Parabolic Hecke eigensheaves

TL;DR

This paper advances the understanding of the Geometric Langlands Conjecture for rank two flat bundles with tame ramification on the projective line, by constructing automorphic D-modules via non-abelian Hodge theory and Fourier-Mukai transforms.

Contribution

It constructs automorphic D-modules predicted by GLC for rank two parabolic bundles on the projective line with five points, using novel geometric and Hodge-theoretic techniques.

Findings

Construction of automorphic D-modules on moduli spaces

Reduction of the problem to classical projective geometry

Application of non-abelian Hodge theory and Fourier-Mukai transform

Abstract

We study the Geometric Langlands Conjecture (GLC) for rank two flat bundles on the projective line $C$ with tame ramification at five points $\{p_{1}, p_{2}, p_{3}, p_{4}, p_{5} \}$. In particular we construct the automorphic $D$-modules predicted by GLC on the moduli space of rank two parabolic bundles on $(C, \{p_{1}, p_{2}, p_{3}, p_{4}, p_{5} \})$. The construction uses non-abelian Hodge theory and a Fourier-Mukai transform along the fibers of the Hitchin fibration to reduce the problem to one in classical projective geometry on the intersection of two quadrics in $\mathbb{P}^{4}$.