Both, as long as we have a continuous $G$ action on $\mathcal{H}$. It does not have a $G$-CW complex associated to it, which is apparently something I need to claim it's a model for EG. I'm trying to figure this bit out.

@DavidRoberts, it works out, under the strong topology $\mathcal{U(H)}$ is a contractible topological group with an injection of $G$. The action of $G$ on $\mathcal{F(U)}$ also factors through $\mathcal{U(H)}$.

I want $G$ to be a Kac-Moody group eventually, but from the paper that @DavidRoberts linked it works for Lie groups using the compact open topology on $\mathcal{U(H)}$. In general $\mathcal{H}$ is a sum of infinitely many copies of each irreducible representation.

I've changed the title. The paper you linked seems to be very helpful at first glance.

Yea, in the case of $SU(n)$, $R_G$ is a polynomial algebra on a suitable basis of symmetric polynomials on $n$-generators, so in this case it reduces to that.

Yea, it's close to a polynomial algebra. It's finitely generated but the exact resolution will probably depend on $G$.

This is not true, btw. I have a counterexample but it's quite long and involved.