Equivariant Poincaré Duality

Question

Let $\Gamma$ be a group acting by smooth orientation-preserving diffeomorphisms on a smooth compact oriented manifold $M$ of dimension $n$. The de Rham complex $\Omega^{\bullet}(M)$ of differential forms and the dual complex of currents $\Omega_{\bullet}(M)$ then are $\Gamma$-spaces. Moreover, the Poincaré duality map $\Omega^{\bullet}(M) \rightarrow \Omega_{n-\bullet}(M)$ is a $\Gamma$-equivariant map. Therefore, for all $p\geq 0$, we get a chain map, $$\mathbb{C}\Gamma^{p+1} \otimes_{\mathbb{C}\Gamma} \Omega^{\bullet}(M) \longrightarrow \mathbb{C}\Gamma^{p+1} \otimes_{\mathbb{C}\Gamma} \Omega_{n-\bullet}(M).$$ Does anyone know if this is a quasi-isomorphism?