Cartan subgroups in connected locally compact groups

Abstract

We define Cartan subgroups in connected locally compact groups, which extends the classical notion of Cartan subgroups in Lie groups. We prove their existence and justify our choice of the definition which differs from the one given by Chevalley on general groups. Apart from proving some properties of Cartan subgroups, we show that the Cartan subgroups of the quotient groups are precisely the images of Cartan subgroups of the ambient group. We establish the so-called `Levi' decomposition of Cartan subgroups which extends W\"ustner's decomposition theorem and our earlier results for Lie groups. We also show that the centraliser of any maximal torus of the radical is connected and its Cartan subgroups are also Cartan subgroups of the ambient group; moreover, every Cartan subgroup arises this way. We prove that Cartan subalgebras defined by Hofmann and Morris in pro-Lie algebras are the same as those corresponding to Cartan subgroups in case of pro-Lie algebras of connected locally compact groups, and that they are nilpotent. We characterise density of the image of a power map in a connected locally compact group in terms of its surjectivity on all Cartan subgroups, and show that weak exponentiality of the group is equivalent to the condition that all its Cartan subgroups are connected.