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Let $F$ be a local non-archimedean field and $\mathcal{O}_F$ its ring ofintegers. Let $\Omega$ be a Bernstein component of the category of smoothrepresentations of $GL_n(F)$, let $(J, \lambda)$ be a Bushnell-Kutzko$\Omega$-type, and let $\mathfrak{Z}_{\Omega}$ be the centre of the Bernsteincomponent $\Omega$. This paper contains two major results. Let $\sigma$ be adirect summand of $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$. We will beginby 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 aZariski-dense set in $\mathrm{Spec}\: \mathfrak{Z}_{\Omega}$. This resultallows 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 withmultiplicity one in $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$.

The endomorphism ring of projectives and the Bernstein centre