Geometric Langlands (de Rham): Equivalence of the Langlands functor LG

Prove that the geometric Langlands functor LG: D-mod¹(Bun_G) → IndCoh_Nilp(LS_G) is an equivalence of categories, where D-mod¹(Bun_G) denotes the category of half-twisted D‑modules on the moduli stack Bun_G of G-bundles on the smooth complete curve X, and IndCoh_Nilp(LS_G) denotes ind‑coherent sheaves on the stack of de Rham G-local systems LS_G with singular support contained in the global nilpotent cone.

Background

The paper constructs the de Rham geometric Langlands functor LG by first defining a coarse functor LG,coarse: D-mod¹(Bun_G) → QCoh(LS_G) via the spectral action and the vacuum Poincaré sheaf, and then upgrading it using cohomological control (Theorem 1.6.2) to land in IndCoh_Nilp(LS_G).

The conjecture asserts that this functor is an equivalence, identifying the automorphic category of half‑twisted D‑modules with the spectral category of ind‑coherent sheaves with nilpotent singular support. The authors also show various versions (restricted, tempered) are logically equivalent and relate the de Rham and Betti settings via Riemann–Hilbert.