Socle localement analytique I
Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 633-685.

Soit L une extension finie de Q p et n un entier >0. À toute filtration de Hodge de poids de Hodge-Tate distincts sur une représentation de rang n suffisamment générique du groupe de Weil-Deligne de L, on associe une représentation localement Q p -analytique semi-simple de longueur finie de GL n (L). On montre plusieurs propriétés de cette représentation. Par exemple, lorsqu’elle possède un réseau stable par GL n (L), alors la filtration de départ est faiblement admissible.

Let L be a finite extension of Q p and n a positive integer. To each Hodge filtration with distinct Hodge-Tate weights on an n-dimensional sufficiently generic representation of the Weil-Deligne group of L, we associate a semi-simple finite length locally Q p -analytic representation of GL n (L). We show several properties of this representation of GL n (L). For instance, if it has an invariant lattice, then the starting Hodge filtration is weakly admissible.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3021
Classification : 11S23, 22E35, 22E50
Mot clés : Représentation localement analytique, filtration de Hodge, socle
Keywords: Locally analytic representation, Hodge filtration, socle
Breuil, Christophe 1

1 Bâtiment 425 C.N.R.S. et Université Paris-Sud 91405 Orsay Cedex France
@article{AIF_2016__66_2_633_0,
     author = {Breuil, Christophe},
     title = {Socle localement analytique {I}},
     journal = {Annales de l'Institut Fourier},
     pages = {633--685},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {2},
     year = {2016},
     doi = {10.5802/aif.3021},
     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3021/}
}
TY  - JOUR
AU  - Breuil, Christophe
TI  - Socle localement analytique I
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 633
EP  - 685
VL  - 66
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3021/
DO  - 10.5802/aif.3021
LA  - fr
ID  - AIF_2016__66_2_633_0
ER  - 
%0 Journal Article
%A Breuil, Christophe
%T Socle localement analytique I
%J Annales de l'Institut Fourier
%D 2016
%P 633-685
%V 66
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3021/
%R 10.5802/aif.3021
%G fr
%F AIF_2016__66_2_633_0
Breuil, Christophe. Socle localement analytique I. Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 633-685. doi : 10.5802/aif.3021. https://aif.centre-mersenne.org/articles/10.5802/aif.3021/

[1] Breuil, Christophe Remarks on some locally p -analytic representations of GL 2 (F) in the crystalline case, Non-abelian fundamental groups and Iwasawa theory (London Math. Soc. Lecture Note Ser.), Volume 393, Cambridge Univ. Press, Cambridge, 2012, pp. 212-238

[2] Breuil, Christophe Vers le socle localement analytique pour GL n II, Math. Ann., Volume 361 (2015) no. 3-4, pp. 741-785 | DOI

[3] Breuil, Christophe; Herzig, Florian Ordinary representations of G( p ) and fundamental algebraic representations, Duke Math. J., Volume 164 (2015) no. 7, pp. 1271-1352 | DOI

[4] Breuil, Christophe; Mézard, Ariane Multiplicités modulaires et représentations de GL 2 (Z p ) et de Gal (Q ¯ p /Q p ) en l=p, Duke Math. J., Volume 115 (2002) no. 2, pp. 205-310 (With an appendix by Guy Henniart) | DOI

[5] Breuil, Christophe; Paškunas, Vytautas Towards a modulo p Langlands correspondence for GL 2 , Mem. Amer. Math. Soc., Volume 216 (2012) no. 1016, vi+114 pages | DOI

[6] Breuil, Christophe; Schneider, Peter First steps towards p-adic Langlands functoriality, J. Reine Angew. Math., Volume 610 (2007), pp. 149-180 | DOI

[7] Bushnell, Colin J. Representations of reductive p-adic groups : localization of Hecke algebras and applications, J. London Math. Soc. (2), Volume 63 (2001) no. 2, pp. 364-386 | DOI

[8] Buzzard, Kevin; Diamond, Fred; Jarvis, Frazer On Serre’s conjecture for mod Galois representations over totally real fields, Duke Math. J., Volume 155 (2010) no. 1, pp. 105-161 | DOI

[9] Colmez, Pierre La série principale unitaire de GL 2 ( p ), Astérisque (2010) no. 330, pp. 213-262

[10] Colmez, Pierre Représentations de GL 2 ( p ) et (φ,Γ)-modules, Astérisque (2010) no. 330, pp. 281-509

[11] Colmez, Pierre; Fontaine, Jean-Marc Construction des représentations p-adiques semi-stables, Invent. Math., Volume 140 (2000) no. 1, pp. 1-43 | DOI

[12] Digne, François; Michel, Jean Representations of finite groups of Lie type, London Mathematical Society Student Texts, 21, Cambridge University Press, Cambridge, 1991, iv+159 pages | DOI

[13] Emerton, Matthew Jacquet modules of locally analytic representations of p-adic reductive groups II. The relation to parabolic induction (à paraître à J. Institut Math. Jussieu.)

