I want to find analog of following two statements.

Let $G$ be a discrete group, $M$ is representation of $G$. Local systems on $BG$ are the same as $G$ representations (because $\pi_1 (BG) =G$). Let $\mathscr{M}$ be a local system corresponding to $M$. $$ H^{\bullet}_{Grp} (G, M) = H^{\bullet} (BG, \mathscr{M})$$

Let $G$ be compact connected Lie group. $\mathfrak{g}$ is corresponding Lie algebra. $\mathbb{R}$ - trivial representation of $\mathfrak{g}$. $$ H^{\bullet}_{Lie} (\mathfrak{g}, \mathbb{R}) = H_{dR}^{\bullet} (G)$$

**Question:** How to express $H^{\bullet}_{Lie} (\mathfrak{g}, M)$ geometrically? Here $M$ is a finite dimensional representation of $\mathfrak{g}$ (if you wish you can assume that it is integrated to Lie group representation).

*Comment 1* :
I am even not sure which geometric object corresponds to representation of $\mathfrak{g}$. Is it bi-D-module (bimodule over differential operators)? Is it $G \times G$ equivariant bundle on $G$?

*Comment 2*:
I want to say in other words what I want. I want a geometric structure on group $G$ considered as a manifold, which counts $H^{\bullet}_{Lie} (\mathfrak{g}, M)$ . It can be sheaf, D-module, whatever. The point is that I want to forget that $G$ is a group. I want just $G$ considered as a manifold with extra geometric structure. And the way to get my cohomology back.