Parabolic Subgroups of Real Direct Limit Groups

TL;DR

This paper extends the classification of parabolic subalgebras from complexified Lie algebras to real direct limit groups, providing geometric criteria for subgroup intersections in infinite-dimensional settings.

Contribution

It generalizes known results on parabolic subalgebras to real direct limit groups and introduces a geometric criterion for subgroup intersections.

Findings

Classification of parabolic subalgebras for real direct limit groups

Geometric criterion for intersections of parabolic subgroups

Analysis of orbit structures on flag ind-manifolds

Abstract

Let $G_R$ be a classical real direct limit Lie group and $g_R$ its Lie algebra. The parabolic subalgebras of the complexification $g_C$ were described by the first two authors. In the present paper we extend these results to $g_R$. This also gives a description of the parabolic subgroups of $G_R$. Furthermore, we give a geometric criterion for a parabolic subgroup $P_C$ of $G_C$ to intersect $G_R$ in a parabolic subgroup. This criterion involves the $G_R$-orbit structure of the flag ind-manifold $G_C/P_C$.