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A rigid irregular connection on the projective line. (English) Zbl 1209.14017

The Langlands correspondence relates automorphic representations of a reductive group \(G\) over the ring of adèles of a global field \(F\) and \(\ell\)-adic representations of the Galois group of \(F\) with values in a dual group of \(G\). In some cases there is a unique irreducible automorphic representation with prescribed local behavior at finitely many points.
A special case of this is when \(F\) is the function field of the projective line \(\mathbb{P}^1\) over a finite field \(k\), and \(G\) is a simple group over \(k\). The problem is then to determine the corresponding family of \(\ell\)-adic representations of the Galois group of \(F\).
The authors construct in this paper a connection \(\nabla\) on the trivial \(G\)-bundle on \(\mathbb{P}^1\) for any simple complex algebraic group \(G\), which is regular outside of the points \(0\) and \(\infty\), has a regular singularity at \(0\) (with principal unipotent monodromy) and an irregular singularity at \(\infty\), with slope the reciprocal of the Coxeter number of \(G\).
They compute the de Rham cohomology of this connection with values in a representation \(V\) of \(G\) and describe the differential Galois group of \(\nabla\) as a subgroup of \(G\).

MSC:

14F40 de Rham cohomology and algebraic geometry
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
11R39 Langlands-Weil conjectures, nonabelian class field theory
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References:

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