Deligne-Lusztig duality on the stack of local systems

Abstract

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