On Moduli of G-bundles over Curves for exceptional G

Let G be a simple and simply connected complex Lie group, g its Lie algebra. In the following, I remove the restriction “G is of classical type or G2” made on G in the papers of Beauville, Laszlo and myself [L-S],[B-L-S] on the moduli of principal G-bundles on a curve. As I will just “patch” the missing technical points, this note should be seen as an appendix to the above cited papers. Let MG,X be the stack of G-bundles on the smooth, connected and projective algebraic curve X of genus g. If ρ : G → SLr is a representation of G, denote by Dρ the pullback of the determinant bundle [D-N] under the morphismMG,X → MSLr,X defined by extension of the structure group. Associate to G the number d(G) and the representation ρ(G) as follows: