Geometric Langlands (Betti): Equivalence of the Betti Langlands functor LG_Betti

Prove that the Betti geometric Langlands functor LG_Betti: Shv¹_Nilp(Bun_G) → IndCoh_Nilp(LS_G^Betti) is an equivalence of categories, where Shv¹_Nilp(Bun_G) is the category of Betti sheaves with singular support in the global nilpotent cone and LS_G^Betti is the stack of Betti G-local systems.

Background

In the Betti setting, the automorphic category is the full subcategory of Betti sheaves on Bun_G with singular support in the global nilpotent cone, and the spectral side uses the stack of Betti local systems. The authors construct a coarse Betti functor and upgrade it, paralleling the de Rham case, leveraging a Betti analog of the vacuum Poincaré construction.

They prove that de Rham and Betti conjectures (including restricted and tempered variants) are logically equivalent via Riemann–Hilbert and auxiliary equivalences, but the conjecture in the Betti form remains explicitly stated as an equivalence of categories.