The n-cohomology of limits of discrete series
type: Item
instance of: scholarly article
title: The n-cohomology of limits of discrete series
zbMATH Open document ID: 0709.22008
DOI: 10.1016/0022-1236(88)90011-0
author: Floyd L. Williams
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: Let G be a connected noncompact real semisimple Lie group with finite center and \(\Gamma\) a cocompact discrete subgroup of G. We assume that G has a compact Cartan subgroup H. The results of this paper are concerned with a cohomological realization of the limits of discrete series representations appearing in \(L^ 2(\Gamma \setminus G)\), where \(L^ 2(\Gamma \setminus G)\) is the space of all square integrable functions on \(\Gamma\setminus G\) with respect to the canonical G-invariant measure. We state those results more precisely after the following notations. Let \({\mathfrak g}\) and \({\mathfrak h}\) be the Lie algebras of G and H, respectively. The root system of (\({\mathfrak g}_ C,{\mathfrak h}_ C)\) will be denoted by \(\Delta\) where \({\mathfrak g}_ C\), \({\mathfrak h}_ C\) are the complexifications of \({\mathfrak g}\), \({\mathfrak h}\). Fix a linear order in \(\Delta\), and denote one half the sum of all positive roots in \(\Delta\) by \(\delta\). Let \(b^*\) be a unitary character of H. Then there exists a unique integral form \(\lambda\) on \({\mathfrak h}_ C\) such that \(b^*(\exp X)=e^{\lambda (X)}\) for all X in \({\mathfrak h}\). We fix a character \(e^{\lambda}\) of H. Let \(\Theta_{\lambda +\delta}\) be Harish- Chandra's tempered invariant eigendistribution on G corresponding to the data \(\lambda +\delta\). If \(\Theta_{\lambda +\delta}\) is not identically zero on G, then \(\Theta_{\lambda +\delta}\) is a character, up to a constant factor, of an irreducible unitary representation \(\pi_{\lambda +\delta}\) of G. \(\pi_{\lambda +\delta}\) is called a discrete [resp. limit of discrete] series representation if \(\lambda +\delta\) is regular [resp. singular]. Let \({\mathcal L}_{\lambda}\to G/H\) be the canonical line bundle to the character \(e^{\lambda}\) of H and \(H_ 2(\Gamma \setminus {\mathcal L}_{\lambda})\) the \(L^ 2\)-cohomology space of the quotient bundle \(\Gamma\setminus {\mathcal L}_{\lambda}\). W. Schmid proved that dim \(H_ 2…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 57T10
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 4165191
zbMATH Keywords: real semisimple Lie groups
zbMATH Keywords: discrete subgroups
zbMATH Keywords: limits of discrete series representations
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90011-0
OpenAlex ID: W2025091762
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description: scientific article; zbMATH DE number 4165191