Mathematical Research Letters

Volume 17 (2010)

Number 1

Any flat bundle on a punctured disc has an oper structure

Pages: 27 – 37

DOI: http://dx.doi.org/10.4310/MRL.2010.v17.n1.a3

Authors

Edward Frenkel (University of California at Berkeley)

Xinwen Zhu (Harvard University)

Abstract

We prove that any flat $G$-bundle, where $G$ is a complex connected reductive algebraic group, on the punctured disc admits the structure of an oper. This result is important in the local geometric Langlands correspondence proposed in \cite{FG}. Our proof uses certain deformations of the affine Springer fibers which could be of independent interest. As a byproduct, we construct representations of affine Weyl groups on the homology of these deformations generalizing representations constructed by Lusztig.

Full Text (PDF format)