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 nonarchimedean locally compact field of residue characteristic $p$, let $R$ be an algebraically closed field of characteristic different from $p$ and let ${\mathcal{ℛ}}_R\left(G\right)$ be the category of smooth representations of $G$ over $R$. In this paper, we prove that a block (indecomposable summand) of ${\mathcal{ℛ}}_R\left(G\right)$ is equivalent to a level-$0$ block (a block in which every simple object has nonzero invariant vectors for the pro-$p$-radical of a maximal compact open subgroup) of ${\mathcal{ℛ}}_R\left(G^{\prime }\right)$, where $G^{\prime }$ is a direct product of groups of the same type of $G$.