Mathematical Research Letters

Volume 17 (2010)

Number 1

Any flat bundle on a punctured disc has an oper structure

Pages: 27 – 37



Edward Frenkel (University of California at Berkeley)

Xinwen Zhu (Harvard University)


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.

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