Parabolic Hecke eigensheaves

We study the Geometric Langlands Conjecture (GLC) for rank two flat bundles on the projective line $C$ with tame ramification at five points ${p_{1}, p_{2}, p_{3}, p_{4}, p_{5} }$\mathrm{.}Inparticularweconstructtheautomorphic$D$-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 ${\mathbb{P}}^4$.