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

TL;DR

This paper studies depth zero representations of p-adic groups over Z[7p], establishing a bijection with tamely ramified Langlands parameters, and explores implications for the local Langlands correspondence and ramification properties.

Contribution

It proves a bijection between blocks of depth zero representations and tamely ramified Langlands parameters, extending to finite groups and connecting to the semi-simple local Langlands correspondence.

Findings

Blocks of depth zero representations correspond to tamely ramified parameters.

Depth zero category is indecomposable for tamely ramified groups.

Depth zero unipotent representations have unramified parameters.

Abstract

We consider the category of depth $0$ representations of a $p$-adic quasi-split reductive group with coefficients in $\overline{\mathbb{Z}}[\frac{1}{p}]$. 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 $\overline{\mathbb{Z}}[\frac{1}{p}]$. 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 semi-simple local Langlands correspondence $\pi\mapsto \varphi_{\pi}$ 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 $\varphi_{\pi}$ to tame inertia in terms of the Deligne-Lusztig parameter of $\pi$ and show, in particular, that $\varphi_{\pi}$ is unramified if $\pi$ is unipotent.