𝐾-invariant Kaehler structures on 𝐾_{𝐂}/𝐍 and the associated line bundles

Abstract

Let K K be a compact semi-simple Lie group, and let N N be a maximal unipotent subgroup of the complexified group K C K_{ \mathbf {C}} . In this paper, we classify all the K K -invariant Kaehler structures on K C / N K_{ \mathbf {C}}/N . For each Kaehler structure ω \omega , let L { \mathbf {L}} be the line bundle with connection whose curvature is ω \omega . We then study the holomorphic sections of L { \mathbf {L}} , which constitute a K K -representation space.