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arXiv · 1803.01584

The endomorphism ring of projectives and the Bernstein centre

Alexandre Pyvovarov
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Abstract

Let $F$ be a local non-archimedean field and $\mathcal{O}_F$ 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 $\mathfrak{Z}_Ω$ be the centre of the Bernstein component $Ω$. This paper contains two major results. Let $σ$ be a direct summand of $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} λ$. We will begin by computing $\mathrm{c\text{--} Ind}_{GL_n(\mathcal{O}_F)}^{GL_n(F)} σ\otimes_{\mathfrak{Z}_Ω}κ(\mathfrak{m})$, where $κ(\mathfrak{m})$ is the residue field at maximal ideal $\mathfrak{m}$ of $\mathfrak{Z}_Ω$, and the maximal ideal $\mathfrak{m}$ belongs to a Zariski-dense set in $\mathrm{Spec}\: \mathfrak{Z}_Ω$. This result allows us to deduce that the endomorphism ring $\mathrm{End}_{GL_n(F)}(\mathrm{c\text{--} Ind}_{GL_n(\mathcal{O}_F)}^{GL_n(F)} σ)$ is isomorphic to $\mathfrak{Z}_Ω$, when $σ$ appears with multiplicity one in $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} λ$.

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Alexandre Pyvovarov. 2019-06-03. The endomorphism ring of projectives and the Bernstein centre. https://arxiv.org/abs/1803.01584

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