  The Dirac operator for a manifold Q, and its chirality operator when Q is
even dimensional, have a central role in noncommutative geometry. We
systematically develop the theory of this operator when Q=G/H, where G and H
are compact connected Lie groups and G is simple. An elementary discussion of
the differential geometric and bundle theoretic aspects of G/H, including its
projective modules and complex, Kaehler and Riemannian structures, is presented
for this purpose. An attractive feature of our approach is that it
transparently shows obstructions to spin- and spin_c-structures. When a
manifold is spin_c and not spin, U(1) gauge fields have to be introduced in a
particular way to define spinors. Likewise, for manifolds like SU(3)/SO(3),
which are not even spin_c, we show that SU(2) and higher rank gauge fields have
to be introduced to define spinors. This result has potential consequences for
string theories if such manifolds occur as D-branes. The spectra and
eigenstates of the Dirac operator on spheres S^n=SO(n+1)/SO(n), invariant under
SO(n+1), are explicitly found. Aspects of our work overlap with the earlier
research of Cahen et al..
