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Dirac cohomology of some Harish-Chandra modules. (English) Zbl 1179.22013
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 [J.-S. Huang and 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.

22E46 Semisimple Lie groups and their representations
18G60 Other (co)homology theories (MSC2010)
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