  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.
