Geometrization of the local Langlands correspondence

TL;DR

This paper develops the geometric Langlands program on the Fargues--Fontaine curve, establishing foundational categories, equivalences, and applications to local Shimura varieties and $L$-parameters, advancing the understanding of local Langlands correspondence.

Contribution

It defines a category of $$-adic sheaves on $ ext{Bun}_G$, proves a geometric Satake equivalence, and studies the stack of $L$-parameters, providing new tools for the local Langlands program.

Findings

Finiteness results for cohomology of local Shimura varieties

Definition of $L$-parameters for smooth representations of $G(E)$

Construction of spectral actions on categories of sheaves

Abstract

Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues--Fontaine curve. In particular, we define a category of $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ of $G$-bundles on the Fargues--Fontaine curve, prove a geometric Satake equivalence over the Fargues--Fontaine curve, and study the stack of $L$-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define $L$-parameters associated with irreducible smooth representations of $G(E)$, a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of $L$-parameters on the category of $\ell$-adic sheaves on $\mathrm{Bun}_G$.