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arXiv 2007.093022020-07math.DGmath.SG

Explicit Pseudo-Kähler Metrics on Flag Manifolds

Thomas Mason, Francois Ziegler

备注：: 20 pages, 2 figures. Accepted version, to appear in Differential Geometry and its Applications
发表信息：: Differ. Geom. Appl. 86 (2023) 101968
DOI：: 10.1016/j.difgeo.2022.101968

英文摘要：

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.

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