Finite multiplicity theorems for induced representations of semisimple Lie groups. II: Applications to generalized Gelfand-Graev representations
This article continues the line of investigations of the first paper of the series (cf. the preceding review Zbl 0734.22004) concerning the finite multiplicity property for induced representations of semisimple Lie groups. Here the main concern is with representations of G induced by representations of a nilpotent subgroup \(N_ 0\) of G. In fact by allowing representations of the nilpotent subgroup to be obtained by the Kirillov construction the author defines a class of representations of G called generalized Gelfand-Graev representations (GGGRs in short). In particular for each nilpotent Ad(G)-orbit \(\omega\) in \({\mathfrak g}\) (the Lie algebra of G) there is a GGGR associated to it. Since these representations are in general not of finite multiplicity, the author refines the preceding construction obtaining the so-called reduced GGGRs, RGGGRs, associated to \(\omega\). These are representations induced by representations of a certain semidirect product \(H_ 0(\times)N_ 0\) in a way resembling Mackey's construction of representations of group extensions. The main application of the above construction, occupying most of the paper is the case of simple Lie groups of hermitian type. The author constructs a family of \(\{\omega_ i\}_{0\leq i\leq k}\) (k being the rank of the associated symmetric space G/K) of Ad(G)-nilpotent orbits in \({\mathfrak g}\) with the property that the left regular representation of G is quasi-equivalent to the direct sum \(\oplus_{0\leq i\leq k}\Gamma_ i\) of GGGRs \(\Gamma_ i\) associated with the orbits of the family. Further the author establishes the finite multiplicity property for RGGGRs obtained from these. Finally the author points out the relevance of these results to the problem of describing embeddings of discrete series representations of G into GGGRs and RGGGRs, which are to be dealt with in subsequent papers of the same author [Adv. Stud. Pure Math. 14, 31-121 (1988) and Jap. J. Math., New Ser. 16, No.1, 31-95 (1990)].
generalized Gelfand- Graev representations
scientific article; zbMATH DE number 4214273