Automorphisms of Moduli Spaces of Symplectic Bundles

Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let M Sp (L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of M Sp (L) is generated by the automorphisms of the form E ↦ E ⊗ M, where Formula: see text, together with the automorphisms induced by automorphisms of X.