Geometrization of the local Langlands correspondence
Résumé
Following the idea of [Far16], we develop the foundations for the geometric Langlands program on the Fargues-Fontaine curve. In particular, we define a category of-adic sheaves on the stack BunG 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-adic sheaves on BunG.
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