Decompositions of Kac-Moody groups

TL;DR

This paper investigates the structural decompositions of split Kac-Moody groups over real or complex fields, establishing Iwasawa decompositions and demonstrating the absence of certain other decompositions in non-spherical cases, impacting the understanding of their symmetric spaces.

Contribution

It provides a detailed analysis of decompositions of Kac-Moody groups, including Iwasawa, polar, and Cartan decompositions, highlighting differences based on the group's type.

Findings

Kac-Moody groups admit refined Iwasawa decompositions.

Non-spherical type groups do not admit polar or Cartan decompositions.

Implications for the geometric structure of Kac-Moody symmetric spaces.

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 $G/K \cong \tau(G)$.