A Rigid Irregular Connection on the Projective Line

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A Rigid Irregular Connection on the Projective Line

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Title: A Rigid Irregular Connection on the Projective Line
Author: Frenkel, Edward; Gross, Benedict H.

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Citation: Frenkel, Edward, and Benedict Gross. 2009. A rigid irregular connection on the projective line. Annals of Mathematics 170(3): 1469-1512.
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Abstract: In this paper we construct a connection ∇ on the trivial G-bundle on P1 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 1, 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.
Published Version: doi:10.4007/annals.2009.170.1469
Other Sources: http://arxiv.org/PS_cache/arxiv/pdf/0901/0901.2163v3.pdf
Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:4211573
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