Dirac cohomology and translation functors
Let \(G\) be a connected reductive Lie group and \(\mathfrak g = \mathfrak k \oplus \mathfrak p\) be the complexified Lie algebra of \(G\). Let \(C(\mathfrak p)\) denote the Clifford algebra corresponding to the nondegenerate invariant bilinear form on \(\mathfrak g\), and \(S\) denote the spin module for \(C(\mathfrak p)\). Finally, let \(D \in U (\mathfrak g) \otimes C(\mathfrak p)\) denote the Dirac operator. Clearly, if \(X\) is a \(\mathfrak g\)-module then \(D\) acts on \(X \otimes S\).NEWLINENEWLINEThe main theorem of this paper is the following. Let \(\mathfrak h\) be a \(\Theta\)-stable Cartan subalgebra of \(\mathfrak g\), and \(X_\lambda\) be a unitarizable \((\mathfrak g, K)\)-module with infinitesimal character \(\chi_\lambda\), where \(\lambda \in \mathfrak h^*\). Let \(F_\nu\) be a finite dimensional \(\mathfrak g\)-module with extremal weight \(\nu\). Suppose that \(X_{\lambda+\nu} \subseteq X_\lambda \otimes F_\nu\) is a \((\mathfrak g, K)\)-submodule with infinitesimal character \(\chi_{\lambda+\nu}\). If the kernel of \(D\) as an operator on \(X_{\lambda+\nu} \otimes S\) is nonzero, then the Dirac cohomology of \(X_\lambda\) is nonzero. In particular, nonvanishing of the Dirac cohomology of \(X_{\lambda+\nu}\) implies nonvanishing of the Dirac cohomology of \(X_\lambda\).NEWLINENEWLINEThe authors also obtain a similar result in which Dirac cohomology is replaced by harmonic spinors and the unitarizability assumption on \(X_\lambda\) is replaced by a positivity condition on an operator associated to \(D\). The authors state that their results generalize some theorems in recent work of Parthasarathy and Mehdi with simpler proofs.
scientific article; zbMATH DE number 6214417