Deformation rings and parabolic induction
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 695-727.

We study deformations of smooth mod p representations (and their duals) of a p-adic reductive group G. Under some mild genericity condition, we prove that parabolic induction with respect to a parabolic subgroup P=LN defines an isomorphism between the universal deformation rings of a supersingular representation σ ¯ of L and of its parabolic induction π ¯. As a consequence, we show that every Banach lift of π ¯ is induced from a unique Banach lift of σ ¯.

Nous étudions les déformations des représentations lisses modulo p (et leurs duaux) d’un groupe réductif p-adique G. Sous une hypothèse de généricité faible, nous prouvons que le foncteur d’induction parabolique relatif à un sous-groupe parabolique P=LN induit un isomorphisme entre l’anneau de déformation universel d’une représentation supersingulière σ ¯ de L et de son induite parabolique π ¯. En conséquence, nous montrons que tout relèvement continu de π ¯ est induit à partir d’un unique relèvement continu de σ ¯.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1046
Classification: 22E50, 11F70
Keywords: $p$-adic reductive groups, smooth representations, $\protect \mathfrak{m}$-adically continuous representations, parabolic induction, deformations
Hauseux, Julien 1; Schmidt, Tobias 2; Sorensen, Claus 3

1 Université de Lille Département de Mathématiques Cité scientifique, Bâtiment M2 59655 Villeneuve d’Ascq Cedex, France
2 Université Rennes, IRMAR - UMR CNRS 6625 35000 Rennes, France
3 Department of Mathematics, UCSD 9500 Gilman Dr. #0112 La Jolla, CA 92093-0112, USA
@article{JTNB_2018__30_2_695_0,
     author = {Hauseux, Julien and Schmidt, Tobias and Sorensen, Claus},
     title = {Deformation rings and parabolic induction},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {695--727},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {2},
     year = {2018},
     doi = {10.5802/jtnb.1046},
     mrnumber = {3891334},
     zbl = {1423.22019},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1046/}
}
TY  - JOUR
AU  - Hauseux, Julien
AU  - Schmidt, Tobias
AU  - Sorensen, Claus
TI  - Deformation rings and parabolic induction
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 695
EP  - 727
VL  - 30
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1046/
DO  - 10.5802/jtnb.1046
LA  - en
ID  - JTNB_2018__30_2_695_0
ER  - 
%0 Journal Article
%A Hauseux, Julien
%A Schmidt, Tobias
%A Sorensen, Claus
%T Deformation rings and parabolic induction
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 695-727
%V 30
%N 2
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1046/
%R 10.5802/jtnb.1046
%G en
%F JTNB_2018__30_2_695_0
Hauseux, Julien; Schmidt, Tobias; Sorensen, Claus. Deformation rings and parabolic induction. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 695-727. doi : 10.5802/jtnb.1046. http://www.numdam.org/articles/10.5802/jtnb.1046/

[1] Abe, Noriyuki On a classification of irreducible admissible modulo p representations of a p-adic split reductive group, Compos. Math., Volume 149 (2013) no. 12, pp. 2139-2168 | MR | Zbl

[2] Abe, Noriyuki; Henniart, Guy; Herzig, Florian; Vignéras, Marie-France A classification of irreducible admissible mod p representations of p-adic reductive groups, J. Am. Math. Soc., Volume 30 (2017) no. 2, pp. 495-559 | MR | Zbl

[3] Abe, Noriyuki; Henniart, Guy; Vignéras, Marie-France Modulo p representations of reductive p-adic groups: functorial properties (2017) (https://arxiv.org/abs/1703.05599v2)

[4] Barthel, Laure; Livné, Ron Irreducible modular representations of GL 2 of a local field, Duke Math. J., Volume 75 (1994) no. 2, pp. 261-292 | MR | Zbl

[5] Bernstein, Joseph N.; Zelevinskiĭ, Andreĭ V. Induced representations of reductive 𝔭-adic groups. I, Ann. Sci. Éc. Norm. Supér., Volume 10 (1977), pp. 441-472 | DOI | Numdam | MR | Zbl

[6] Breuil, Christophe The emerging p-adic Langlands programme, Proceedings of the International Congress of Mathematicians, Volume II, World Scientific; Hindustan Book Agency, 2010, pp. 203-230 | Zbl

[7] Colmez, Pierre Représentations de GL 2 ( p ) and (ϕ,Γ)-modules, Représentations p-adiques de groupes p-adiques II: Représentations de GL 2 ( p ) et (ϕ,Γ)-modules (Astérisque), Volume 330, Société Mathématique de France, 2010, pp. 281-509 | Zbl

