  Given an action of a Lie group on a smooth manifold, we discuss the induced
action on the Hochschild cohomology of smooth functions, and notions of
invariance on this space. Depending on whether one considers invariance of
cochains or invariance of cohomology classes, two different spaces of
invariants arise. We perform a general comparison of these notions, give an
interpretation of the lower orders of the invariant cohomology spaces and
conclude as our main result that for proper group actions both spaces are
isomorphic. As a corollary and a geometric interpretation, an invariant version
of the Hochschild-Kostant-Rosenberg theorem is given, identifying the
cohomology of invariant cochains with invariant multivector fields. Using this
theorem, we shortly discuss the invariant Hochschild cohomology in the case of
homogeneous spaces.
