Unipotent nearby cycles and the cohomology of shtukas

We give cases in which nearby cycles commute with pushforward from sheaves on the moduli stack of shtukas to a product curves over finite field. The proof systematically uses property that taking Satake Beilinson–Drinfeld Grassmannian parahoric reduction is central functor together ‘Zorro's lemma’ argument similar Xue [ Smoothness cohomology stacks , Preprint (2020), arXiv:2012.12833]. As an application, for automorphic forms at level, we characterize image tame inertia under Langlands correspondence terms two-sided cells.