  The spectrum and degeneracies of the Dirac operator are analysed on compact
coset spaces when there is a non-zero homogeneous background gauge field which
is compatible with the symmetries of the space, in particular when the gauge
field is derived from the spin-connection. It is shown how the degeneracy of
the lowest Landau level in the recently proposed higher dimensional quantum
Hall effect is related to the Atiyah-Singer index theorem for the Dirac
operator on a compact coset space.
