Stepwise Square Integrable Representations: the Concept and Some\n Consequences

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