On the Hochschild homology of convolution algebras of proper Lie groupoids

TL;DR

This paper investigates the Hochschild homology of convolution algebras associated with proper Lie groupoids, introducing a sheaf-theoretic approach and confirming Brylinski's conjecture for circle actions.

Contribution

It develops a localization technique for Hochschild homology sheaves of proper Lie groupoids and verifies Brylinski's conjecture in specific cases involving circle actions.

Findings

Hochschild homology sheaf at each stalk is quasi-isomorphic to the linearized case

Localization result for Hochschild homology sheaves

Verification of Brylinski's conjecture for smooth circle actions

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.