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Simpson, in 1994, introduced the notion of $\Lambda$-modules and constructedthe corresponding moduli space, where $\Lambda$ is a sheaf of rings ofdifferential operators. Higgs bundles, connections and $\lambda$-connections(as defined by Delgine) are particular cases of $\Lambda$-modules. In thisarticle the concept of parabolic $\Lambda$-modules over a curve is introducedand their moduli space is built. As an application, we construct the parabolicHodge moduli space parameterizing parabolic $\lambda$-connections.

Moduli space of parabolic $\Lambda$-modules over a curve