Bohr--Sommerfeld Lagrangians of moduli spaces of Higgs bundles

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