A rigid irregular connection on the projective line

Abstract

In this paper we construct 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 the point $0$, with principal unipotent monodromy, and has an irregular singularity at the point $\infty$, with slope $1/h$, the reciprocal of the Coxeter number of $G$. The connection $\nabla$, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic $0$ counterpart of a hypothetical family of $\ell$-adic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. These $\ell$-adic representations, and their characteristic $0$ counterparts, have been constructed in some cases by Deligne and Katz. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. It provides an example of the geometric Langlands correspondence with wild ramification. We compute the de Rham cohomology of our connection with values in a representation $V$ of $G$, and describe the differential Galois group of $\nabla$ as a subgroup of $G$.