Rigidity of proper almost-homogeneous domains in positive flag manifolds

Abstract

We show that, inside the Shilov boundary of any given Hermitian symmetric space of tube type, there is, up to isomorphism, only one proper domain such that every point on its boundary belongs to the closure of an orbit under its automorphism group. This gives a classification of all closed proper manifolds locally modelled on such Shilov boundaries, and provides a positive answer, in the case of flag manifolds admitting a $\Theta$-positive structure, to a rigidity question of Limbeek and Zimmer.