  Let $G$ be a $p$-adic group that splits over an unramified extension. We
decompose $Rep_{\Lambda}^{0}(G)$, the abelian category of smooth level $0$
representations of $G$ with coefficients in
$\Lambda=\overline{\mathbb{Q}}_{\ell}$ or $\overline{\mathbb{Z}}_{\ell}$, into
a product of subcategories indexed by inertial Langlands parameters. We
construct these categories via systems of idempotents on the Bruhat-Tits
building and Deligne-Lusztig theory. Then, we prove compatibilities with
parabolic induction and restriction functors and the local Langlands
correspondence.
