An explicit geometric Langlands correspondence for the projective line minus four points

This thesis concerns the tamely ramified geometric Langlands correspondence in rank 2 on P¹_{F_q} minus four distinct points. For each irreducible pure rank 2 local system E with unipotent monodromy on P¹_{F_q} \ D, where D = {\infty, 0, 1, t} ⊂ P¹1_{F_q}(F_q)$ is a set of four distinct points, we construct the associated Hecke eigensheaf on the moduli space of parabolic rank 2 vector bundles on P¹. The proof of this seems to be new and relies on the observations that the support of the cusp forms can be obtained from a single parabolic vector bundle through Hecke transforms, and on a symmetry that results from this. In addition, we construct a basis of the q-dimensional space of cusp forms in a single degree and provide an explicit formula for the action of the Hecke operator on the cusp forms in terms of that basis.


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