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

Keywords
equivalence of categories, blocks, modular representations of p-adic reductive groups, type theory, semisimple types, Hecke algebras, level-0 representations
Mathematical Subject Classification 2010
Primary: 20C20
Secondary: 22E50
Milestones
Received: 31 July 2017
Revised: 8 May 2018
Accepted: 12 June 2018
Published: 27 October 2018
Authors
Gianmarco Chinello
Dipartimento di Matematica e Applicazioni
Università degli Studi di Milano-Bicocca
Milano
Italy