Equivalence of categories between coefficient systems and systems of idempotents

Abstract

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $Re{p}_R(G)$, 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 $R={\overline{\mathbb{Z}}}_{\ell }$, $\ell \ne p$, by Dat for $G{L}_n$ and the author for a more general group. Wang proved in the case of $G{L}_n$ 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 $G{L}_n$ 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.

17 pages, in English