Dirac cohomology of some Harish-Chandra modules
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.
scientific article; zbMATH DE number 5565884
scientific article