Microsheaves from Hitchin fibers via Floer theory

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.