Schubert varieties and cycle spaces

Abstract

Let $G_0$ be a real semisimple Lie group. It acts naturally on every complex flag manifold $Z=G/Q$ of its complexification. Given an Iwasawa decomposition $G_0 = K_0 A_0 N_0$, a $G_0$-orbit $γ⊂Z$, and the dual $K$-orbit $κ⊂Z$, Schubert varieties are studied and a theory of Schubert slices for arbitrary $G_0$-orbits is developed. For this, certain geometric properties of dual pairs $(γ,κ)$ are underlined. Canonical complex analytic slices contained in a given $G_0$-orbit γ which are transversal to the dual $K_0$-orbit $\gamma \cap \kappa$ are constructed and analyzed. Associated algebraic incidence divisors are used to study complex analytic properties of certain cycle domains. In particular, it is shown that the linear cycle space $Ω_W$($D$) is a Stein domain that contains the universally defined Iwasawa domain $Ω_I$. This is one of the main ingredients in the proof that $Ω_W(D)=Ω_{AG}$ for all but a few Hermitian exceptions. In the Hermitian case, $Ω_W(D)$ is concretely described in terms of the associated bounded symmetric domain.