Let $\hat{\mathfrak{g}}$ be an affine Kac-Moody algebra which is the central extension of $\mathfrak{g}[t,t^{-1}]$(polynomial version). Consider Iwahori subalgebra $I$. My question is whether $I$ corresponds to some Cartan decomposition for $\hat{\mathfrak{g}}$
Further question is if considering "generalized Verma module" corresponding to $I$, i.e. Define $M(\lambda):=U(\hat{\mathfrak{g}})\otimes_{U(I)}\mathbb{C}_\lambda$. Does $M(\lambda)$ in the category $O$ correspoding to this "Iwahori-induction"?