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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.

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Pacific Journal of Mathematics

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