Deligne--Lusztig duality on the moduli stack of bundles

TL;DR

This paper proves a conjecture connecting duality functors and Eisenstein series on the moduli stack of G-bundles over a curve, advancing understanding of D-modules in geometric representation theory.

Contribution

It establishes the Deligne-Lusztig duality conjecture for D-modules on Bun_G(X) and proves a second adjointness property for related functors.

Findings

Confirmed the Deligne-Lusztig duality conjecture for D-modules.

Established a second adjointness result for enhanced functors.

Linked pseudo-identity functors to Eisenstein series and constant term functors.

Abstract

Let $Bun_G(X)$ be the moduli stack of $G$-torsors on a smooth projective curve $X$ for a reductive group $G$. We prove a conjecture made by Drinfeld-Wang and Gaitsgory on the Deligne-Lusztig duality for D-modules on $Bun_G(X)$. This conjecture relates Drinfeld-Gaitsgory's pseudo-identity functors to the enhanced Eisenstein series and geometric constant term functors on $DMod(Bun_G(X))$. We also prove a "second adjointness" result for these enhanced functors.