On convexity and the Iwasawa decomposition of split real and complex Kac–Moody groups

We prove an analogue of Kostant’s convexity theorem for split real or complex Kac–Moody groups associated to free, cofree and cotorsion-free root data. The result can be seen as a first step towards describing the multiplication map in a Kac–Moody group in terms of Iwasawa coordinates. Our method involves a detailed analysis of the geometry of regular Weyl group orbits in the Cartan subalgebra of a real Kac–Moody algebra. It provides an alternative proof of Kostant convexity for a large class of semisimple Lie groups and extends a linear variant of Kostant’s theorem for Kac–Moody algebras that was established by Kac and Peterson in 1984.