I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 of this PDF). Let $\mathcal{K}_J$ be a parabolic subgroup of $\mathcal{K}$.
I would like to use the system $\{\mathcal{K}^{*n}\}$ as a model for $E\mathcal{K}_J$ and compare it to the system $\{\mathcal{K}_J^{*n}\}$. Specifically I want to show that $\varprojlim K_{\mathcal{K}_J}( \mathcal{K}^{*n}) \cong \varprojlim K_{\mathcal{K}_J}( \mathcal{K}_J^{*n})$. Normally the two systems are required to be compact so that at each stage so that you can construct an isomorphism (like page 5 of Atiyah-Segal's completion theorem ).
Can I use the standard $\mathcal{K}_J$-CW complex of $\mathcal{K}$ to filter $\{\mathcal{K}_J^{*n}\}$. Lets call the $m$-th stage $\mathcal{K}_{J,m}$. It is compact. Can I use the system $\{\mathcal{K}_{J,m}^{*n}\}$, which is compact at every stage as an intermediary between $\{\mathcal{K}_J^{*n}\}$ and $\{\mathcal{K}^{*n}\}$ to show that $\varprojlim K_{\mathcal{K}_J}( \mathcal{K}^{*n}) \cong \varprojlim K_{\mathcal{K}_J}( \mathcal{K}_J^{*n})$?
