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.

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