Riemann–Hilbert for tame complex parahoric connections

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. The corresponding Betti data involves pairs (M, P) consisting of the local monodromy M ∈ G and a (weighted) parabolic subgroup P ⊂ G such that M ∈ 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.