Bounded symmetric domains and derived geometric structures

Wilhelm, Kaup

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 13 (2002) no. 3-4, pp. 243-257.

Résumé

Every homogeneous circular convex domain $D \subset \mathbb{C}^{n}$ (a bounded symmetric domain) gives rise to two interesting Lie groups: The semi-simple group $G = Aut(D)$ of all biholomorphic automorphisms of $D$ and its isotropy subgroup $K \subset GL(n,\mathbb{C})$ at the origin (a maximal compact subgroup of $G$). The group $G$ acts in a natural way on the compact dual $X$ of $D$ (a certain compactification of $\mathbb{C}^{n}$ that generalizes the Riemann sphere in case $D$ is the unit disk in $\mathbb{C}$). Various authors have studied the orbit structure of the $G$-space $X$, here we are interested in the Cauchy-Riemann structure of the $G$-orbits in $X$ (which in general are only real-analytic submanifolds of $X$). Also, we discuss certain $K$-orbits in the Grassmannian of all linear subspaces of $\mathbb{C}^{n}$ that are closely related to the geometry of the bounded symmetric domain $D$.