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

Chinello, Gianmarco

doi:10.2140/ant.2018.12.1675

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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

DOI: 10.2140/ant.2018.12.1675

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 $ℛ_{R} (G)$ 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 $G$ .

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

Vol. 12, No. 7, 2018