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 $\theta$-connection. We show that under geometric Langlands correspondence, $\theta$-connection corresponds to the Hecke eigensheaf attached the rigid automorphic datum, assuming the stable grading is inner and its Kac coordinate $s_0$ is positive. We provide numerous applications of the main result including physical and cohomological rigidity of $\theta$-connections, global oper structures, and a de Rham analog of Reeder-Yu's predictions on epipelagic Langlands parameters.