The endomorphism ring of projectives and the Bernstein centre

Abstract: Let $F$ be a local non-archimedean field and $\mathcal{O}F$ its ring of integers. Let $\Omega$ be a Bernstein component of the category of smooth representations of $GL_n(F)$, let $(J, \lambda)$ be a Bushnell-Kutzko $\Omega$-type, and let $\mathfrak{Z}{\Omega}$ be the centre of the Bernstein component $\Omega$. This paper contains two major results. Let $\sigma$ be a direct summand of $\mathrm{Ind}J^{GL_n(\mathcal{O}_F)} \lambda$. We will begin by computing $\mathrm{c\text{--} Ind}{GL_n(\mathcal{O}F)}^{GL_n(F)} \sigma\otimes{\mathfrak{Z}{\Omega}}\kappa(\mathfrak{m})$, where $\kappa(\mathfrak{m})$ is the residue field at maximal ideal $\mathfrak{m}$ of $\mathfrak{Z}{\Omega}$, and the maximal ideal $\mathfrak{m}$ belongs to a Zariski-dense set in $\mathrm{Spec}: \mathfrak{Z}{\Omega}$. This result allows us to deduce that the endomorphism ring $\mathrm{End}{GL_n(F)}(\mathrm{c\text{--} Ind}{GL_n(\mathcal{O}_F)}^{GL_n(F)} \sigma)$ is isomorphic to $\mathfrak{Z}{\Omega}$, when $\sigma$ appears with multiplicity one in $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$.