On the Hochschild homology of proper Lie groupoids
Markus Pflaum
University of Colorado, Boulder, USAHessel Posthuma
University of Amsterdam, NetherlandsXiang Tang
Washington University, St. Louis, USA
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Abstract
We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and 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, which is the transformation groupoid of a linear action of a compact isotropy group on a vector space. We then explain Brylinski’s ansatz to compute the Hochschild homology of the transformation groupoid of a compact group action on a manifold. We verify Brylinski’s conjecture for the case of smooth circle actions that the Hochschild homology is given by basic relative forms on the associated inertia space.
Cite this article
Markus Pflaum, Hessel Posthuma, Xiang Tang, On the Hochschild homology of proper Lie groupoids. J. Noncommut. Geom. 17 (2023), no. 1, pp. 101–162
DOI 10.4171/JNCG/467