  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 $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$.
