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.