Grassmannian perspectives of classical Lie groups and Cartan involutions

Abstract

Classical noncompact reductive Lie group $G$ admits a compactification $\overline{G}$ as a Riemannian symmetric space by He. First, we provide a unified construction of these compactifications via Grassmannian geometry and realize the group structures in terms of the geometry of configurations of linear subspaces. Second, we show that the Cartan involution $\rho$ on $G$ extends uniquely to an isometric involution $\bar{\rho }$ on $\overline{G}$ and ${\overline{G}}^{\bar{\rho }}=G^{\rho }=K$, the maximal compact subgroup of $G$. Third, we show that $\eta (g)=\rho (g{)}^{-1}$ extends uniquely to an isometric involution $\bar{\eta }$ on $\overline{G}$ and ${\overline{G}}^{\bar{\eta }}=G_c/K$, the compact symmetric space dual to $(G^{\eta }{)}_0=G/K$. This provides a natural generalization of the classical Borel embeddings $G/K\hookrightarrow G_c/K$. Furthermore, $K$ and $G_c/K$ form a complementary pair of reflective submanifolds in $\overline{G}$.