  For any two degrees coprime to the rank, we construct a family of ring
isomorphisms parameterized by GSp(2g) between the cohomology of the moduli
spaces of stable Higgs bundles which preserve the perverse filtrations. As
consequences, we prove two structural results concerning the cohomology of
Higgs moduli which are predicted by the P=W conjecture in non-abelian Hodge
theory: (1) Galois conjugation for character varieties preserves the perverse
filtrations for the corresponding Higgs moduli spaces. (2) The restriction of
the Hodge-Tate decomposition for a character variety to each piece of the
perverse filtration for the corresponding Higgs moduli space gives also a
decomposition.
  Our proof uses reduction to positive characteristic and relies on the
non-abelian Hodge correspondence in characteristic p between Dolbeault and de
Rham moduli spaces.
