Any flat bundle on a punctured disc has an oper structure

TL;DR

This paper proves that every flat G-bundle on a punctured disc can be endowed with an oper structure, advancing the understanding of local geometric Langlands correspondence and related algebraic structures.

Contribution

It establishes the existence of oper structures on all flat G-bundles on a punctured disc, using deformations of affine Springer fibers, and constructs new representations of affine Weyl groups.

Findings

Every flat G-bundle admits an oper structure.

Deformations of affine Springer fibers are used in the proof.

New representations of affine Weyl groups are constructed.

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 arXiv:math/0508382. 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.