Kaehler structures on Kc/(P,P)

TL;DR

This paper classifies K-invariant Kaehler structures on certain homogeneous spaces derived from compact semi-simple Lie groups, analyzing their geometric quantization and representation-theoretic properties.

Contribution

It provides a classification of Kaehler structures on Kc/(P,P) and explores their moment maps, line bundles, and representation multiplicities.

Findings

Classification of K-invariant Kaehler structures

Analysis of moment maps and pre-quantum line bundles

Study of irreducible representation multiplicities

Abstract

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.