  We introduce a Hopf algebroid associated to a proper Lie group action on a
smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is
equal to the de Rham cohomology of invariant differential forms. When the
action is cocompact, we develop a generalized Hodge theory for the de Rham
cohomology of invariant differential forms. We prove that every cyclic
cohomology class of the Hopf algebroid is represented by a generalized harmonic
form. This implies that the space of cyclic cohomology of the Hopf algebroid is
finite dimensional. As an application of the techniques developed in this
paper, we discuss properties of the Euler characteristic for a proper cocompact
action.
