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 $\operatorname {Rep}_R(G)$, 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.