For an exam I need to prove the following:
Assume that $\epsilon$ and $\epsilon'$ are both Killing spinors. Show that $\nabla_\mu \bar{\epsilon}'\gamma_\nu \epsilon = -\bar{\epsilon}' \gamma_\nu \epsilon / L$.
We are working in a $D$-dimensional AdS space. The condition for the Killing spinors is $\hat{D}_\mu \epsilon = 0$ with $\hat{D}_\mu = D_\mu - \frac{1}{2L} \gamma_\mu = \partial_\mu + \frac{1}{4}\omega_{\mu a b} \gamma^{ a b} - \frac{1}{2L} \gamma_\mu$.
Does anybody know how to prove this? Even though the problem doesn't seem that complicated, I have been stuck at it for quite a while and can't solve it. It is problem 22.15 from the book Supergravity by Van Proeyen and Freedman.
Thanks!