Geometric Langlands for Irregular Theta Connections and Epipelagic Representations

TL;DR

This paper explores the geometric Langlands correspondence for irregular theta connections associated with epipelagic representations, establishing a correspondence with Hecke eigensheaves and demonstrating applications in rigidity and oper structures.

Contribution

It proves the correspondence between theta connections and Hecke eigensheaves for certain stable gradings, extending the geometric Langlands framework to irregular, epipelagic cases.

Findings

Establishes the geometric Langlands correspondence for theta connections.

Shows physical and cohomological rigidity of theta connections.

Provides a de Rham analog of Reeder-Yu's predictions.

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 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.