Let G be a real semisimple
group. Two important invariants are associated with the equivalence class of an
irreducible unitary representation of G, namely, the associated variety of the
annihilator in the universal enveloping algebra and Howe’s N-spectrum, where N is a
nilpotent subgroup of G. The associated variety is defined in a purely algebraic way.
The N-spectrum is defined analytically. In this paper, we prove some results about
the relation between associated variety and N-associated variety, where
N is a subgroup of G. We then relate N-associated variety with Howe’s
N-spectrum when N is abelian. This enables us to compute Howe’s rank
in terms of the associated variety. The relationship between Howe’s rank
and the associated variety has been established by Huang and Li, at about
the same time this paper was first written, using the result of Matomoto
on Whittaker vectors. It can also be derived from works of Przebinda and
Daszkiewicz–Kraśkiewicz–Przebinda. Our approach is independent and
more self-contained. It does not involve Howe’s correspondence in the stable
range.
Keywords
classical groups of type I, associated variety, spectral
measure, unitary representations, N-spectrum, wave front set, filtered
algebra
