Abstract: Let g be a reductive Lie algebra over C. For any simple weight module of g, we show that its Dirac cohomology is vanished unless it is a highest weight module. This completes the calculation of Dirac cohomology for simple weight modules since the Dirac cohomology of simple highest weight modules was carried out in our previous work. We also show that the Dirac index pairing of two simple weight modules agrees with their Euler-Poincare pairing. The analogue of this result for Harish-Chandra modules is a consequence of the Kazhdan's orthogonality conjecture which was proved by Jingsong Huang and Binyong Sun.
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