Mappings preserving cosets of subgroups and the Fourier-Stieltjes algebra

Abstract

I will outline two parallel families of results. Suppose that \(G_1\) and \(G_2\) are connected Lie groups, and that  \(\phi: G_1 \to G_2\) is a bijection. In many cases it is possible to prove that, if \(\phi\) sends cosets of subgroups to cosets of subgroups, or if composition with \(\phi\)  maps the Fourier-Stieltjes algebra \(B(G_2)\) to \(B(G_1)\), then \(\phi\) is composed of some or all of the following: a group isomorphism; a translation, and reflection. Curiously, nilpotent Lie groups are the hardest to deal with.