On moduli of G-bundles on a curve for exceptional G

Let X be a complex, smooth, projective and connected curve, G be a simple a nd simple connected complex algebraic group and M-G,M-X be the stack of G-b undles on S. We show, using the decomposition formulas of Tsuchiya-Ueno-Yam ada [T-U-Y] and Faltings [F], the existence of certain line bundles on M-G, M-X conjectured in [L-S]. The result is then applied to the question of loc al factoriality of the coarse moduli space of semi-stable G-bundles. (C) El sevier, Paris.