A comparison of endomorphism algebras

TL;DR

This paper explicitly constructs an isomorphism between two types of progenerators in the representation theory of p-adic groups and compares their endomorphism algebras described via affine Hecke algebras.

Contribution

It provides an explicit isomorphism between compactly induced and parabolically induced progenerators and compares their endomorphism algebras in the context of Bernstein blocks.

Findings

Explicit isomorphism between the two progenerators.

Comparison of endomorphism algebras via affine Hecke algebras.

Unified understanding of endomorphism algebra descriptions.

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.