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arXiv · 2009.03216

On the Hochschild homology of convolution algebras of proper Lie groupoids

Markus J. Pflaum·Hessel B. Posthuma·Xiang Tang
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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 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.

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BibTeXRIS

Markus J. Pflaum, Hessel B. Posthuma, Xiang Tang. 2020-09-07. On the Hochschild homology of convolution algebras of proper Lie groupoids. https://arxiv.org/abs/2009.03216

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