Deligne–Lusztig duality on the moduli stack of bundles

Abstract

Let Bun G ⁡ ( X ) \operatorname {Bun}_{G}(X) be the moduli stack of G G -torsors on a smooth projective curve X X for a reductive group G G . We prove a conjecture made by Drinfeld-Wang and Gaitsgory on the Deligne–Lusztig duality for D-modules on Bun G ⁡ ( X ) \operatorname {Bun}_{G}(X) . This conjecture relates the pseudo-identity functors given by Gaitsgory [Ann. Sci. Éc. Norm. Supér. (4) (2017)] and Drinfeld and Gaitsgory [Camb. J. Math. 3 (2015), pp. 19–125] to the enhanced Eisenstein series and geometric constant term functors given by Gaitsgory [Astérisque (2015)]. We also prove a “second adjointness” result for these enhanced functors.