Small unitary representations and

We will introduce a notion of small unitary representation
for a semisimple Lie group. This is a natural generalization
of the notion of minimal representation. Previous methods
to study small representations, including the work of Howe,
Sahi and Weissman, are based on abelian harmonic analysis.
Using Kostant’s cascade of strongly orthogonal roots, we
show how a new definition can be given in terms of
Kirillov’s method of orbits. As a result, we obtain refined
estimates on the decay of matrix coefficients, which improve
recent results by Li, Zhu and OH.