  In this paper we study irreducible unitary representations of GL(n,R) and
prove a number of results. Our first result establishes a precise connection
between the annihilator of a representation and the existence of degenerate
Whittaker functionals, for both smooth and K-finite vectors, thereby
generalizing results of Kostant, Matumoto and others. Our second result relates
the annihilator to the sequence of highest derivatives, as defined in this
setting by one of the authors. Based on those results, we suggest a new notion
of rank of a smooth admissible representation of GL(n,R), which for
unitarizable representations refines Howe's notion of rank. Our third result
computes the highest derivatives for (almost) all unitary representations in
terms of the Vogan classification. We also indicate briefly the analogous
results over complex and p-adic fields.
