Blocks of the category of smooth $\ell$-modular representations of $GL(n,F)$ and its inner forms: reduction to level-$0$

Abstract: Let $G$ be an inner form of a general linear group over a non-archimedean locally compact field of residue characteristic $p$, let $R$ be an algebraically closed field of characteristic different from $p$ and let $\mathscr{R}_R(G)$ be the category of smooth representations of $G$ over $R$. In this paper, we prove that a block (indecomposable summand) of $\mathscr{R}_R(G)$ is equivalent to a level-$0$ block (a block in which every object has non-zero invariant vectors for the pro-$p$-radical of a maximal compact open subgroup) of $\mathscr{R}_R(G')$, where $G'$ is a direct product of groups of the same type of $G$.