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 $\overline{G}$.