Invariant connections and PBW theorem for homogeneous spaces of Lie groupoids

We investigate homogeneous spaces of Lie groupoids. Given a closed wide Lie subgroupoid $A$ of a Lie groupoid $L$, i.e. a Lie groupoid pair, we introduce an associated associated Atiyah class in $A$-cohomology, prove its invariance under Morita equivalence of Lie groupoid pairs, and interpret it 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 a relative Poincaré-Birkhoff-Witt map for inclusions of Lie algebroids 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 of a regular foliation as an obstruction to simultaneous linearization of all the monodromies.

Along the way, a general theory of connections and connection forms on equivariant principal bundles of Lie groupoids is developed. Also, a computational substitute to the adjoint action (which only exists "up to homotopy") is suggested.

(This is joint work with Camille Laurent-Gengoux.)