Explicit Pseudo-K\"ahler Metrics on Flag Manifolds

DOI10.1016/J.DIFGEO.2022.101968arXiv2007.09302MaRDI QIDQ6345342

Publication date: 17 July 2020

Abstract: The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) K"ahler 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*-K"ahler 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 $U_{n} / (U_{n_{1}} i m e s c d o t s i m e s U_{n_{k}})$ has exactly $k!$ invariant complex structures, a count which seems to have hitherto escaped attention.

Mathematics Subject Classification ID

Grassmannians, Schubert varieties, flag manifolds (14M15) Kähler manifolds (32Q15) Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics (53C50) Homogeneous complex manifolds (32M10) Coadjoint orbits; nilpotent varieties (17B08)

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