  Let $G$ be a $p$-adic group which splits over an unramified extension and
$Rep_{\Lambda}^{0}(G)$ the abelian category of smooth level $0$ representations
of $G$ with coefficients in $\Lambda=\overline{\mathbb{Q}}_{\ell}$ or
$\overline{\mathbb{Z}}_{\ell}$. We study the finest decomposition of
$Rep_{\Lambda}^{0}(G)$ into a product of subcategories that can be obtained by
the method introduced in an article of Lanard (arXiv:1703.08689), which is
currently the only one available when $\Lambda=\overline{\mathbb{Z}}_{\ell}$
and $G$ is not an inner form of $GL_n$. We give two descriptions of it, a first
one on the group side \`a la Deligne-Lusztig, and a second one on the dual side
\`a la Langlands. We prove several fundamental properties, like for example the
compatibility with parabolic induction and restriction or the compatibility
with the local Langlands correspondence. The factors of this decomposition are
not blocks, but we show how to group them to obtain "stable" blocks.
