On stable Cartan subgroups of Lie groups

TL;DR

This paper investigates the existence and properties of stable Cartan subgroups in real Lie groups under automorphisms, providing characterizations, explicit automorphism identifications, and stability results for subgroups.

Contribution

It establishes the existence of b3-stable Cartan subgroups in connected real Lie groups and characterizes their behavior in quotients and subgroups, with explicit automorphism examples for classical Lie algebras.

Findings

Existence of b3-stable Cartan subgroups in connected real Lie groups.

Characterization of b3-stable Cartan subgroups in quotient groups.

Explicit automorphisms fixing representatives of non-conjugate Cartan subalgebras.

Abstract

Let $G$ be a connected real Lie group with associated Lie algebra $\mathfrak g$, and let ${\rm Aut}(G)$ be the group of (Lie) automorphisms of $G$. It is noted here that, given a super-solvable subgroup $\Gamma\subset {\rm Aut}(G)$ of semisimple automorphisms, there exists a $\Gamma$-stable Cartan subgroup, by using a result of Borel and Mostow. We characterize the $\Gamma$-stable Cartan subgroups (with induced action) in the quotient group modulo a $\Gamma$-stable closed normal subgroup as the images of the $\Gamma$-stable Cartan subgroups in the ambient group. It is well known that a semisimple automorphism of $\mathfrak g$ always fixes a Cartan subalgebra of $\mathfrak g$. Conversely, if we take a representative from each non-conjugate class of Cartan subalgebras in a real Lie algebra, we show that there exists a non-identity automorphism that fixes these representatives. We explicitly identify such automorphisms in the case of classical simple Lie algebras. As a consequence, we deduce an analogous result for semisimple Lie groups. Moreover, given a $\Gamma$-stable Cartan subgroup $H$ of $G$, and a $\Gamma$-stable closed connected normal subgroup $M$ of $G$, we prove that there exists a $\Gamma$-stable Cartan subgroup $H_M$ of $M$ such that $H\cap M\subset H_M$.