5. Fukaya category of Hitchin fibration

Problem 5.1.
Compute the Fukaya category of the total space of the Hitchin fibration, and compare with representations of the double affine Hecke algebra (DAHA). 
The above problem is fairly ambitious: so what would be a first step? The next problem would give some situations where we could perform calculations of the Fukaya category.
Problem 5.2.
Characterize situation where the central fibre of the Hitchin fibration has known, smooth components, intersecting transversely.
Cite this as: AimPL: Singular geometry and Higgs bundles in string theory, available at http://aimpl.org/singularhiggs.