The endomorphism ring of projectives and the Bernstein centre

Pyvovarov, Alexandre

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 $\Omega$ une composante de Bernstein de la catégorie des représentations lisses de ${\mathrm{GL}}_n\left(F\right).$ Soient $(J,\lambda )$ un $\Omega$-type de Bushnell–Kutzko et ${ℨ}_{\Omega }$ le centre de Bernstein de la composante $\Omega$. Soit $\sigma$ un facteur direct de ${\mathrm{Ind}}_J^{{\mathrm{GL}}_n\left({𝒪}_F\right)}\lambda$. Nous commençons par calculer $\mathrm{c}\text{--}{\mathrm{Ind}}_{{\mathrm{GL}}_n\left({𝒪}_F\right)}^{{\mathrm{GL}}_n\left(F\right)}\sigma {\otimes }_{{ℨ}_{\Omega }}\kappa \left(𝔪\right)$, où $\kappa \left(𝔪\right)$ est le corps résiduel de ${ℨ}_{\Omega }$ en un idéal maximal $𝔪$, et $𝔪$ appartient à un ensemble Zariski dense dans $\mathrm{Spec}{ℨ}_{\Omega }$.

Ce résultat nous permet ensuite de déduire que l’anneau des endomorphismes ${\mathrm{End}}_{{\mathrm{GL}}_n\left(F\right)}\left(\mathrm{c}\text{--}{\mathrm{Ind}}_{{\mathrm{GL}}_n\left({𝒪}_F\right)}^{{\mathrm{GL}}_n\left(F\right)}\sigma \right)$ est isomorphe à ${ℨ}_{\Omega }$, si $\sigma$ apparait avec multiplicité un dans ${\mathrm{Ind}}_J^{{\mathrm{GL}}_n\left({𝒪}_F\right)}\lambda$.

Let $F$ be a local non-archimedean field and ${𝒪}_F$ its ring of integers. Let $\Omega$ be a Bernstein component of the category of smooth representations of ${\mathrm{GL}}_n\left(F\right)$, let $(J,\lambda )$ be a Bushnell–Kutzko $\Omega$-type, and let ${ℨ}_{\Omega }$ be the centre of the Bernstein component $\Omega$. Let $\sigma$ be a direct summand of ${\mathrm{Ind}}_J^{{\mathrm{GL}}_n\left({𝒪}_F\right)}\lambda$. We will begin by computing $\mathrm{c}\text{--}{\mathrm{Ind}}_{{\mathrm{GL}}_n\left({𝒪}_F\right)}^{{\mathrm{GL}}_n\left(F\right)}\sigma {\otimes }_{{ℨ}_{\Omega }}\kappa \left(𝔪\right)$, where $\kappa \left(𝔪\right)$ is the residue field at maximal ideal $𝔪$ of ${ℨ}_{\Omega }$, and the maximal ideal $𝔪$ belongs to a Zariski-dense set in $\mathrm{Spec}{ℨ}_{\Omega }$.

This result will allow us to deduce that the endomorphism ring ${\mathrm{End}}_{{\mathrm{GL}}_n\left(F\right)}\left(\mathrm{c}\text{--}{\mathrm{Ind}}_{{\mathrm{GL}}_n\left({𝒪}_F\right)}^{{\mathrm{GL}}_n\left(F\right)}\sigma \right)$ is isomorphic to ${ℨ}_{\Omega }$, when $\sigma$ appears with multiplicity one in ${\mathrm{Ind}}_J^{{\mathrm{GL}}_n\left({𝒪}_F\right)}\lambda$.

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

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