Restricted shtukas and $\Psi$-factorizable sheaves

We show that Lusztig's theories of two-sided cells and non-unipotent representations of a reductive group over a finite field are compatible with the V. Lafforgue's automorphic-to-galois direction of the Langlands correspondence. To do this, we extend cases where nearby cycles commutes with pushforward from sheaves on the moduli space of shtukas to a product of curves to include certain depth $0$ cases. More generally, we introduce the notion of $\Psi$-factorizability to study nearby cycles over general bases, whereby a sheaf is $\Psi$-factorizable if its nearby cycles are the same as iterated nearby cycles with respect to arbitrary compositions of specializations on the base. The Satake sheaves on Beilinson-Drinfeld grassmannians and their cohomology sheaves on curves are nontrivial examples of $\Psi$-factorizable sheaves. This notion allows us to adapt arguments from Xue. As an application, for certain automorphic forms in depth zero attached to a Langlands parameter, we characterize the image of the tame generator of this parameter in terms of semisimple orbits and two-sided cells attached to representations, extending ideas of Lusztig-Yun and Bezrukavnikov-Finkelberg-Ostrik.