[14] Emerton, Matthew Jacquet modules of locally analytic representations of p-adic reductive groups. I. Construction and first properties, Ann. Sci. École Norm. Sup. (4), Volume 39 (2006) no. 5, pp. 775-839 | DOI

[15] Emerton, Matthew Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties, Astérisque (2010) no. 331, pp. 355-402

[16] Fontaine, Jean-Marc Représentations l-adiques potentiellement semi-stables, Astérisque (1994) no. 223, pp. 321-347 Périodes p-adiques (Bures-sur-Yvette, 1988)

[17] Fontaine, Jean-Marc Représentations p-adiques semi-stables, Astérisque (1994) no. 223, pp. 113-184 With an appendix by Pierre Colmez, Périodes p-adiques (Bures-sur-Yvette, 1988)

[18] Gee, Toby Automorphic lifts of prescribed types, Math. Ann., Volume 350 (2011) no. 1, pp. 107-144 | DOI

[19] Gee, Toby On the weights of mod p Hilbert modular forms, Invent. Math., Volume 184 (2011) no. 1, pp. 1-46 | DOI

[20] Gee, Toby; Kisin, Mark The Breuil-Mézard conjecture for potentially Barsotti-Tate representations, Forum Math. Pi, Volume 2 (2014), e1, 56 pages | DOI

[21] Herzig, Florian The weight in a Serre-type conjecture for tame n-dimensional Galois representations, Duke Math. J., Volume 149 (2009) no. 1, pp. 37-116 | DOI

[22] Hu, Yongquan Normes invariantes et existence de filtrations admissibles, J. Reine Angew. Math., Volume 634 (2009), pp. 107-141 | DOI

[23] Humphreys, James E. Representations of semisimple Lie algebras in the BGG category 𝒪, Graduate Studies in Mathematics, 94, American Mathematical Society, Providence, RI, 2008, xvi+289 pages | DOI

[24] Kohlhaase, Jan Invariant distributions on p-adic analytic groups, Duke Math. J., Volume 137 (2007) no. 1, pp. 19-62 | DOI

[25] Liu, Ruochuan Locally analytic vectors of some crystabelian representations of GL 2 ( p ), Compos. Math., Volume 148 (2012) no. 1, pp. 28-64 | DOI

[26] Orlik, Sascha; Schraen, Benjamin The Jordan-Hölder series of the locally analytic Steinberg representation, Doc. Math., Volume 19 (2014), pp. 647-671

[27] Orlik, Sascha; Strauch, Matthias On the irreducibility of locally analytic principal series representations, Represent. Theory, Volume 14 (2010), pp. 713-746 | DOI

[28] Orlik, Sascha; Strauch, Matthias On Jordan-Hölder series of some locally analytic representations, J. Amer. Math. Soc., Volume 28 (2015) no. 1, pp. 99-157 | DOI

[29] Prasad, Dipendra Locally algebraic representations of p-adic groups, Represent. Theory, Volume 5 (2001), pp. 111-128 appendice à U(g)-finite locally analytic representations (Schneider P., Teitelbaum J.) | DOI

[30] Schein, Michael M. Weights in Serre’s conjecture for Hilbert modular forms : the ramified case, Israel J. Math., Volume 166 (2008), pp. 369-391 | DOI

[31] Schmidt, Tobias Analytic vectors in continuous p-adic representations, Compos. Math., Volume 145 (2009) no. 1, pp. 247-270 | DOI

[32] Schneider, P.; Teitelbaum, J. Banach-Hecke algebras and p-adic Galois representations, Doc. Math. (2006) no. Extra Vol., pp. 631-684

[33] Schneider, Peter Nonarchimedean functional analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002, vi+156 pages | DOI

[34] Schneider, Peter; Teitelbaum, Jeremy Locally analytic distributions and p-adic representation theory, with applications to GL 2 , J. Amer. Math. Soc., Volume 15 (2002) no. 2, p. 443-468 (electronic) | DOI

[35] Schneider, Peter; Teitelbaum, Jeremy Algebras of p-adic distributions and admissible representations, Invent. Math., Volume 153 (2003) no. 1, pp. 145-196 | DOI

[36] Schraen, Benjamin Représentations p-adiques de GL 2 (L) et catégories dérivées, Israel J. Math., Volume 176 (2010), pp. 307-361 | DOI

[37] Zelevinsky, A. V. Induced representations of reductive p-adic groups. II. On irreducible representations of GL (n), Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 2, pp. 165-210

Cité par Sources :