Euler–Poincaré–Dirac index conjecture for Harish-Chandra modules

Let $\widetilde K$ be the spin cover of a maximal compact subgroup $K$, and let $\mathsf{I}_{\mathrm{Dir}}(X)$ denote the Dirac index of a finite-length Harish-Chandra module $X$. Write $[\,\ ]_{\widetilde K}$ for the usual pairing of virtual finite-dimensional representations of $\widetilde K$, and let $\langle X,Y\rangle_{\mathrm{Dir}}=[\mathsf{I}_{\mathrm{Dir}}(X),\mathsf{I}_{\mathrm{Dir}}(Y)]_{\widetilde K}$ be the Dirac pairing. Euler–Poincaré–Dirac index conjecture. For any finite-length Harish-Chandra modules $X$ and $Y$ with infinitesimal character,

$$\mathbf{EP}(X,Y)=[\mathsf{I}_{\mathrm{Dir}}(X),\mathsf{I}_{\mathrm{Dir}}(Y)]_{\widetilde K}=\langle X,Y\rangle_{\mathrm{Dir}}.$$

The conjecture arises because the formal Euler–Poincaré computation uses potentially infinite-dimensional Hom spaces, so the equality of the two extreme terms is not justified. Its content is that the Euler–Poincaré pairing factors through Dirac indices, and hence through Dirac cohomology.