Let $M$ be a Riemannian manifold. The Hodge decomposition tells that $$ \Omega^*(M) = \mathrm{im} \ d \oplus \mathrm{im} \ d^* \oplus \mathscr H^*(M) $$ where $d^*$ is the adjoint operator of the exterior derivative $d$. Namely, a differential form $\omega$ has a unique decomposition as a sum in the form $\omega=d\alpha_1+d^*\alpha_2+\gamma$ where $\gamma$ is a harmonic form in the sense that $\Delta \gamma=0$ for $\Delta=dd^*+d^*d$.
We further assume that there is an action of a compact Lie group $G$ with Lie algebra $\mathfrak g$, and we can consider the space $ \Omega^*_G(M)= (\mathbb C[\mathfrak g]\otimes \Omega^*(M))^G $ of $G$-equivariant differential forms (i.e. $G$-invariant polynomial maps $\mathfrak g\to \Omega^*(M)$). Also, we can define the equivariant exterior differential $d_{\mathfrak g}$. Then, the equivariant cohomology $H^*_G(M)$ is the cohomology of the complex $(\Omega^*_G(M), d_{\mathfrak g})$.
By analogy, we could probably define $\Delta_{\mathfrak g}=d_{\mathfrak g}d^*_{\mathfrak g}+d^*_{\mathfrak g}d_{\mathfrak g}$ and the space of $G$-equivariant harmonic forms $\mathscr H^*_{\mathfrak g}(M)$; then, perhaps we could also show $\mathscr H_{\mathfrak g}^*(M)\cong H_G^*(M)$ and a similar Hodge decomposition $$ \Omega^*_G(M)=\mathrm{im} \ d_{\mathfrak g}\oplus \mathrm{im} \ d_{\mathfrak g}^* \oplus \mathscr H^*_{\mathfrak g}(M)$$
This looks natural and standard. If what I guess was true, does anyone know a good reference? Thanks.
