Geometric Langlands for irregular theta connections and epipelagic representations

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