The Arinkin-Gaitsgory temperedness conjecture

TL;DR

This paper proves the Arinkin-Gaitsgory temperedness conjecture unconditionally, establishing the independence of auxiliary data in the definition of tempered D-modules on Bun_G, and confirming their conjectured equivalence with quasi-coherent sheaves on LocSys_{reve{G}}.

Contribution

It provides the first unconditional proof of the conjecture, removing reliance on unavailable technology and confirming the independence of auxiliary data in the definition.

Findings

Confirmed the conjectural equivalence between tempered D-modules and quasi-coherent sheaves.

Proved independence of auxiliary data in the definition of tempered D-modules.

Established foundational results for the geometric Langlands program.

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_{\check{G}}. 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.