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

Masoud Kamgarpour, Daniel S. Sage

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.

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

Abstract

A fundamental theorem of Katz \cite{Katz87} determines the differential Galois groups of rank connections on algebraic curves with slope at a singularity, where . We extend this result to -connections, where is a simple algebraic group and the slope is , with the Coxeter number of and . This allows us to compute the differential Galois groups of a broad class of -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 -connections whose differential Galois groups realise all reductive subgroups of maximal degree.
Paper Structure (94 sections, 41 theorems, 102 equations, 3 tables)

This paper contains 94 sections, 41 theorems, 102 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Reductive subgroups of maximal degree
  4. Main result
  5. Normal forms of Coxeter singularities
  6. Applications to Coxeter connections
  7. Applications to inverse differential Galois theory
  8. Relevance in the geometric Langlands program
  9. Relevance in the Deligne--Simpson problem
  10. Future directions
  11. Structure of the paper
  12. Acknowledgments
  13. Group-theoretic preliminaries
  14. Notation
  15. Pinned automorphisms
  16. ...and 79 more sections

Key Result

Theorem 1

Let $\nabla$ be a rank $n$ connection on $X-S$ with trivial determinant and connected geometric monodromy. Assume $\nabla$ has an irregular singularity at some $s\in S$ with slope $r/n$, where $\gcd(r,n)=1$. Then $G_\nabla = \mathrm{Sp}_n$ if $n$ is even and $\nabla$ is self-dual; otherwise, $G_\nab

Theorems & Definitions (83)

  • Theorem 1: Katz87
  • Definition 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • Remark 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Theorem 10
  • ...and 73 more