  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.
