On the Dirac cohomology of complex Lie group representations (Q1949286)

Consider a real reductive Lie group \(G\) with an involution \(\Theta\). Let \(K=G^\Theta\) be a maximal compact subgroup. Define \(\widetilde{K}\) as a spin double of \(K\). Define the Dirac cohomology of a Harish-Chandra module \(X\) as a \(\widetilde{K}\)-module \[ H_D(X)=\mathrm{Ker}_D/(\mathrm{Im}D \cap\mathrm{Ker}D), \] where \(D\) is a Dirac operator on \(X\otimes S_G\), and \(S_G\) is a spin module over the Clifford algebra of \(\nu\), where the Lie algebra \(\nu\) comes from the decomposition of the Lie algebra \(g\) induced by the involution \(\Theta\): \[ g=k\oplus\nu. \] In the paper under review the classification of unitary \(G\)-modules with non-zero Dirac cohomologies is reduced to the classification of spherical ones with non-zero Dirac cohomologies on Levi level.