A rigid irregular connection on the projective line | Annals of Mathematics

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 ∞, 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.

KEYWORDS

SHARE & LIKE

COMMENTS

ABOUT THE AUTHOR

数学年刊(Annals of Mathematics)

0 Following 0 Fans 0 Projects 674 Articles

SIMILAR ARTICLES

Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin

阅读更多

Abstract For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we

阅读更多

Abstract Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let

阅读更多

Abstract We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely

阅读更多

Abstract We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We

阅读更多

Abstract This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal

阅读更多

Abstract We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge

阅读更多

Abstract We prove that isoparametric hypersurfaces with (g,m)=(6,2) are homogeneous, which answers Dorfmeister-Neher’s conjecture affirmatively and so

阅读更多

Abstract We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorificat

阅读更多

Abstract If F(x,y)∈ℤ[x,y] is an irreducible binary form of degree k≥3, then a theorem of Darmon and Granville implies that the generalized superellipt

阅读更多