
Journal of Lie Theory 31 (2021), No. 2, 557574 Copyright Heldermann Verlag 2021 Invariant Hochschild Cohomology of Smooth Functions Lukas Miaskiwskyi EWI/DIAM, Delft University of Technology, Delft, The Netherlands l.t.miaskiwskyi@tudelft.nl 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 HochschildKostantRosenberg 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. Keywords: Differential geometry, homological algebra, deformation quantization. MSC: 13D03, 16W22. [ Fulltextpdf (158 KB)] for subscribers only. 