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.
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Xiang Tang, Yi-Jun Yao, Weiping Zhang, Hopf cyclic cohomology and Hodge theory for proper actions. J. Noncommut. Geom. 7 (2013), no. 3, pp. 885–905DOI 10.4171/JNCG/138