. Schneider & J. Teitelbaum, Algebras of p -adic distributions and admissible representations, Invent. Math.153 (2003), 145–196. Zbl1028.11070MR1990669
  51. [51] P. Schneider & J. Teitelbaum, Duality for admissible locally analytic representations, Represent. Theory9 (2005), 297–326. Zbl1146.22301MR2133762
  52. [52] B. Schraen, Représentations p -adiques de GL 2 ( L ) et catégories dérivées, Israel J. Math.176 (2010), 307–361. Zbl1210.11066MR2653197
  53. [53] E. de Shalit, The p -adic monodromy-weight conjecture for p -adically uniformized varieties, Compos. Math.141 (2005), 101–120. Zbl1087.14019MR2099771
  54. [54] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Math. 38, Cambridge Univ. Press, 1994. Zbl0797.18001MR1269324
  55. [55] N. Yoneda, On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I8 (1960), 507–576. Zbl0163.26902MR225854

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.