Orthogonal bundles on curves and theta functions

Let M be the moduli space of principal So(r)-bundles on a curve C, and L the determinant bundle on M. We define in isomorphism of H-0(M, L) onto the dual of the space of r-th order theta functions on the Jacobian of C. This isomorphism identifies the rational map M -> vertical bar L vertical bar* defined by the linear system vertical bar L vertical bar with the map M -> vertical bar r circle minus vertical bar which associates to a quadratic bundle (E, q) the theta divisor Theta(E). The two components M+ and M- of M are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous question for Sp(2r)-bundles.