Linear dynamical systems of nilpotent Lie groups

We study the topology of orbits of dynamical systems defined by finite-dimensional representations of nilpotent Lie groups. Thus, the following dichotomy is established: either the interior of the set of regular points is dense in the representation space, or the complement of the set of regular points is dense, and then the interior of that complement is either empty or dense in the representation space. The regular points are by definition the points whose orbits are locally compact in their relative topology. We thus generalize some results from the recent literature on linear actions of abelian Lie groups. As an application, we determine the generalized $ax+b$-groups whose $C^*$-algebras are antiliminary, that is, no closed 2-sided ideal is type~I.