## Algebra & Number Theory

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

Gianmarco Chinello

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

#### Article information

Source
Algebra Number Theory, Volume 12, Number 7 (2018), 1675-1713.

Dates
Revised: 8 May 2018
Accepted: 12 June 2018
First available in Project Euclid: 9 November 2018

https://projecteuclid.org/euclid.ant/1541732438

Digital Object Identifier
doi:10.2140/ant.2018.12.1675

Mathematical Reviews number (MathSciNet)
MR3871507

Zentralblatt MATH identifier
06976299

#### Citation

Chinello, Gianmarco. Blocks of the category of smooth $\ell$-modular representations of GL$(n,F)$ and its inner forms: reduction to level 0. Algebra Number Theory 12 (2018), no. 7, 1675--1713. doi:10.2140/ant.2018.12.1675. https://projecteuclid.org/euclid.ant/1541732438

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