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

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 $\pi \mapsto {\phi }_{\pi }$ 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 ${\phi }_{\pi }$ to tame inertia in terms of the Deligne-Lusztig parameter of $\pi$ and show, in particular, that ${\phi }_{\pi }$ is unramified if $\pi$ is unipotent.

We added applications to the Fargues-Scholze semisimple correspondence. We proved that this correspondence takes depth 0 representations to tamely ramified parameters. We also bound the restriction of a parameter to tame inertia in terms of the Deligne-Lusztig parameter of the representation and show, in particular, that the parameter associated with a unipotent representation is unramified