  Let K be a compact semi-simple Lie group. We classify K-invariant Kaehler
structures on the space Kc/(P,P), where Kc is the complexification of K, P is a
parabolic subgroup of Kc, and (P,P) the commutator subgroup. For each Kaehler
structure, we study its moment map and associated pre-quantum line bundle for
geometric quantization. Some holomorphic sections of the line bundle form a
unitary K-representation space, and we study the multiplicity of its
irreducible subrepresentations.
