Let $\pi:X\rightarrow Y$ a double cover of Riemann surfaces, and consider a $SL_n-$bundle $E$ with a perfect $\mathcal O_Y-$bilinear pairing $$\psi:\pi_*E\times\pi_*E\rightarrow\pi_*\mathcal O_X$$ which verifies $\psi(av,w)=\psi(v,\sigma(a)w)$ and $\psi(v,w)=\sigma(\psi(w,v))$, where $\sigma:X\rightarrow X$ is the canonical involution.
My question is:
- Is this the same as given a perfet bilinear form on $E$?
- and How could we translate this propriety to the transition functions of $E$?
Thanks