Grassmannian perspectives of classical Lie groups and Cartan involutions
Yunxia Chen, Naichung Conan Leung
TL;DR
This work provides a unified Grassmannian framework for realizing classical noncompact Lie groups $G$ and their symmetric spaces via the Grassmannian compactification $\overline{G}$ of maximal isotropic or isotropic-analog subspaces. It introduces double graph-like and isotropic Grassmannians to encode the group structure and Lie algebra, and shows that the Cartan involution $\rho$ extends to an isometry $\bar{\\rho}$ on $\overline{G}$ with fixed point set $K$, while the companion involution $\\eta=\\rho^{-1}$ extends to $\\bar{\\eta}$ with fixed set $G_c/K$, yielding a generalized Borel embedding $G/K\\hookrightarrow G_c/K$ and a complementary pair of reflective submanifolds. The paper also demonstrates that $G/K$ can be realized as the space-like part of the compact dual Grassmannian, linking noncompact symmetric spaces to their compact duals and providing Cayley-transform relations within the Grassmannian setting. The results apply across all ten families of classical groups, with explicit geometric descriptions and an appendix detailing the group operation via the double graph-like Grassmannian. This geometric perspective unifies compactifications, Cartan involutions, and Borel-type embeddings in a common Grassmannian framework, enriching the toolkit for studying classical symmetric spaces.
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 $ρ$ on $G$ extends uniquely to an isometric involution $\barρ$ on $\overline{G}$ and $\overline{G}^{\barρ} = G^ρ = K$, the maximal compact subgroup of $G$. Third, we show that $η(g) = ρ(g)^{-1}$ extends uniquely to an isometric involution $\barη$ on $\overline{G}$ and $\overline{G}^{\barη} = G_c/K$, the compact symmetric space dual to $(G^η)_0 = G/K$. 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 $\overline{G}$.