The endomorphism ring of projectives and the Bernstein centre

Let $F$ be a local non-archimedean field and ${\mathcal{O}}_F$ its ring of integers. Let $\mathrm{\Omega }$ be a Bernstein component of the category of smooth representations of $G{L}_n(F)$, let $(J,\lambda )$ be a Bushnell-Kutzko $\mathrm{\Omega }$-type, and let ${\mathfrak{Z}}_{\mathrm{\Omega }}$ be the centre of the Bernstein component $\mathrm{\Omega }$. This paper contains two major results. Let $\sigma$ be a direct summand of ${\mathrm{I}\mathrm{n}\mathrm{d}}_J^{G{L}_n({\mathcal{O}}_F)}\lambda$. We will begin by computing $$\mathrm{c}\text{–}\mathrm{I}\mathrm{n}\mathrm{d}${GL_n($\mathcal{O}$F)}^{GL_n(F)} \sigma\otimes{\mathfrak{Z}{\Omega}}\kappa(\mathfrak{m})$,where$\kappa(\mathfrak{m})$istheresiduefieldatmaximalideal$\mathfrak{m}$of$\mathfrak{Z}{\Omega}$,andthemaximalideal$\mathfrak{m}$ belongs to a Zariski-dense set in $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}{\mathfrak{Z}}_{\mathrm{\Omega }}$. This result allows us to deduce that the endomorphism ring $$\mathrm{E}\mathrm{n}\mathrm{d}${GL_n(F)}($\mathrm{c}\text{–}\mathrm{I}\mathrm{n}\mathrm{d}$_{GL_n($\mathcal{O}$F)}^{GL_n(F)} \sigma)$isisomorphicto$\mathfrak{Z}{\Omega}$,when$\sigma$ appears with multiplicity one in ${\mathrm{I}\mathrm{n}\mathrm{d}}_J^{G{L}_n({\mathcal{O}}_F)}\lambda$.