Explicit pseudo-Kähler metrics on flag manifolds

Abstract

The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) Kähler structure, famously used to realize the group's irreducible representations in holomorphic sections of appropriate line bundles (Borel-Weil theorem). Less studied are the (indefinite) invariant pseudo-Kähler structures they also admit, which can be used to realize the same representations in higher cohomology of the sections (Bott's theorem). Using “eigenflag” embeddings, we give a very explicit description of these metrics in the case of the unitary group. As a byproduct we show that Un/(Un1×⋯×Unk) has exactly k! invariant complex structures, a count which seems to have hitherto escaped attention.