#### Vol. 12, No. 7, 2018

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

### Gianmarco Chinello

Vol. 12 (2018), No. 7, 1675–1713
##### 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 ${\mathsc{ℛ}}_{R}\left(G\right)$ be the category of smooth representations of $G$ over $R$. In this paper, we prove that a block (indecomposable summand) of ${\mathsc{ℛ}}_{R}\left(G\right)$ 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 ${\mathsc{ℛ}}_{R}\left({G}^{\prime }\right)$, where ${G}^{\prime }$ is a direct product of groups of the same type of $G$.

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