# Dirac cohomology and Euler-Poincar\'e pairing for weight modules

## TL;DR

This paper investigates Dirac cohomology for simple weight modules over reductive Lie algebras, showing vanishing results unless modules are highest weight, and relates Dirac index pairing to Euler-Poincaré pairing.

## Contribution

It completes the calculation of Dirac cohomology for simple weight modules and establishes the equality of Dirac index pairing and Euler-Poincaré pairing for modules with infinitesimal characters.

## Key findings

- Dirac cohomology vanishes for non-highest weight simple modules.
- Dirac index pairing equals Euler-Poincaré pairing for weight modules.
- Results extend to Harish-Chandra modules via Kazhdan's orthogonality.

## Abstract

Let $\mathfrak{g}$ be a reductive Lie algebra over $\mathbb{C}$. For any simple weight module of $\mathfrak{g}$ with finite-dimensional weight spaces, 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 weight modules which have infinitesimal characters agrees with their Euler-Poincar\'{e} pairing. The analogue of this result for Harish-Chandra modules is a consequence of the Kazhdan's orthogonality conjecture which was settled by the first named author and Binyong Sun.