A vanishing theorem for co-Higgs bundles on the moduli space of bundles

We consider smooth moduli spaces of semistable vector bundles of fixed rank and determinant on a compact Riemann surface $X$ of genus at least $3$. The choice of a Poincar\'e bundle for such a moduli space $M$ induces an isomorphism between $X$ and a component of the moduli space of semistable sheaves over $M$. We prove that $h^0(M, \text{End}({\mathcal E})\otimes TM)= 1$ for a vector bundle $\mathcal E$ on $M$ coming from this component. Furthermore, there are no nonzero integrable co-Higgs fields on $\mathcal E$.