On the notion of conductor in the local geometric Langlands correspondence

TL;DR

This paper establishes a geometric analogue of the local Langlands correspondence, showing that the conductor of a categorical representation exceeds the irregularity of the associated meromorphic connection, extending classical number theory results.

Contribution

It introduces a geometric analogue of the conductor comparison in the local Langlands correspondence, relating categorical representations and meromorphic connections.

Findings

Conductor of categorical representation is greater than irregularity.

Extends classical number theory results to geometric setting.

Bridges representation theory and algebraic geometry.

Abstract

Under the local Langlands correspondence, the conductor of an irreducible representation of $\Gl_n(F)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.