Invariant Hochschild Cohomology of Smooth Functions

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Abstract: 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.