A comparison of endomorphism algebras

Abstract: Let $F$ be a non-archimedean local field and $G$ be a connected reductive group over $F$. For a Bernstein block in the category of smooth complex representations of $G(F)$, we have two kinds of progenerators: the compactly induced representation $\text{ind}{K}^{G(F)} (\rho)$ of a type $(K, \rho)$, and the parabolically induced representation $I{P}^{G}(\Pi^{M})$ of a progenerator $\Pi^{M}$ of a Bernstein block for a Levi subgroup $M$ of $G$. In this paper, we construct an explicit isomorphism of these two progenerators. Moreover, we compare the description of the endomorphism algebra $\text{End}{G(F)}\left(\text{ind}{K}^{G(F)} (\rho)\right)$ for a depth-zero type $(K, \rho)$ by Morris with the description of the endomorphism algebra $\text{End}{G(F)}\left(I{P}^{G}(\Pi^{M})\right)$ by Solleveld, that are described in terms of affine Hecke algebras.