Blocks of the category of smooth ℓ-modular representations of GL(n, F) and its inner forms: Reduction to level 0

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 R R (G) be the category of smooth representations of G over R. In this paper, we prove that a block (indecomposable summand) of R R (G) 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 R R (G ′ ), where G ′ is a direct product of groups of the same type of G.