Basic Forms and Orbit Spaces: a Diffeological Approach
  If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential forms in the diffeological sense. 
Yael Karshon and Jordan Watts
math.GT math.DG math.SG