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 Rep R ⁡ ( G ) \operatorname {Rep}_R(G) , the category of smooth representations of a p p -adic group G G with coefficients in R R . In particular, they were used to construct level 0 decompositions when R = Z ¯ ℓ R=\overline {\mathbb {Z}}_{\ell } , ℓ ≠ p \ell \neq p , by Dat for G L n GL_{n} and the author for a more general group. Wang proved in the case of G L n 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 G L n 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.