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 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 applications of the main result on cohomological rigidity of $\theta$-connections, global oper structures, and a de Rham analog of Reeder-Yu's predictions on epipelagic Langlands parameters.

This version corrected a mistake on the nonvanishing of Hecke eigensheaf and some typos in the published version on PLMS. We removed physical rigidity from the main theorem and added some new results in the appendix