Construction of the Poincare sheaf for higher genus curves

Let $X$ be a smooth projective curve of genus $g$ and $L$ be a degree $l$ line bundle on $X$ with $l\ge 2g-1$. Denote the stack of rank two Higgs bundles on $X$ with value in $L$ by $\mathcal{H}iggs$ and the semistable part by $\mathcal{H}igg{s}_{ss}$. Let $H$ be the Hitchin base. In this paper we will construct the Poincare sheaf $\mathcal{P}$ on $\mathcal{H}iggs{\times }_H\mathcal{H}igg{s}_{ss}$ which is a maximal Cohen-Macaulay sheaf and flat over $\mathcal{H}igg{s}_{ss}$. In particular this includes the locus of nonreduced spectral curves.