Variations of Hodge structure and orbits in flag varieties

Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$–orbits in flag varieties $G\mathrm{/}P$. We investigate Hodge–theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths–Yukawa coupling to the variety of lines on $G\mathrm{/}P$ (under a minimal homogeneous embedding), construct a large class of polarized $G_{\mathbb{R}}$–orbits in $G\mathrm{/}P$, and compute the associated Hodge–theoretic boundary components. An emphasis is placed throughout on adjoint flag varieties and the corresponding families of Hodge structures of levels two and four.