Quantization, reduction, and flag manifolds

Let K be a compact connected semi-simple Lie group, let G be its complexification, and let G = KAN be an Iwasawa decomposition. Let B be the Borel subgroup containing A and N. Let P be a parabolic subgroup of G containing B, and (P, P) its commutator subgroup. In this paper, we perform geometric quantization and symplectic reduction to the pseudo-Kähler forms on the complex homogeneous space G/(P, P). The reduced space is a disjoint union of copies of the flag manifold G/P, and this allows us to study the signatures of the K-invariant pseudo-Kähler forms on G/P via symplectic reduction. We also discuss the connectivity of the reduced space.