General correspondence between epipelagic Hecke eigensystems and θ-connections

Establish that, in the de Rham geometric Langlands setting, for any admissible parahoric subgroup P of G((t)) and any stable functional φ in V_P^*, the Hecke eigen Gˇ-local system E_φ on P^1 \ {0,∞} obtained from the epipelagic automorphic datum (P_opp, 1; P(1), L_φ) is isomorphic to the θ-connection ∇_X attached to the corresponding stable grading of g and stable vector X in gˇ_1 that matches φ under the canonical identification V_P^*//L_P ≃ gˇ_1//Gˇ_0.

Background

The paper recalls a bijection between stable gradings of g and its Langlands dual gˇ, under which stable L_P-orbits in V_P^* correspond to stable Gˇ0-orbits in gˇ_1. It then cites a conjecture (due to Yun, recorded in [Che17]) asserting that, in de Rham geometric Langlands, the Hecke eigenvalue Eφ of the rigid epipelagic automorphic datum coincides with the θ-connection ∇_X determined by the matching stable grading and stable vector.

The authors prove this correspondence for a large family of inner stable gradings with positive Kac coordinate s_0 on both sides (Theorem 5), but the conjecture remains open in full generality, notably beyond these hypotheses.