Invariant connections and PBW theorem for Lie groupoid pairs

Given a closed wide Lie subgroupoid A of a Lie groupoid L, i.e. a Lie groupoid pair, we interpret the associated Atiyah class as the obstruction to the existence of L-invariant fibrewise affine connections on the homogeneous space L/A. For Lie groupoid pairs with vanishing Atiyah class, we show that the left A-action on the quotient space L/A can be linearized. In addition to giving an alternative proof of a result of Calaque about the Poincare–Birkhoff–Witt map for Lie algebroid pairs with vanishing Atiyah class, this result specializes to a necessary and sufficient condition for the linearization of dressing actions, and gives a clear interpretation of the Molino class as an obstruction to simultaneous linearization of all the monodromies. In the course of the paper, a general theory of connections and connection forms on Lie groupoid principal bundles is developed. Also, a computational substitute to the adjoint action (which only exists " up to homotopy ") is suggested .