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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"?

Thanks

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  • $\begingroup$ I'm not sure what counts as a "Cartan decomposition;" of course, you can define a Cartan involution using that on the finite part and $t\mapsto t^{-1}$. However, invariants won't be the Lie algebra of something compact. $\endgroup$
    – Ben Webster ♦
    Commented Dec 9, 2011 at 14:47
  • $\begingroup$ To your second question: yes. The Iwahori is exactly the subalgebra spanned by positive root spaces, so induction from it is a Verma module. $\endgroup$
    – Ben Webster ♦
    Commented Dec 9, 2011 at 14:49
  • $\begingroup$ Thank you very much! Yes, I realized that one can write down this decomposition which is very similar to "standard cartan decomposition" for affine Kac-Moody algebra(the only difference is replace $t^{-1}\mathbb{C}[t^{-1}]$ by $t\mathbb{C}[t]$ in standard one) and Cartan involution is like what you said: finite part and $t\mapsto t^{-1}$ $\endgroup$
    – Shizhuo Zhang
    Commented Dec 9, 2011 at 15:02

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