Stability Of The Parabolic Poincar\'e Bundle

Given a compact Riemann surface $X$ and a moduli space $M_{\alpha}(\Lambda)$ of parabolic stable bundles on it of fixed determinant of complete parabolic flags, we prove that the Poincar\'e parabolic bundle on $X\times M_{\alpha}(\Lambda)$ is parabolic stable with respect to a natural polarization on $X\times M_{\alpha}(\Lambda)$.