Geometric realization of discrete series for semisimple symmetric spaces
type: Item
instance of: scholarly article
title: Geometric realization of discrete series for semisimple symmetric spaces
zbMATH Open document ID: 0685.22008
DOI: 10.1007/BF01393969
author: S. P. Wang
author: Yue Lin L. Tong
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143684
review text: Let G be a real semisimple Lie group, \(\sigma\) an involution of G and H the identity component of the group of \(\sigma\)-fixed points in G. Let K be a \(\sigma\)-stable maximal compact subgroup of G and assume that \(rank(H\setminus G)=rank(K\cap H\setminus K)\); then \(H\setminus G\) has countably many discrete series representations [cf. \textit{M. Flensted- Jensen}, Ann. Math., II. Ser. 111, 253-311 (1980; Zbl 0462.22006) and \textit{T. Oshima} and \textit{T. Matsuki}, Adv. Stud. Pure Math. 4, 331-390 (1984; Zbl 0577.22012)]. Let \(\Gamma\) \(\setminus {\mathcal D}\) be a locally symmetric space, where \({\mathcal D}=G/K\), and where \(\Gamma\) is a cocompact discrete subgroup of G such that \(\Gamma\) \(\cap H\setminus H\) is compact. The authors consider a suitable subset \(\Lambda\) of the set of discrete series parameters (basically the restriction comes down to sufficient regularity). For \(\lambda\in \Lambda\) the associated discrete series representation \({\mathcal V}_{\lambda}\subset L^ 2(H\setminus G)\) occurs with multiplicity one. It has a unique minimal K-type: the associated isotopic component of \({\mathcal V}_{\lambda}\) contains Flensted-Jensen's generating function \(\psi_{\lambda}\). From \(\psi_{\lambda}\) the authors construct a harmonic form \({\tilde \omega}{}_{\lambda}\) on G with values in a finite dimensional G-module V,\(\tau\) (of lowest weight \(\rho\)-\(\lambda)\). This form is left H-invariant and the closed span of its right G-translates realizes the contragredient of \({\mathcal V}_{\lambda}\). Moreover, let E be the (locally constant) vector bundle associated with the principal fiber bundle \(\Gamma\) \(\setminus G\to \Gamma \setminus {\mathcal D}\) and \(\tau\) \(| K\), and let \(p^*E\) denote its pull back under the canonical projection p: \({\mathcal D}\to \Gamma \setminus {\mathcal D}\). Then by construction \({\tilde \omega}{}_{\lambda}\) is the pull-back of a \(p^*E\)-valued closed form \(\omega_{\lambda}\) on \({\mathca…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 4122250
zbMATH Keywords: real semisimple Lie group
zbMATH Keywords: maximal compact subgroup
zbMATH Keywords: discrete series representations
zbMATH Keywords: locally symmetric space
zbMATH Keywords: minimal K-type
zbMATH Keywords: generating function
zbMATH Keywords: harmonic form
zbMATH Keywords: vector bundle
zbMATH Keywords: principal fiber bundle
zbMATH Keywords: cohomology class
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4122250