Non-abelian Hodge moduli spaces and homogeneous affine Springer fibers

Starting from a homogeneous affine Springer fiber $F{l}_{\psi }$, we construct three moduli spaces that correspond to the Dolbeault, de Rham and Betti aspects of a hypothetical Simpson correspondence with wild ramifications. We show that $F{l}_{\psi }$ is homeomorphic to the central Lagrangian fiber in the Dolbeault space, prove that the Dolbeaut and de Rham spaces both have the same cohomology as $F{l}_{\psi }$, and construct a map from the de Rham space to the Betti space which we conjecture to be an analytic isomorphism.