  Fix a non-stacky component of the moduli of stable Higgs bundles, on which
the Hitchin fibration is proper. We show that any smooth Hitchin fiber
determines a microsheaf on the global nilpotent cone, that distinct fibers give
rise to orthogonal microsheaves, and that the endomorphisms of the microsheaf
is isomorphic to the cohomology of the Hitchin fiber. These results are
consequences of recent advances in Floer theory. Natural constructions on our
microsheaves provide plausible candidates for Hecke eigensheaves for the
geometric Langlands correspondence.
