# Deligne-Lusztig duality on the stack of local systems

## TL;DR

This paper explores the geometric Langlands conjecture, demonstrating how Deligne-Lusztig duality relates to the divergence phenomena on the stack of local systems, and introduces a spectral functor with specific properties.

## Contribution

It redefines Deligne-Lusztig duality in the geometric Langlands setting and characterizes the spectral Deligne-Lusztig functor as a projection followed by a D-module action.

## Key findings

- DL_G^spec is the projection onto QCoh(LS_G)
- The Steinberg D-module acts as a dualizing sheaf for semisimple local systems
- DL_G^spec is fully faithful on compact objects

## 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 $\check{G}$-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 acting on the spectral Langlands DG category IndCoh_N(LS_G). We prove that DL_G^\spec is the projection IndCoh_N(LS_G) \to QCoh(LS_G), followed by the action of a coherent D-module St_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_G^\spec, while far from being conservative, is fully faithful on the subcategory of compact objects.