Dirac cohomology of some Harish-Chandra modules
type: Item
instance of: scholarly article
title: Dirac cohomology of some Harish-Chandra modules
zbMATH Open document ID: 1179.22013
author: Jing-Song Huang
author: Yi-Fang Kang
author: Pavle Pandžić
published in: Transformation Groups
publication date: 2009-06-17T00:00:00Z
review text: Let \(G\) be a connected real reductive Lie group with Cartan involution \(\theta\), such that \(K=G^{\theta}\) is a maximal compact subgroup of \(G\). For example, \(G\) can be a connected semisimple Lie group with finite centre. In the paper under review the authors determine the Dirac cohomology of irreducible unitary Harish-Chandra modules \(A_{\mathfrak q}(\lambda)\). This family of modules includes all irreducible unitary representations with nonzero \(({\mathfrak g},K)\)-cohomology and all irreducible unitary Harish-Chandra modules with strongly regular infinitesimal character. As an illustration of the applications the authors relate their results with those of Vogan and Zuckerman on the \(({\mathfrak g},K)\)-cohomology of \(A_{\mathfrak q}(\lambda)\) modules. The results of the present paper can be considered as a supplement to [\textit{J.-S. Huang} and \textit{P. Pandžić}, J. Am. Math. Soc. 15, No.~1, 185--202 (2002; Zbl 0980.22013)], where two of the authors obtained a necessary and sufficient condition for some Harish-Chandra modules to have nonzero Dirac cohomology.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 18G60
zbMATH DE Number: 5565884
zbMATH Keywords: Dirac operator
zbMATH Keywords: Dirac cohomology
zbMATH Keywords: Harish-Chandra modules
zbMATH Keywords: unitary representation
zbMATH Keywords: infinitesimal character
MaRDI profile type: Publication
full work available at URL: s00031-008-9036-7
OpenAlex ID: W1975897881
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DOI: 10.1007/S00031-008-9036-7
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description: scientific article; zbMATH DE number 5565884