Let $C$ be a Riemann surface of genus at least 2, and consider the gauge group $GL_n$ or $SL_n$. The data of a Higgs bundle $(E,\phi)$ or a holomorphic connection $(E,\nabla)$ together with a sub-line bundle $L$ of $E$ defines a divisor on $C$. We will show that fixing such divisors defines Lagrangians in the moduli spaces of Higgs bundles and flat connections. Furthermore, this construction defines a Lagrangian correspondence between the moduli spaces of Higgs bundles (flat connections) and the Hilbert scheme of points in $T*C$ (its twisted version, respectively). We will also briefly discuss how these constructions fit in the context of the geometric Langlands correspondence and (a reduction of) the Kapustin-Witten equations.