Reflexive symmetric differentials and quotients of bounded symmetric domains

For each classical irreducible bounded symmetric domain \(\mathcal {D}\) $\mathcal{D}$ , Klingler has computed the minimum number \(m_{\mathcal {D}}\) $m_{\mathcal{D}}$ such that any smooth projective quotient \(X=\mathcal {D}/\Gamma \) $X=\mathcal{D}/\mathrm{\Gamma }$ , for \(\Gamma \in \text {Aut}^0(\mathcal {D})\) $\mathrm{\Gamma }\in {\text{Aut}}^0(\mathcal{D})$ , satisfies \(H^0(X,\textrm{Sym}^i\Omega ^1_X)=0\) $H^0(X,{\text{Sym}}^i{\mathrm{\Omega }}_X^1)=0$ for \(0<i<m_{\mathcal {D}}\) $0<i<m_{\mathcal{D}}$ . In this article, we extend Klingler’s result to the case when X is normal and projective. This, together with a normal version of Arapura’s result about the relationship between the vanishing of global symmetric differentials on X and the rigidity of finite dimensional representations of \(\pi _1(X)\) ${\pi }_1(X)$ , gives rigidity statements for representations of \(\pi _1(X)\) ${\pi }_1(X)$ and \(\pi _1(X_{reg})\) ${\pi }_1(X_{\mathit{reg}})$ in a low dimensional range, when X is a normal projective quotient of a bounded symmetric domain.