The fine structure of translation functors

Let $E$ be a simple finite-dimensional representation of a semisimple Lie algebra with extremal weight $\nu$ and let $0 \neq e \in E_{\nu}$. Let $M(\tau)$ be the Verma module with highest weight $\tau$ and $0 \neq v_{\tau} \in M(\tau)_{\tau}$. We investigate the projection of $e \otimes v_{\tau} \in E \otimes M(\tau)$ on the central character $\chi(\tau+\nu)$. This is a rational function in $\tau$ and we calculate its poles and zeros. We then apply this result in order to compare translation functors.