Decompositions of Kac-Moody groups

Abstract

Let $G$ be a split (minimal) Kac-Moody group over $\mathbb{R}$ or $\mathbb{C}$ with maximal torus $T$, and let $\theta$ be a Cartan-Chevalley involution of $G$, twisted by complex conjugation, and satisfying that $\theta (T)=T$. Furthermore, let $K$ be the subgroup fixed by $\theta$, and $\tau :G\to G,g\mapsto g\theta (g{)}^{-1}$. Let $A:=\tau (T)$. In this note, we show resp. revisit that $G$ admits a (refined) Iwasawa decompositions $G=UAK$. We also show that if $G$ is of non-spherical type, then it never admits a polar decomposition $G=\tau (G)K$ nor a Cartan decompositions $G=KAK$. This has implications for the geometrical structure of the Kac-Moody symmetric space $G/K\cong \tau (G)$.