If I have a Lie group (or a Kac-Moody group) $G$ such that it's the amalgated product of it's proper parabolic subgroups $P_J$, i.e. $G = \text{colim} P_J$, then could I use this to compute the equivariant K-theory of a $G$-space $X$. There is a map $$ K_G(X) \to \lim K_{P_J}(X)$$
I know that this question reduces to computing the representation ring of $G$ from it's parabolic subgroups. In terms of discrete groups this doesn't work so well.
