The Orbit Space and Basic Forms of a Proper Lie Groupoid

A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is generalized to the case of a proper Lie groupoid, in which the orbit space is equipped with the quotient diffeological structure. As an application of this, we obtain a de Rham theorem for the de Rham complex on the orbit space.
Comment: To appear in Trends in Mathematics, Research Perspectives: Proceedings of the 12th ISAAC Congress, Aveiro, Portugal, 2019, Birkh"auser. V4: Many straightforward details removed, shortening the paper to 9 pages. Two new corollaries of the main theorem given. Terminology updated, using the language of del Hoyo-Fernandes