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

## TL;DR

This paper demonstrates that blocks of smooth modular representations of an inner form of a general linear group over a non-archimedean field can be reduced to level-0 blocks of related groups, simplifying their classification.

## Contribution

It establishes an equivalence between blocks of representations of an inner form and level-0 blocks of product groups, reducing complex problems to a more manageable setting.

## Key findings

- Blocks of $ ext{Rep}_R(G)$ are equivalent to level-0 blocks of $ ext{Rep}_R(G')$
- Reduction simplifies the classification of smooth modular representations
- Provides a structural understanding of representation categories for inner forms

## Abstract

Let $G$ be an inner form of a general linear group over a non-archimedean locally compact field of residue characteristic $p$, let $R$ be an algebraically closed field of characteristic different from $p$ and let $\mathscr{R}_R(G)$ be the category of smooth representations of $G$ over $R$. In this paper, we prove that a block (indecomposable summand) of $\mathscr{R}_R(G)$ is equivalent to a level-$0$ block (a block in which every object has non-zero invariant vectors for the pro-$p$-radical of a maximal compact open subgroup) of $\mathscr{R}_R(G')$, where $G'$ is a direct product of groups of the same type of $G$.