  We study nontrivial entropy invariants in the class of parabolic flows on
homogeneous spaces, quasi-unipotent flows. We show that topological complexity
(ie, slow entropy) can be computed directly from the Jordan block structure of
the adjoint representation. Moreover using uniform polynomial shearing we are
able to show that the metric orbit growth (ie, slow entropy) coincides with the
topological one, establishing hence variational principle for quasi-unipotent
flows (this also applies to the non-compact case). Our results also apply to
sequence entropy. We establish criterion for a system to have trivial
topological complexity and give some examples in which the measure-theoretic
and topological complexities do not coincide for uniquely ergodic systems,
violating the intuition of the classical variational principle.
