Differential Galois groups of G-connections with Coxeter singularities

2023/09/21 by Masoud Kamgarpour, Kamgarpour, Masoud, Daniel S. Sage +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2309.11742

openalex publication_date 2023/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A fundamental theorem of Katz \citeKatz87 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.

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