Stepwise Square Integrable Representations: the Concept and Some Consequences

TL;DR

This paper surveys recent advances in the theory of stepwise square integrable representations of nilpotent Lie groups, focusing on their implications for harmonic analysis, geometry, and representation theory.

Contribution

It introduces new developments in Plancherel formulas and matrix coefficient growth for nilpotent Lie groups, expanding understanding of their representations and applications.

Findings

Enhanced Plancherel formulas for nilpotent Lie groups

New insights into matrix coefficient growth

Implications for geometry of weakly symmetric spaces

Abstract

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.