  Starting from a homogeneous affine Springer fiber $Fl_{\psi}$, 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
$Fl_{\psi}$ 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 $Fl_{\psi}$, and construct a map from the de Rham space to the Betti space
which we conjecture to be an analytic isomorphism.
