Pseudoconcavity of flag domains: The method of supporting cycles

TL;DR

This paper investigates the geometric property of pseudoconcavity in flag domains of complex Lie groups, showing that most are pseudoconcave unless they are products involving Hermitian symmetric domains, with estimates provided for their degree.

Contribution

It establishes that flag domains without non-constant holomorphic functions are pseudoconcave and provides explicit estimates of pseudoconcavity degree using root invariants.

Findings

Flag domains with only constant holomorphic functions are pseudoconcave.

Explicit pseudoconcavity degree estimates are given for domains in Grassmannians.

Most flag domains are pseudoconcave unless they are products involving Hermitian symmetric domains.

Abstract

A flag domain of a real from $G_0$ of a complex semismiple Lie group $G$ is an open $G_0$-orbit $D$ in a (compact) $G$-flag manifold. In the usual way one reduces to the case where $G_0$ is simple. It is known that if $D$ possesses non-constant holomorphic functions, then it is the product of a compact flag manifold and a Hermitian symmetric bounded domain. This pseudoconvex case is rare in the geography of flag domains. Here it is shown that otherwise, i.e., when $\mathcal{O}(D)\cong\mathbb{C}$, the flag domain $D$ is pseudoconcave. In a rather general setting the degree of the pseudoconcavity is estimated in terms of root invariants. This estimate is explicitly computed for domains in certain Grassmannians.