  The spectral side of the (conjectural) Betti geometric Langlands
correspondence concerns sheaves on the character stack of an algebraic curve;
in particular, the categories in question are manifestly invariant under
deformations of the curve. By contrast the same invariance is certainly not
manifest, and is presently not known, for their automorphic counterparts, in
particular because the singularities of the global nilpotent cone may vary
significantly with the complex structure of the curve.
  Here we establish the corresponding invariance statement for the category of
microsheaves on the open subset of stable Higgs bundles on nonstacky components
where all semistables are stable, e.g. for coprime rank and degree or for a
punctured curve with generic parabolic weights. The proof uses the known global
symplectic geometry of the Higgs moduli space to invoke recent results on the
invariance of microlocal sheaves.
