Connections on Lie groupoids and Chern-Weil theory

Let $\mathbb{X}=[X_1\rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $\mathcal{H}\subset T{X}_1$ transversal to the fibers of the source map. Under the assumption that the distribution $\mathcal{H}$is integrable, we define a version of de Rham cohomology for the pair $(\mathbb{X},\mathcal{H})$, and we study connections on principal $G$-bundles over $(\mathbb{X},\mathcal{H})$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern-Weil theory and describe characteristic classes of principal G-bundles over a pair $(\mathbb{X},\mathcal{H})$.