Rigidity in automorphic representations and local systems

TL;DR

This paper introduces the concept of rigidity for automorphic representations over global function fields, explicitly constructs their Langlands parameters as local systems, and discusses examples, especially for $GL_{2}$.

Contribution

It defines rigidity in automorphic representations, constructs associated local systems explicitly, and provides new examples, advancing understanding in the Langlands program for function fields.

Findings

Explicit construction of Langlands parameters as local systems.

Identification of rigid automorphic representations.

New examples for $GL_{2}$ discussed in detail.

Abstract

We introduce the notion of rigidity for automorphic representations of groups over global function fields. We construct the Langlands parameters of rigid automorphic representations explicitly as local systems over open curves. We expect these local systems to be rigid. Examples of rigid automorphic representations from previous work are reviewed and more examples for $GL_{2}$ are discussed in details.