Moduli spaces of orthogonal and symplectic bundles over an algebraic curve

We prove that, given a smooth projective curve C of genus g >= 2, the forgetful morphism.Mo(r) -> M(GLr) (respectively M(Sp2r) -> M(GL2r)) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.