Equivariant basic cohomology of Lie groupoids

This paper develops equivariant basic cohomology for Lie groupoids equipped with weak actions of Lie groups. The weak action is encoded by a Kan fibration over the classifying groupoid, and the basic complex of the fiber is shown to carry the structure needed for Weil and Cartan models. The construction is compared with Bott--Shulman--Stasheff cohomology, where the equivariant theory is obtained from the quotient groupoid. For orbifolds, basic forms are interpreted as orbifold differential forms, and the resulting equivariant basic cohomology is used to formulate differential-geometric constructions such as equivariant integration and localization. The paper also studies the induced weak action on the inertia groupoid and uses it to define an equivariant refinement of the Chen--Ruan cohomology ring. In this framework the sectorwise equivariant cohomology, obstruction bundle, equivariant Euler class, Gysin maps and three-point functions are assembled into an equivariant Chen--Ruan product whenever the corresponding pairing is nondegenerate.