  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.
