Deligne–Lusztig duality on the stack of local systems

Abstract

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