Riemann-Hilbert for tame complex parahoric connections

TL;DR

This paper establishes a local Riemann-Hilbert correspondence for tame meromorphic connections with parahoric structures, linking differential equations to monodromy data and constructing associated quasi-Hamiltonian structures.

Contribution

It introduces a new Riemann-Hilbert correspondence compatible with parahoric level structures, extending classical cases to more general parabolic G-bundles.

Findings

Established a correspondence between connections and monodromy data involving pairs (M,P).

Constructed quasi-Hamiltonian structures on spaces of enriched monodromy data.

Extended the multiplicative Brieskorn-Grothendieck-Springer resolution to the parabolic case.

Abstract

A local Riemann-Hilbert correspondence for tame meromorphic connections on a curve compatible with a parahoric level structure will be established. Special cases include logarithmic connections on G-bundles and on parabolic G-bundles, where G is a complex reductive group. The corresponding Betti data involves pairs (M,P) consisting of the local monodromy M in G and a (weighted) parabolic subgroup P of G such that M is in P, as in the multiplicative Brieskorn-Grothendieck-Springer resolution (extended to the parabolic case). The natural quasi-Hamiltonian structures that arise on such spaces of enriched monodromy data will also be constructed.