Differential Galois groups of $G$-connections with Coxeter singularities

TL;DR

This paper extends Katz's theorem to compute differential Galois groups of $G$-connections with Coxeter singularities, broadening understanding in geometric Langlands and inverse Galois theory.

Contribution

It generalizes Katz's results to $G$-connections with Coxeter singularities, enabling explicit Galois group computations for a wide class of connections.

Findings

Computed Galois groups for Coxeter and Frenkel--Gross connections

Provided explicit constructions for Galois groups realizing all reductive subgroups

Extended the theory to connections with Coxeter singularities

Abstract

A fundamental theorem of Katz \cite{Katz87} determines the differential Galois groups of rank $n$ connections on algebraic curves with slope $r/n$ at a singularity, where $\gcd(r,n)=1$. We extend this result to $G$-connections, where $G$ is a simple algebraic group and the slope is $r/h$, with $h$ the Coxeter number of $G$ and $\gcd(r,h)=1$. This allows us to compute the differential Galois groups of a broad class of $G$-connections that have been central to recent advances in the geometric Langlands program and the Deligne--Simpson problem -- namely, Coxeter connections, generalised Frenkel--Gross connections, and Airy connections. We apply our results to inverse differential Galois theory by giving uniform and explicit constructions of $G$-connections whose differential Galois groups realise all reductive subgroups of maximal degree.