Spectral action in Betti Geometric Langlands

Abstract

Let $X$ be a smooth projective curve, $G$ a reductive group, and $Bu{n}_G(X)$ the moduli of $G$-bundles on $X$. For each point of $X$, the Satake category acts by Hecke modifications on sheaves on $Bu{n}_G(X)$. 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 $Lo{c}_{G^{\vee }}(X)$ 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.

30 pages

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