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 $G{L}_n(F)$, 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 ${\mathrm{Ind}}_J^{G{L}_n({\mathcal{O}}_F)}\lambda$. We will begin by computing ${\mathrm{c}\text{–}\mathrm{Ind}}_{G{L}_n({\mathcal{O}}_F)}^{G{L}_n(F)}\sigma {\otimes }_{{\mathfrak{Z}}_{\Omega }}\kappa (\mathfrak{m})$, 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 ${\mathrm{End}}_{G{L}_n(F)}({\mathrm{c}\text{–}\mathrm{Ind}}_{G{L}_n({\mathcal{O}}_F)}^{G{L}_n(F)}\sigma )$ is isomorphic to ${\mathfrak{Z}}_{\Omega }$, when $\sigma$ appears with multiplicity one in ${\mathrm{Ind}}_J^{G{L}_n({\mathcal{O}}_F)}\lambda$.

29 pages. Rewritten introduction. Changed abstract. Minor revisions made