Finite multiplicity theorems for induced representations of semisimple Lie groups. II: Applications to generalized Gelfand-Graev representations
type: Item
instance of: scholarly article
title: Finite multiplicity theorems for induced representations of semisimple Lie groups. II: Applications to generalized Gelfand-Graev representations
zbMATH Open document ID: 0734.22005
author: Hiroshi Yamashita
published in: Journal of Mathematics of Kyoto University
publication date: 1988-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22D30
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 4214273
zbMATH Keywords: finite multiplicity property
zbMATH Keywords: induced representations
zbMATH Keywords: semisimple Lie groups
zbMATH Keywords: nilpotent subgroup
zbMATH Keywords: Kirillov construction
zbMATH Keywords: generalized Gelfand- Graev representations
zbMATH Keywords: simple Lie groups of hermitian type
zbMATH Keywords: nilpotent orbits
zbMATH Keywords: left regular representation
zbMATH Keywords: discrete series representations
reviewed by: A. Strasburger
MaRDI profile type: Publication
Wikidata QID: Q115240399
full work available at URL: https://doi.org/10.1215/kjm/1250520400
OpenAlex ID: W1494704398
DOI: 10.1215/KJM/1250520400
description: scientific article; zbMATH DE number 4214273