A rigid irregular connection on the projective line

By E. Frenkel, B. Gross. 2009

82 citations recorded by the source corpus.

In this paper we construct a connection ∇ on the trivial G-bundle on P 1 for any simple complex algebraic group G, which is regular outside of the points 0 and ∞, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point ∞, with slope 1/h, the reciprocal of the Coxeter number of G. The connection ∇, 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 l-adic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. These l- 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 ∇ as a subgroup of G.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f0e815a4688c9f4682448820754c3b185c58c6fa

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Kloosterman sheaves for reductive groups (2010, background): “The predictions about the conjectural local system KlǦ(φ) made in [15] served as a guideline for our work.”
Arithmetic invariants of discrete Langlands parameters (2010, methodology): “Then in chapter 5 we compute the leading term of Γ ϕ and its implications for ϕ, arriving at the notions of minimal Swan conductor and simple wild parameters.”