  We develop a relative version of Kostant's harmonic theory and use this to
prove a relative version of Kostant's theorem on Lie algebra (co)homology.
These are associated to two nested parabolic subalgebras in a semisimple Lie
algebra. We show how relative homology groups can be used to realize
representations with lowest weight in one (regular or singular) affine Weyl
orbit. In the regular case, we show how all the weights in the orbit can be
realized as relative homology groups (with different coefficients). These
results are motivated by applications to differential geometry and the
construction of invariant differential operators.
