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 -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 -local systems. We also show that DL_G^\spec, while far from being conservative, is fully faithful on the subcategory of compact objects.
Keywords
Cite
@article{arxiv.1906.00934,
title = {Deligne-Lusztig duality on the stack of local systems},
author = {Dario Beraldo},
journal= {arXiv preprint arXiv:1906.00934},
year = {2021}
}