Unipotent nearby cycles and the cohomology of shtukas

TL;DR

This paper investigates conditions under which nearby cycles commute with pushforward in the context of shtukas, using Satake sheaves and a Zorro's lemma argument, with applications to automorphic forms and the Langlands correspondence.

Contribution

It establishes new cases where nearby cycles commute with pushforward for shtukas and characterizes the image of tame inertia in the Langlands correspondence at parahoric level.

Findings

Nearby cycles commute with pushforward in specific shtuka settings.

Characterization of tame inertia image in the Langlands correspondence.

Use of Zorro's lemma for smoothness of cohomology sheaves.

Abstract

We give cases in which nearby cycles commutes with pushforward from sheaves on the moduli stack of shtukas to a product of curves over a finite field. The proof systematically uses the property that taking nearby cycles of Satake sheaves on the Beilinson-Drinfeld Grassmannian with parahoric reduction is a central functor together with a "Zorro's lemma" argument similar to that given by Xue to prove the smoothness of cohomology sheaves at unramified places. As an application, for automorphic forms at the parahoric level, we characterize the image of tame inertia under the Langlands correspondence in terms of two-sided cells.