Depth zero representations over ℤ

Dat, Jean-François; Lanard, Thomas

doi:10.2140/ant.2026.20.1451

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Depth zero representations over $\overline{\mathbb{Z}}\bigl[\frac{1}{p}\bigr]$

Jean-François Dat and Thomas Lanard

Vol. 20 (2026), No. 7, 1451–1478

DOI: 10.2140/ant.2026.20.1451

Abstract

We consider the category of depth $0$ representations of a $p$ -adic quasisplit reductive group with coefficients in $\bar{ℤ} [\frac{1}{p}]$ . We prove that the blocks of this category are in natural bijection with the connected components of the space of tamely ramified Langlands parameters for $G$ over $\bar{ℤ} [\frac{1}{p}]$ . As a particular case, this depth $0$ category is thus indecomposable when the group is tamely ramified. Along the way we prove a similar result for finite reductive groups. As an application, we deduce that the semisimple local Langlands correspondence $π \mapsto φ_{π}$ constructed by Fargues and Scholze takes depth $0$ representations to tamely ramified parameters, using a motivic version of their construction recently announced by Scholze. We also bound the restriction of $φ_{π}$ to tame inertia in terms of the Deligne–Lusztig parameter of $π$ and show, in particular, that $φ_{π}$ is unramified if $π$ is unipotent.

Keywords

representations of $p$-adic groups, blocks, local Langlands correspondence, depth 0 representations

Mathematical Subject Classification

Primary: 22E50

Secondary: 11S37

Milestones

Received: 23 May 2024

Revised: 7 February 2025

Accepted: 27 June 2025

Published: 14 August 2026

Authors

Jean-François Dat
Institut de Mathématiques de Jussieu Sorbonne Université – Université de Paris – CNRS Paris France

Thomas Lanard
Laboratoire de Mathematiques de Versailles (LMV) Université Paris-Saclay Versailles France

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Vol. 20, No. 7, 2026