An explicit local geometric Langlands for supercuspidal representations: the toral case

TL;DR

This paper proposes a conjecture linking local geometric Langlands correspondence with supercuspidal representations, confirming it for toral cases and connecting it to global Airy connections and Hecke eigensheaves.

Contribution

It formulates a conjecture on local geometric Langlands for supercuspidal representations, refined for regular cases, and confirms it for toral supercuspidal representations.

Findings

Confirmed the conjecture for toral supercuspidal representations.

Established correspondence between global Airy connections and Hecke eigensheaves.

Connected irreducible isoclinic connections with Langlands parameters.

Abstract

We formulate a conjecture on local geometric Langlands for supercuspidal representations using Yu's data and Feigin-Frenkel isomorphism. We refine our conjecture for a large family of regular supercuspidal representations defined by Kaletha, and then confirm the conjecture for toral supercuspidal representations of Adler whose Langlands parameters turn out to be exactly all the irreducible isoclinic connections. As an application, we establish the conjectural correspondence between global Airy connections for reductive groups and the family of Hecke eigensheaves constructed by Jakob-Kamgarpour-Yi.