  There are some new developments on Plancherel formula and growth of matrix
coefficients for unitary representations of nilpotent Lie groups. These have
several consequences for the geometry of weakly symmetric spaces and analysis
on parabolic subgroups of real semisimple Lie groups, and to (infinite
dimensional) locally nilpotent Lie groups. Many of these consequences are still
under development. In this note I'll survey a few of these new aspects of
representation theory for nilpotent Lie groups and parabolic subgroups.
