Bohr–Sommerfeld Lagrangians of moduli spaces of Higgs bundles

Let XX be a compact connected Riemann surface of genus at least two. Let MH(r,d)MH(r,d) denote the moduli space of semistable Higgs bundles on XX of rank rr and degree dd. We prove that the compact complex Bohr–Sommerfeld Lagrangians of MH(r,d)MH(r,d) are precisely the irreducible components of the nilpotent cone in MH(r,d)MH(r,d). This generalizes to Higgs GG-bundles and also to the parabolic Higgs bundles.