[8] Emerton, Matthew Local-global compatibility in the p-adic Langlands programme for GL 2/ (Draft dated March 23, 2011)

[9] Emerton, Matthew Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties, Représentations p-adiques de groupes p-adiques III: Méthodes globales et géométriques (Astérisque), Volume 331, Société Mathématique de France, 2010, pp. 355-402 | Numdam | MR | Zbl

[10] Emerton, Matthew Ordinary parts of admissible representations of p-adic reductive groups II. Derived functors, Représentations p-adiques de groupes p-adiques III: Méthodes globales et géométriques (Astérisque), Volume 331, Société Mathématique de France, 2010, pp. 403-459 | Numdam | MR | Zbl

[11] Hauseux, Julien Extensions entre séries principales p-adiques et modulo p de G(F), J. Inst. Math. Jussieu, Volume 15 (2016), pp. 225-270 | DOI | MR | Zbl

[12] Hauseux, Julien On the exactness of ordinary parts over a local field of characteristic p, Pac. J. Math., Volume 295 (2018) no. 1, pp. 17-30 | DOI | MR | Zbl

[13] Hauseux, Julien Parabolic induction and extensions, Algebra Number Theory, Volume 12 (2018) no. 4, pp. 779-831 | DOI | MR | Zbl

[14] Hauseux, Julien; Schmidt, Tobias; Sorensen, Claus Functorial properties of generalised Steinberg representations, J. Number Theory, Volume 195 (2019), pp. 312-329 | DOI | MR | Zbl

[15] Herzig, Florian The classification of irreducible admissible mod p representations of a p-adic GL n , Invent. Math., Volume 186 (2011) no. 2, pp. 373-434 | DOI | MR | Zbl

[16] Kashiwara, Masaki; Schapira, Pierre Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, 332, Springer, 2006, x+497 pages | MR | Zbl

[17] Kisin, Mark Deformations of G p and GL 2 ( p ) representations, Représentations p-adiques de groupes p-adiques II: Représentations de GL 2 ( p ) et (ϕ,Γ)-modules (Astérisque), Volume 330, Société Mathématique de France, 2010, pp. 511-528 | Zbl

[18] Paškūnas, Vytautas Extensions for supersingular representations of GL 2 ( p ), Représentations p-adiques de groupes p-adiques III: Méthodes globales et géométriques (Astérisque), Volume 331, Société Mathématique de France, 2010, pp. 317-353 | Zbl

[19] Paškūnas, Vytautas The image of Colmez’s Montreal functor, Publ. Math., Inst. Hautes Étud. Sci., Volume 118 (2013), pp. 1-191 | DOI | MR | Zbl

[20] Schikhof, Wilhelmus Hendricus A perfect duality between p-adic Banach spaces and compactoids, Indag. Math., New Ser., Volume 6 (1995) no. 3, pp. 325-339 | DOI | MR | Zbl

[21] Schmidt, Tobias On unitary deformations of smooth modular representations, Isr. J. Math., Volume 193 (2013), pp. 15-46 | DOI | MR | Zbl

[22] Schneider, Peter p-adic Lie Groups, Grundlehren der Mathematischen Wissenschaften, 344, Springer, 2011, xi+254 pages | MR | Zbl

[23] Shatz, Stephen S. Profinite groups, Arithmetic, and Geometry, Annals of Mathematics Studies, 67, Princeton University Press, 1972, x+252 pages | MR | Zbl

[24] Sorensen, Claus Deformations of principal series modulo p for GL n (2015) (unpublished)

[25] Tits, Jacques Reductive groups over local fields, Automorphic Forms, Representations, and L-functions (Proceedings of Symposia in Pure Mathematics), Volume 33, American Mathematical Society, 1979, pp. 29-69 | DOI | Zbl

[26] Vignéras, Marie-France The right adjoint of the parabolic induction, Arbeitstagung Bonn 2013: In Memory of Friedrich Hirzebruch (Progress in Mathematics), Volume 319, Birkhäuser, 2016, pp. 405-425 | DOI | MR | Zbl

[27] Wilson, John S. Profinite Groups, London Mathematical Society Monographs. New Series, 19, Clarendon Press, 1998, ix+284 pages | MR | Zbl

[28] Yekutieli, Amnon On Flatness and Completion for Infinitely Generated Modules over Noetherian Rings, Commun. Algebra, Volume 39 (2011) no. 11, pp. 4221-4245 | DOI | MR | Zbl

Cited by Sources: