  The solution of the eigenvalue problem of the Laplacian on a general
homogeneous space G/H is given. Here, G is a compact, semisimple Lie group, H
is a closed subgroup of G, and the rank of H is equal to the rank of G. It is
shown that the multiplicity of the lowest eigenvalue of the Laplacian on G/H is
just the degeneracy of the lowest Landau level for a particle moving on G/H in
the presence of the background gauge field. Moreover, the eigenspace of the
lowest eigenvalue of the Laplacian on G/H is, up to a sign, equal to the
G-equivariant index of the Dirac operator of Kostant on G/H.
