The endomorphism ring of projectives and the Bernstein centre

Alexandre Pyvovarov

doi:10.5802/jtnb.1111

Alexandre Pyvovarov ¹

¹ Morningside Center of Mathematics No.55 Zhongguancun Donglu Academy of Mathematics and Systems Science Beijing Haidian District 100190, China

Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 49-71.

Abstract
Résumé

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 $ℨ_{Ω}$ 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 $c -- {Ind}_{{GL}_{n} (𝒪_{F})}^{{GL}_{n} (F)} σ \otimes_{ℨ_{Ω}} κ (𝔪)$ , where $κ (𝔪)$ is the residue field at maximal ideal $𝔪$ of $ℨ_{Ω}$ , and the maximal ideal $𝔪$ belongs to a Zariski-dense set in $Spec ℨ_{Ω}$ .

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 $ℨ_{Ω}$ , when $σ$ appears with multiplicity one in ${Ind}_{J}^{{GL}_{n} (𝒪_{F})} λ$ .

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 $ℨ_{Ω}$ 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 $c -- {Ind}_{{GL}_{n} (𝒪_{F})}^{{GL}_{n} (F)} σ \otimes_{ℨ_{Ω}} κ (𝔪)$ , où $κ (𝔪)$ est le corps résiduel de $ℨ_{Ω}$ en un idéal maximal $𝔪$ , et $𝔪$ appartient à un ensemble Zariski dense dans $Spec ℨ_{Ω}$ .

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 $ℨ_{Ω}$ , si $σ$ apparait avec multiplicité un dans ${Ind}_{J}^{{GL}_{n} (𝒪_{F})} λ$ .

Received: 2018-12-16
Revised: 2020-02-17
Accepted: 2020-04-25
Published online: 2020-08-21

DOI: 10.5802/jtnb.1111

Classification: 22E50, 11F70
Keywords: smooth representations, $p$-adic groups, Bernstein centre

Author's affiliations:

Alexandre Pyvovarov ¹

¹ Morningside Center of Mathematics No.55 Zhongguancun Donglu Academy of Mathematics and Systems Science Beijing Haidian District 100190, China

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {The endomorphism ring of projectives and the {Bernstein} centre},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Alexandre Pyvovarov. The endomorphism ring of projectives and the Bernstein centre. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 49-71. doi : 10.5802/jtnb.1111. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1111/

References
Cited by

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

[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 | Zbl

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

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

[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) | DOI | MR | Zbl

[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) | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

[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) | DOI | Numdam | MR | Zbl

[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 | DOI | MR | Zbl

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

[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, 17, Société Mathématique de France, 2010, vi+332 pages | MR | Zbl

[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) | DOI | MR | Zbl

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

[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 | DOI | Numdam | MR | Zbl

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