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

TL;DR

This paper investigates a detailed decomposition of level 0 representations of certain p-adic groups, providing dual descriptions and establishing key compatibilities with induction, restriction, and Langlands correspondence.

Contribution

It introduces a new, refined decomposition method for level 0 representations of p-adic groups that extends previous approaches to more general cases, including non-inner forms of GL_n.

Findings

Provides a Deligne-Lusztig type description of the decomposition.

Establishes compatibility with parabolic induction and restriction.

Shows how to group factors into stable blocks.

Abstract

Let $G$ be a $p$-adic group which splits over an unramified extension and $Rep_{\Lambda}^{0}(G)$ the abelian category of smooth level $0$ representations of $G$ with coefficients in $\Lambda=\overline{\mathbb{Q}}_{\ell}$ or $\overline{\mathbb{Z}}_{\ell}$. We study the finest decomposition of $Rep_{\Lambda}^{0}(G)$ into a product of subcategories that can be obtained by the method introduced in an article of Lanard (arXiv:1703.08689), which is currently the only one available when $\Lambda=\overline{\mathbb{Z}}_{\ell}$ and $G$ is not an inner form of $GL_n$. We give two descriptions of it, a first one on the group side \`a la Deligne-Lusztig, and a second one on the dual side \`a la Langlands. We prove several fundamental properties, like for example the compatibility with parabolic induction and restriction or the compatibility with the local Langlands correspondence. The factors of this decomposition are not blocks, but we show how to group them to obtain "stable" blocks.