Associated Varieties and Howe’s N-spectrum

Let G be a real semisimple group. There are two important invariants 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 (see Definition 1.2, Theorem. 0.1.) where N is a subgroup of G. We then relate N -associated variety with Howe’s N -spectrum when N is Abelian (see Theorem. 0.2). This enables us to compute Howe’s rank in terms of the associated variety (see Theorems 0.3, 0.4). The relationship between Howe’s rank and the associated variety has been established by Huang-Li about the same time this paper was firstly written, using the result of Matomoto on Whittaker vectors. It can also be derived from works of Przebinda and DaszkiewiczKraśkiewicz-Przebinda. Our approach is independent and more self-contained. It does not involve Howe’s correspondence in the stable range.