  In the present paper we consider two related problems, i.e. the description
of geodesics and the calculation of the spectrum of the Laplace-Beltrami
operator on a flag manifold. We show that there exists a family of invariant
metrics such that both problems can be solved simply and explicitly. In order
to determine the spectrum of the Laplace-Beltrami operator, we construct
natural, finite-dimensional approximations (of spin chain type) to the Hilbert
space of functions on a flag manifold.
