The endomorphism ring of projectives and the Bernstein centre
Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 49-71.

Soient F un corps local non-archimédien et 𝒪 F son anneau des entiers. Soit Ω une composante de Bernstein de la catégorie des représentations lisses de GL n (F). Soient (J,λ) un Ω-type de Bushnell–Kutzko et Ω le centre de Bernstein de la composante Ω. Soit σ un facteur direct de Ind J GL n (𝒪 F ) λ. Nous commençons par calculer c--Ind GL n (𝒪 F ) GL n (F) σ Ω κ(𝔪), où κ(𝔪) est le corps résiduel de Ω en un idéal maximal 𝔪, et 𝔪 appartient à un ensemble Zariski dense dans Spec Ω .

Ce résultat nous permet ensuite de déduire que l’anneau des endomorphismes End GL n (F) (c--Ind GL n (𝒪 F ) GL n (F) σ) est isomorphe à Ω , si σ apparait avec multiplicité un dans Ind J GL n (𝒪 F ) λ.

Let F be a local non-archimedean field and 𝒪 F its ring of integers. Let Ω be a Bernstein component of the category of smooth representations of GL n (F), let (J,λ) be a Bushnell–Kutzko Ω-type, and let Ω be the centre of the Bernstein component Ω. Let σ be a direct summand of Ind J GL n (𝒪 F ) λ. We will begin by computing c--Ind GL n (𝒪 F ) GL n (F) σ Ω κ(𝔪), where κ(𝔪) is the residue field at maximal ideal 𝔪 of Ω , and the maximal ideal 𝔪 belongs to a Zariski-dense set in Spec Ω .

This result will allow us to deduce that the endomorphism ring End GL n (F) (c--Ind GL n (𝒪 F ) GL n (F) σ) is isomorphic to Ω , when σ appears with multiplicity one in Ind J GL n (𝒪 F ) λ.

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DOI : https://doi.org/10.5802/jtnb.1111
Classification : 22E50,  11F70
Mots clés : smooth representations, p-adic groups, Bernstein centre
@article{JTNB_2020__32_1_49_0,
     author = {Alexandre Pyvovarov},
     title = {The endomorphism ring of projectives and the {Bernstein} centre},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {49--71},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {1},
     year = {2020},
     doi = {10.5802/jtnb.1111},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1111/}
}
Alexandre Pyvovarov. The endomorphism ring of projectives and the Bernstein centre. Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 49-71. doi : 10.5802/jtnb.1111. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1111/

[1] Hyman Bass Algebraic K-theory, Mathematics Lecture Note Series, W. A. Benjamin, 1968, xx+762 pages | MR 0249491 | Zbl 0174.30302

[2] Joseph N. Bernstein Le “centre” de Bernstein, Representations of reductive groups over a local field (Travaux en Cours), Hermann, 1984, pp. 1-32 | MR 771671 | Zbl 0599.22016

[3] Nicolas Bourbaki Éléments de mathématique. Algèbre commutative. Chapitres 1 à 4, Masson, 1985, 362 pages | MR 782296 | Zbl 0547.13001

[4] Nicolas Bourbaki Éléments de mathématique. Algèbre commutative. Chapitres 5 à 7, Masson, 1985, 351 pages | MR 782297 | Zbl 0547.13002

[5] Nicolas Bourbaki Algebra II. Chapters 4–7, Elements of Mathematics, Springer, 2003, viii+461 pages (translated from the 1981 French edition by P. M. Cohn and J. Howie, Reprint of the 1990 English edition) | Article | MR 1994218 | Zbl 1017.12001

[6] Nicolas Bourbaki Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples, Springer, 2012, x+489 pages (second revised edition of the 1958 edition) | Article | MR 3027127 | Zbl 1245.16001

[7] Colin J. Bushnell; Philip C. Kutzko The admissible dual of GL (N) via compact open subgroups, Annals of Mathematics Studies, Volume 129, Princeton University Press, 1993, xii+313 pages | Article | MR 1204652 | Zbl 0787.22016

[8] Colin J. Bushnell; Philip C. Kutzko Smooth representations of reductive p-adic groups: structure theory via types, Proc. Lond. Math. Soc., Volume 77 (1998) no. 3, pp. 582-634 | Article | MR 1643417 | Zbl 0911.22014

[9] Colin J. Bushnell; Philip C. Kutzko Semisimple types in GL n , Compos. Math., Volume 119 (1999) no. 1, pp. 53-97 | Article | MR 1711578 | Zbl 0933.22027

[10] Ana Caraiani; Matthew Emerton; Toby Gee; David Geraghty; Vytautas Paškūnas; Sug Woo Shin Patching and the p-adic local Langlands correspondence, Camb. J. Math., Volume 4 (2016) no. 2, pp. 197-287 | Article | MR 3529394 | Zbl 1403.11073

[11] Jean-François Dat Caractères à valeurs dans le centre de Bernstein, J. Reine Angew. Math., Volume 508 (1999), pp. 61-83 | Article | MR 1676870 | Zbl 0938.22016

[12] Jean-François Dat Types et inductions pour les représentations modulaires des groupes p-adiques, Ann. Sci. Éc. Norm. Supér., Volume 32 (1999) no. 1, pp. 1-38 (with an appendix by Marie-France Vignéras) | Article | Numdam | MR 1670599 | Zbl 0935.22013

[13] David Helm The Bernstein center of the category of smooth W(k)[ GL n (F)]-modules, Forum Math. Sigma, Volume 4 (2016), e11, 98 pages | Article | MR 3508741 | Zbl 1364.11097

[14] Chris Jantzen On the Iwahori-Matsumoto involution and applications, Ann. Sci. Éc. Norm. Supér., Volume 28 (1995) no. 5, pp. 527-547 | Article | Numdam | MR 1341660 | Zbl 0840.22030

[15] Alexandre Pyvovarov On the Breuil-Schneider conjecture: Generic case (2018) (https://arxiv.org/abs/1803.01610)

[16] David Renard Représentations des groupes réductifs p-adiques, Contributions in Mathematical and Computational Sciences, Volume 17, Société Mathématique de France, 2010, vi+332 pages | MR 2567785 | Zbl 1186.22020

[17] Peter Schneider; Ernst-Wilhelm Zink K-types for the tempered components of a p-adic general linear group, J. Reine Angew. Math., Volume 517 (1999), pp. 161-208 (with an appendix by Schneider and U. Stuhler) | Article | MR 1728541 | Zbl 0934.22021

[18] Richard G. Swan Projective modules over Laurent polynomial rings, Trans. Am. Math. Soc., Volume 237 (1978), pp. 111-120 | Article | MR 0469906 | Zbl 0404.13006

[19] Andreĭ V. Zelevinsky Induced representations of reductive 𝔭-adic groups. II. On irreducible representations of GL (n), Ann. Sci. Éc. Norm. Supér., Volume 13 (1980) no. 2, pp. 165-210 | Article | Numdam | MR 584084 | Zbl 0441.22014