Stability of symplectic and orthogonal Poincaré bundles

Abstract

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