The Arinkin–Gaitsgory temperedness conjecture

Abstract

Arinkin and Gaitsgory defined a category of tempered 𝐷-modules on Bun 𝐺 that is conjecturally equivalent to the category of quasi-coherent (not ind-coherent!) sheaves on LocSys Ǧ . However, their definition depends on the auxiliary data of a point of the curve; they conjectured that their definition is independent of this choice.Beraldo has outlined a proof of this conjecture that depends on some technology that is not currently available. Here we provide a short, unconditional proof of the Arinkin-Gaitsgory conjecture. M S C 2 0 2 0 14D24 (primary), 14F08, 14F10 (secondary) Contents 1.1.1Let 𝑋 be a geometrically connected, smooth, and projective curve over a field 𝑘 of characteristic 0. Let 𝐺 be a split reductive group over 𝑘. Let Bun 𝐺 denote the moduli stack of 𝐺-bundles on 𝑋, and let 𝐷(Bun 𝐺 ) denote the DG category of 𝐷-modules on Bun 𝐺 .