Stability of the Parabolic Picard Sheaf

Let $X$ be a smooth irreducible complex projective curve of genus $g\,\geq\, 2$, and let $D\,=\,x_1+\dots+x_r$ 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 $\{\{k^i_j\}_{j=1}^{m_i}\}_{i=1}^r$. Assume that the greatest common divisor of the collection of integers $\{\text{degree}(L),\, \{\{k^i_j\}_{j=1}^{m_i}\}_{i=1}^r\}$ is $1$; this condition ensures that there is a Poincar\'e parabolic vector bundle on $X\times U_{\alpha}(L)$. The direct image, to $U_{\alpha}(L)$, of the vector bundle underlying the Poincar\'e parabolic vector bundle is called the parabolic Picard sheaf. We prove that the parabolic Picard sheaf is stable.