Moduli spaces of irregular singular connections

TL;DR

This paper develops a geometric theory of fundamental strata to analyze irregular singular connections on the projective line, linking them to the Langlands correspondence and constructing a symplectic moduli space.

Contribution

It introduces the concept of regular strata for irregular connections and constructs a symplectic moduli space incorporating these structures.

Findings

Defined regular strata for irregular singular connections

Generalized regular semisimple leading term condition

Constructed a symplectic moduli space of connections

Abstract

In the geometric version of the Langlands correspondence, irregular singular point connections play the role of Galois representations with wild ramification. In this paper, we develop a geometric theory of fundamental strata to study irregular singular connections on the projective line. Fundamental strata were originally used to classify cuspidal representations of the general linear group over a local field. In the geometric setting, fundamental strata play the role of the leading term of a connection. We introduce the concept of a regular stratum, which allows us to generalize the condition that a connection has regular semisimple leading term to connections with non-integer slope. Finally, we construct a symplectic moduli space of meromorphic connections on the projective line that contain a regular stratum at each singular point.