Local-global compatibility over function fields

The Langlands program predicts a relationship between automorphic representations of a reductive group G and Galois representations valued in its L-group. For general G over a global function field, the automorphic-to-Galois direction has been constructed by V. Lafforgue. More recently, for general G over a nonarchimedean local field, a similar correspondence has been constructed by Fargues–Scholze.

We present a proof that the V. Lafforgue and Fargues–Scholze correspondences are compatible, generalizing local-global compatibility from class field theory. As a consequence, the correspondences of Genestier–Lafforgue and Fargues–Scholze agree, which answers a question of Fargues–Scholze, Hansen, Harris, and Kaletha.