Associated varieties and Gelfand-Kirillov dimensions for the discrete series of a semisimple Lie group
type: Item
instance of: scholarly article
title: Associated varieties and Gelfand-Kirillov dimensions for the discrete series of a semisimple Lie group
zbMATH Open document ID: 0829.22024
author: Hiroshi Yamashita
published in: Proceedings of the Japan Academy. Series A
publication date: 1996-01-22T00:00:00Z
review text: In this article the author describes the associated variety of the Harish-Chandra \(({\mathfrak g},K)\)-module of discrete series as the cone \(K_\mathbb{C} {\mathfrak p}_-\). Here \(K_\mathbb{C}\) denotes the complexification of a maximal compact subgroup of a connected semisimple Lie group \(G\) with finite center, and \({\mathfrak p}_-\) the sum of the root spaces associated with a negative system of noncompact roots. One of the tools used in the outline of the proof the author gives is a result of \textit{R. Hotta} and \textit{R. Parthasarathy} [Invent. Math. 26, 133-178 (1974; Zbl 0298.22013)] that shows a graduation module of a discrete series as the kernel of an operator graduation of the Schmid operator. As the author mentions, the result can be also obtained using \(\mathcal D\)- module theory combining some results of \textit{W. Borho} and \textit{J.-L. Brylinski} [Invent. Math. 80, 1-68 (1985; Zbl 0577.22014)] and \textit{W. Schmid} [Harmonic Analysis on Reductive Groups, Proc. Conf., Brunswick/ME (USA) 1989, Prog. Math. 101, 235-275 (1991; Zbl 0751.22003)]. As consequences, the author shows that the associated ideal coincides with its radical, and that the associated variety of a discrete series is presented in terms of the symbol mapping of the Schmid operator. Finally, the author gives a formula for the Gelfand-Kirillov dimension for discrete series of \(\text{SU} (p,q)\).
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 605461
zbMATH Keywords: associated variety of \(U(g)\)-modules
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: Harish- Chandra \(({\mathfrak g},K)\)-module
zbMATH Keywords: complexification
zbMATH Keywords: maximal compact subgroup
zbMATH Keywords: connected semisimple Lie group
zbMATH Keywords: Schmidt operator
Wikidata QID: Q115219933
MaRDI profile type: Publication
full work available at URL: pjaa.70.50
OpenAlex ID: W2095034696
cites work: Differential operators on homogeneous spaces. III: Characteristic varieties of Harish Chandra modules and of primitive ideals
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cites work: Embeddings of discrete series into induced representations of semisimple Lie groups. II: Generalized Whittaker models for \(SU(2,2)\)
cites work: Criteria for the finiteness of restriction of \(U({\mathfrak g})\)-modules to subalgebras and applications to Harish-Chandra modules
DOI: 10.3792/PJAA.70.50
description: scientific article; zbMATH DE number 605461