Geometrization of the local Langlands correspondence

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 ${\mathrm{Bun}}_G$.

356 pages, v4: accepted version. v3: improved discussion of spectral action with integral coefficients (now also applying in the usual Betti setting), including new results (Theorem VIII.0.3) on preservation of good filtrations ("Donkin subgroups"). Comments welcome!

References in corpus (4)

• Higher Topos Theory
• Cycles on Shimura varieties via geometric Satake
• Smoothness of cohomology sheaves of stacks of shtukas
• A $(-q)$-analogue of weight multiplicities

Cited by in corpus (9)

• Diamantine Picard functors of rigid spaces
• The Jacobson--Morozov morphism for Langlands parameters in the relative setting
• Local parameters of supercuspidal representations
• Geometrization of the Satake transform for mod $p$ Hecke algebras
• The de Rham-Fargues-Fontaine cohomology
• The plectic conjecture over function fields
• Averaging functors in Fargues' program for GL_n
• Progrès récents sur les représentations supercuspidales
• The Arinkin-Gaitsgory temperedness conjecture