Epipelagic representations and rigid local systems

TL;DR

This paper constructs automorphic and Galois representations with epipelagic local components over function fields, leading to new conjecturally rigid, wildly ramified local systems that generalize Kloosterman sheaves, and analyzes their monodromy.

Contribution

It introduces new automorphic and Galois representations with epipelagic local components over function fields, expanding the class of rigid local systems.

Findings

Constructed automorphic representations with epipelagic local components.

Attached Galois representations that generalize Kloosterman sheaves.

Computed monodromy for classical groups.

Abstract

We construct automorphic representations for quasi-split groups $G$ over the function field $F=k(t)$ one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified $^{L}G$-local systems over $\mathbb{P}^{1}-\{0,\infty\}$ that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ng\^o and the author. We study the monodromy of these local systems and compute all examples when $G$ is a classical group.