We present a proof that V. Lafforgue's global Langlands correspondence is compatible with Fargues–Scholze's semisimplified local Langlands correspondence. By globalizing local representations, this has the following consequences:
・Fargues–Scholze's construction canonically lifts to a non-semisimplified correspondence in characteristic ≥ 5,
・Genestier–Lafforgue's correspondence agrees with Fargues–Scholze's.
The proof relies on a formal model for the moduli of local shtukas with multiple legs, which we construct using algebraization theorems for analytic local shtukas. |