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

A fundamental theorem of Katz \cite{Katz87} determines the differential Galois groups of rank $n$ connections on algebraic curves with slope $r\mathrm{/}n$ 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 $r\mathrm{/}h$, 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.