Stability of the Parabolic Picard Sheaf

Let $X$ be a smooth irreducible complex projective curve of genus $g\ge 2$, and let $D=x_1+\cdots +x_r$ be a reduced effective divisor on $X$. Denote by $U_{\alpha }(L)$ the moduli space of stable parabolic vector bundles on $X$ of rank $n$, determinant $L$ of degree $d$ with flag type $\{\{k_j^i{\}}_{j=1}^{m_i}{\}}_{i=1}^r$. Assume that the greatest common divisor of the collection of integers $\{\text{degree}(L),\{\{k_j^i{\}}_{j=1}^{m_i}{\}}_{i=1}^r\}$ is $1$; this condition ensures that there is a Poincaré parabolic vector bundle on $X\times U_{\alpha }(L)$. The direct image, to $U_{\alpha }(L)$, of the vector bundle underlying the Poincaré parabolic vector bundle is called the parabolic Picard sheaf. We prove that the parabolic Picard sheaf is stable.