Parametrization and reduction to depth zero of $\overline{\mathbb{Z}}[\frac{1}{p}]$-blocks of tame $p$-adic groups

Abstract

Let $G$ be a reductive group over a non-archimedean local field $F$ of residue characteristic $p$. We consider pairs $(\phi,I)$ consisting of a "wild inertia" Langlands parameter $\phi: P_F \longrightarrow \hat{G}$ whose centralizer $C_{\hat{G}}(\phi)$ is a Levi subgroup of $\hat{G}$, and a cohomological invariant $I$ whose definition is inspired by the theory of endoscopy. Assuming that $p$ is odd and not a torsion prime of $G$ nor of $\hat{G}$, we associate to each such pair $(\phi,I)$ a Serre subcategory $\mathrm{Rep}^{\phi,I}(G(F))$ of the category of smooth $\overline{\mathbb{Z}}[\frac{1}{p}]$-representations of $G(F)$. Then we construct an equivalence between this Serre subcategory and the category of depth-zero $\overline{\mathbb{Z}}[\frac{1}{p}]$-representations of a twisted Levi subgroup $G_{\phi,I}$ of $G$, which is dual to $C_{\hat{G}}(\phi)$. This pattern for reduction to depth zero fits well with the conjectural (categorical) local Langlands correspondence. When $G$ is tamely ramified and $p$ does not divide the order of its Weyl group, then the above Serre subcategories provide the block decomposition of the category of all smooth $\overline{\mathbb{Z}}[\frac{1}{p}]$-representations of $G(F)$. In this case, we thus obtain a reduction-to-depth-zero process for smooth representations of $G(F)$ valued in any algebraically closed field of characteristic different from $p$. When that field has characteristic 0, this recovers some of the recent results of Adler--Fintzen--Mishra--Ohara. When that field is $\overline{\mathbb{F}}_{\ell}$, we use our results together with Zhu's unipotent categorical correspondence to produce a fully faithful embedding of $D\mathrm{Rep}_{\overline{\mathbb{F}}_{\ell}}(\mathrm{GL}_{n}(F))$ into a suitable category of coherent sheaves on the moduli space of $n$-dimensional $\overline{\mathbb{F}}_{\ell}$-representations of the Weil group.