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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.

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