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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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/

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