Geometric Langlands for Irregular Theta Connections and Epipelagic Representations

From a stable vector of a stable grading on a simple Lie algebra, Yun defined a rigid automorphic datum that encodes a epipelagic representation, and also an irregular connection on the projective line called θ-connection. We show that under geometric Langlands correspondence, θ-connection corresponds to the Hecke eigensheaf attached the rigid automorphic datum, assuming the stable grading is inner and its Kac coordinate s₀ is positive. We provide numerous applications of the main result including physical and cohomological rigidity of θ-connections, global oper structures, and a de Rham analog of Reeder-Yu's predictions on epipelagic Langlands parameters.