Depth-zero local Langlands via shtukas in mixed characteristic

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 $\pi$ be an irreducible smooth depth-zero representation of $G(E)$ over ${\bar{\mathbb{F}}}_{\mathrm{\ell }}$ where $\mathrm{\ell }$ is a prime number which does not divide $p\mathrm{\mid }{\pi }_{0}Z(G)\mathrm{\mid }$. Let ${\phi }_{\pi }^{\mathrm{FS}}$ be the Fargues-Scholze $L$-parameter associated to $\pi$. We describe the restriction of ${\phi }_{\pi }^{\mathrm{FS}}$ 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 ${\bar{\mathbb{Q}}}_{\mathrm{\ell }}$.