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 : 10.5802/jtnb.1111
Classification : 22E50, 11F70
Mots clés : smooth representations, $p$-adic groups, Bernstein centre
Pyvovarov, Alexandre 1

1 Morningside Center of Mathematics No.55 Zhongguancun Donglu Academy of Mathematics and Systems Science Beijing Haidian District 100190, China
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Pyvovarov, Alexandre. 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. http://archive.numdam.org/articles/10.5802/jtnb.1111/

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