Spectral Action in Betti Geometric Langlands

Let X be a smooth projective curve, G a reductive group, and BunG(X) the moduli of G-bundles on X. For each point of X, the Satake category acts by Hecke modifications on sheaves on BunG(X). We show that, for sheaves with nilpotent singular support, the action is locally constant with respect to the point of X. This equips sheaves with nilpotent singular support with a module structure over perfect complexes on the Betti moduli LocG∨ (X) of dual group local systems. In particular, we establish the “automorphic to Galois” direction in the Betti Geometric Langlands correspondence—to each indecomposable automorphic sheaf, we attach a dual group local system—and define the Betti version of V. Lafforgue’s excursion operators