Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy" to see that $H^\bullet(BG)$ has a Hodge structure. In fact, since $H^\bullet(BG)\to H^\bullet(BT)$ is injective, it's easy to see that this structure is pure and Tate.
For $G=U(n)/{\operatorname{GL}(n,\mathbb{C})}$ or more generally a product of $U(n)$'s (so, for example, a torus, which I used above), there's a straightforward explanation of this point: $BG$ is an ind-projective variety, the Grassmannian of $n$-planes in $\mathbb{C}^{\infty}$. For a construction I'm trying to use, my life will be much easier if I can figure out a similar presentation for other groups.
Is this something special about $U(n)$, or can $BG$ for other compact groups also be thought of as a limit of projective (or just Kähler) varieties? Is there a notion of complex structure on $BG$ where the Hodge-to-DeRham spectral sequence degenerates?
$\operatorname{SO(n)}$ in particular seems like it might be an issue, since it's a real Stiefel manifold.
