Grassmannian perspectives of classical Lie groups and Cartan involutions

TL;DR

This paper explores Grassmannian geometric methods to compactify classical noncompact reductive Lie groups, extending involutions and revealing geometric structures related to symmetric spaces and classical embeddings.

Contribution

It provides a unified Grassmannian-based construction of compactifications, extends Cartan involutions to these compactifications, and generalizes classical Borel embeddings with geometric insights.

Findings

Compactifications realized via Grassmannian geometry

Extension of Cartan involution to compactification

Generalization of classical Borel embeddings

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 $\overline{G}$.