In the first part of the talk we study the Hochschild homology of the convolution algebra of a proper Lie groupoid by constructing the convolution sheaf over the space of orbits. Using localization methods essentially going back to Teleman we prove that the Hochschild homology sheaf at each stalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolution algebra of its linearization. In the second part of the talk the Hochschild homology theory of transformation groupoids of compact Lie groups will be considered. We interpret Brylinski's basic relative forms in terms of certain Grauert-Grothendieck forms on the inertia space of the groupoid. For the case of smooth circle actions we show that the Hochschild homology of the corresponding transformation groupoid coincides with the basic relative forms thus verifying a conjecture by Brylinski in this case.

Joint work with H. Posthuma and X. Tang