Stability of symplectic and orthogonal Poincaré bundles

Let LL be a line bundle on a smooth complex projective curve XX. Let MLML be the moduli space of regularly stable orthogonal or symplectic bundles of rank rr on XX with fixed determinant LL. There is a Poincaré projective bundle on X×MLX×ML. It defines a principal PSp(r,C)PSp(r,C) (respectively, PS0(r,C)PS0(r,C)) bundle PP on X×MLX×ML in the symplectic (respectively, orthogonal) case. For a fixed point xx on XX, let PxPx be its restriction to {x}×ML{x}×ML. We prove that the principal bundle PxPx is stable. As a corollary, PP is stable with respect to any polarization on X×MLX×ML.