Equivalence of categories between coefficient systems and systems of idempotents

TL;DR

This paper proves an equivalence of categories between coefficient systems and systems of idempotents for connected reductive groups over non-archimedean local fields, generalizing previous results for GL_n.

Contribution

It extends Wang's equivalence of categories from GL_n to all connected reductive groups over non-archimedean local fields.

Findings

Establishes an equivalence between coefficient systems and idempotent systems for general reductive groups.

Generalizes previous results from GL_n to a broader class of groups.

Provides tools for analyzing level zero representations and blocks in p-adic groups.

Abstract

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $Rep_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to construct level 0 decompositions when $R=\overline{\mathbb{Z}}_{\ell}$, $\ell \neq p$, by Dat for $GL_n$ and the author for a more general group. Wang proved in the case of $GL_n$ that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of $GL_n$ and a unipotent block of another group. In this paper, we generalize Wang's equivalence of category to a connected reductive group on a non-archimedean local field.