Hochschild cohomology of the skew group ring D(X)#G in the complex analytic case

Question

Let $X$ be a n-dimensional complex (not necessarily compact) manifold, let $G$ be a finite subgroup of $Aut(X)$ acting by biholomorphic maps on $X$ and let $D(X)$ be the ring of differential operators on $X$. I would like to compute the Hochschild cohomology group $HH^{m}(D(X)\rtimes G,D(X)\rtimes G)$, $m\geq0$ of the skew-group ring $D(X)\rtimes G$, where $\rtimes$ denotes the smash product. In http://arxiv.org/pdf/math/0406499v3.pdf Etingof computes the above cohomology group in the case of an algebraic G-variety X. Does anyone know anything about that? How am I supposed to proceed here. Thanks in advance for your advices and hints.

Answer 1

Here is my solution to the problem I posted yesterday. I would make use of the fact that $D(X)$ is a locally convex algebra. Assuming that one can define a bornology on it (see http://arxiv.org/pdf/0706.0027.pdf) the smash product $D(X)\rtimes G$ can be made into a bornological algebra, too. Then we may use the fact that the Hochschild cohomology of any bornological algebra is defined in terms of the Ext-functor. The rest is as in Etingof's paper http://arxiv.org/pdf/math/0406499v3.pdf.