Remarks on the signatures of invariant pseudo-Kähler metrics on generalized flag manifolds

Abstract

A pseudo-Kähler manifold is a natural generalization of a Kähler manifold. It is well-known that any generalized flag manifold has pseudo-Kähler metrics. Moreover, there exists a $T$-root system corresponding to a generalized flag manifold. In this paper, we investigate the signatures of invariant pseudo-Kähler metrics on a generalized flag manifold of which the $T$-root system becomes one of the irreducible reduced root systems (in general, a $T$-root system is not an irreducible reduced root system).