2025/07/09 by Kumar, Parteek, Mandal, Arunava, Singh, Shashank Vikram
#22E15 #22E25 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2507.07027
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 Γ⊂ \rm Aut(G) of semisimple automorphisms, there exists a Γ-stable Cartan subgroup, by using a result of Borel and Mostow. We characterize the Γ-stable Cartan subgroups (with induced action) in the quotient group modulo a Γ-stable closed normal subgroup as the images of the Γ-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 Γ-stable Cartan subgroup H of G, and a Γ-stable closed connected normal subgroup M of G, we prove that there exists a Γ-stable Cartan subgroup HM of M such that H∩ M⊂ HM.