Classification of admissible nilpotent orbits in the classical real Lie algebras

Abstract

M. Duflo introduced the notion of admissibility of a linear form on a real Lie algebra and constructed certain unitary representations from admissible linear forms with good polarization in [D]. D. Vogan has wanted to find a notion of “unipotent representations” of real reductive groups which corresponds to the one of Lusztig’s unipotent representations of finite reductive groups. He pointed out that a unipotent representation should be a constituent of a representation attached to an admissible nilpotent coadjoint orbit by some “orbit method” (cf. [Vl, V2]). But for real reductive groups there have not yet been found the correct definition of unipotent representations and hence the orbit method which should exist (i.e., a correspondence between admissible nilpotent coadjoint orbits and representations which contain unipotent representations as constituents). However, it seems that these notions must play an important role for the classification of unitary representations. The purpose of this paper is to determine all the admissible nilpotent orbits in real classical (non-compact) Lie groups. Here a real classical Lie group means a Lie group in the following list (RG):