Deligne–Lusztig duality on the stack of local systems

In the setting of geometric Langlands conjecture, we argue that phenomenon divergence at infinity on Bun_G (that is, difference between $!$-extensions and $*$-extensions) is controlled, Langlands-dually, by locus semisimple $\check{G}$-local systems. To see this, first rephrase question in terms Deligne-Lusztig duality then study functor DL_G^\spec acting spectral DG category IndCoh_N(LS_G). We prove projection IndCoh_N(LS_G) \to QCoh(LS_G), followed action a coherent D-module St_G which call {Steinberg} D-module. might be regarded as dualizing sheaf $G$-local also show DL_G^\spec, while far from being conservative, fully faithful subcategory compact objects.