# Sur les $\ell$-blocs de niveau z\'ero des groupes $p$-adiques

## TL;DR

This paper decomposes the category of smooth level 0 representations of a p-adic group into components associated with inertial Langlands parameters, using idempotents and Deligne-Lusztig theory, and explores their compatibilities.

## Contribution

It introduces a new decomposition of level 0 representation categories for p-adic groups via idempotents and establishes compatibility with key functors and the local Langlands correspondence.

## Key findings

- Decomposition of representation categories indexed by inertial parameters
- Construction of categories using idempotents and Deligne-Lusztig theory
- Compatibility with parabolic induction, restriction, and Langlands correspondence

## Abstract

Let $G$ be a $p$-adic group that splits over an unramified extension. We decompose $Rep_{\Lambda}^{0}(G)$, the abelian category of smooth level $0$ representations of $G$ with coefficients in $\Lambda=\overline{\mathbb{Q}}_{\ell}$ or $\overline{\mathbb{Z}}_{\ell}$, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat-Tits building and Deligne-Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.