Maximal isotropy subgroups for absolutely irreducible representations of compact Lie groups

Abstract

We answer some long-standing questions concerning absolutely irreducible representations of compact Lie groups. Such representations provide the natural setting for steady-state equivariant bifurcation theory. We prove the existence of maximal isotropy subgroups for which there are no branches of equilibria or relative equilibria. Also, we obtain examples of complex and quaternionic maximal isotropy subgroups. A consequence of this is the existence of primary branches of nontrivial relative equilibria (rotating waves).