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 $U_n/(U_{n_1}\times \cdots \times U_{n_k})$ has exactly $k!$ invariant complex structures, a count which seems to have hitherto escaped attention.

20 pages, 2 figures. Accepted version, to appear in Differential Geometry and its Applications