Differential Galois groups of $G$-connections with Coxeter singularities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914339213017088 |
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| author | Kamgarpour, Masoud Sage, Daniel S. |
| author_facet | Kamgarpour, Masoud Sage, Daniel S. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_11742 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Differential Galois groups of $G$-connections with Coxeter singularities Kamgarpour, Masoud Sage, Daniel S. Algebraic Geometry Representation Theory 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. |
| title | Differential Galois groups of $G$-connections with Coxeter singularities |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2309.11742 |