Parabolic Hecke eigensheaves

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 $\mathbb{P}^{4}$.