Conformal blocks, parahoric torsors and Borel-Weil-Bott

Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a parahoric group scheme on $X$ as in \cite{pr}. Via the principle of Hecke correspondences, we set-up relationships between the cohomology of lines bundles on various moduli stacks of torsors. This approach gives a proof of \cite[Conjecture 3.7]{pr} for group schemes $\mathcal G$ as above in characteristic zero. This further gives as a consequence, the principle of propagation of vacua. We give a direct proof of the independence of central charge on base points. Projective flatness is recovered as a corollary of Faltings construction of the Hitchin connection. Using C.Teleman's basic results (\cite{bwb}), we deduce the analogous result that cohomology of line bundles on the stack of principal $G$-bundles vanish in all degrees except possibly one. Results on twisted vacua \cite{hongkumar} are obtained as immediate consequences.