Explicit Pseudo-K\"ahler Metrics on Flag Manifolds

TL;DR

This paper explicitly describes invariant pseudo-K"ahler metrics on flag manifolds associated with unitary groups, revealing new complex structures and their role in representation theory through geometric and algebraic analysis.

Contribution

It provides an explicit description of pseudo-K"ahler metrics on flag manifolds and uncovers the exact count of invariant complex structures, advancing understanding of geometric structures in representation theory.

Findings

Explicit pseudo-K"ahler metrics on flag manifolds are constructed.

The number of invariant complex structures on certain flag manifolds is exactly k!.

Pseudo-K"ahler structures realize representations in higher cohomology, extending classical results.

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