Nearby cycles and the cohomology of shtukas

Abstract

V. Lafforgue constructed Langlands parameters from automorphic forms for any reductive group over a function field using excursion operators. Our aim is to give a general approach for proving certain local-global compatibilities satisfied by these Langlands parameters. The main consequence for the Langlands correspondence is to show that Lafforgue's construction is compatible with Lusztig's theory of character sheaves at a given point of a smooth curve over a finite field. Namely, using the theory of character sheaves, one attaches a torus character and a two-sided cell to an irreducible representation of a reductive group over a finite field. If our automorphic form lives in an isotypic component determined by this irreducible representation, we show that the torus character and two-sided cell determine the semisimple and unipotent parts of the image of the tame generator under the Langlands correspondence, respectively.

One key step is showing that nearby cycles commute with pushforward of certain perverse sheaves from the stack of global shtukas to a power of a curve. The main technical ingredient is the notion of what we call $\Psi$-factorizability, where nearby cycles over a general base are independent of the composition of specializations chosen, and the $\Psi$-factorizability statements we make give some answers to a question raised by Genestier-Lafforgue. To compute the action of framed excursion operators, we instead compute in monodromic affine Hecke categories. Ultimately, this reduces certain questions in the global function field Langlands program to questions in local geometric Langlands.