  The coadjoint orbits of compact Lie groups carry many K\"ahler structures,
which include a Riemannian metric and a complex structure. We provide a fairly
explicit formula for the Levi-Civita connection of the Riemannian metric, and
we use the complex structure to give a fairly explicit construction of a
canonical Dirac operator for the Riemannian metric, in a way that avoids use of
the Spin^c groups. Substantial parts of our results apply to compact
almost-Hermitian homogeneous spaces, and to other connections besides the
Levi-Civita connection. For these other connections we give a criterion that is
both necessary and sufficient for their Dirac operator to be formally
self-adjoint.
  We hope to use the detailed results given here to clarify statements in the
literature of high-eneregy physics concerning "Dirac operators" for matrix
algebras that converge to coadjoint orbits. To facilitate this we employ here
only global methods -- we never use local coordinate charts, and we use the
cross-section modules of vector bundles.
