Depth-zero local Langlands via shtukas in mixed characteristic

Abstract

Assuming a smoothness property for mixed-characteristic shtuka spaces and hyperbolic localization for twisted sheaves, we determine the inertial restriction of the Fargues-Scholze $L$-parameter for irreducible depth-zero representations. Let $E$ be a non-archimedean local field of residue characteristic $p$, let $G$ be an unramified connected reductive group over $E$, and let $π$ be an irreducible smooth depth-zero representation of $G(E)$ over $\overline{\mathbb{F}}_{\ell}$ where $\ell$ is a prime number which does not divide $p|π_0Z(G)|$. Let $φ_π^{\operatorname{FS}}$ be the Fargues-Scholze $L$-parameter associated to $π$. We describe the restriction of $φ_π^{\operatorname{FS}}$ to the inertia subgroup in terms of Deligne-Lusztig theory. We prove this by computing the inertia action on the cohomology of shtuka spaces with depth-zero level structure. As an application, we prove that the Fargues--Scholze $L$-parameters agree with DeBacker--Reeder $L$-parameters over $\overline{\mathbb{Q}}_{\ell}$.