Dirac cohomology and orthogonality relations for weight modules (Q2171876)

Let \(\mathfrak{g}\) be a complex reductive Lie algebra. Let \(\mathfrak{h}\) be a Cartan subalgebra of \(\mathfrak{g}\). Let \(\mathcal{M}(\mathfrak{g}, \mathfrak{h})\) be the category of finitely generated admissible \((\mathfrak{g}, \mathfrak{h})\) modules. An object in \(\mathcal{M}(\mathfrak{g}, \mathfrak{h})\) is called a weight module. Let \(M, N\) be two weight modules. Theorem 6.1 of the paper proves that the Euler-Poincare pairing between \(M\) and \(N\) equals the spinor pairing between them. This can be viewed as an analogue of Harish-Chandra's orthogonality relations of discrete series in the setting of weight modules. Theorem 7.1 further proves that the Euler-Poincare pairing between \(M\) and \(N\) equals the Dirac index pairing between them. Then the authors show in Theorem 7.9 that a simple weight module has non-zero Dirac cohomology if and only if it is a weight module with respect to a Borel subalgebra. As a consequence, Dirac cohomology of any simple weight module is determined.