The line bundles on the moduli of parabolicG - bundles over curves and

Let X be a complex, smooth, complete and connected curve and G be a complex simple and simply connected algebraic group. We compute the Picard group of the stack of quasi-parabolic G-bundles over X, describe explicitly its generators in case for classical G and G 2 and then identify the corresponding spaces of global sections with the vacua spaces of Tsuchiya, Ueno and Yamada. The method uses the uniformization theorem which describes these stacks as double quotients of certain innnite dimensional algebraic groups. We describe also the dualizing bundle of the stack of G-bundles and show that it admits a unique square root, which we construct explicitly. If G is not simply connected, the square root depends on the choice of a theta-characteristic. These results about stacks allow to recover the Drezet-Narasimhan theorem (for the coarse moduli space) and to show an analogous statement when G = Sp 2r. We prove also that the coarse moduli spaces of semi-stable SO r-bundles are not locally factorial for r 7.