On convexity and the Iwasawa decomposition of split real and complex Kac-Moody groups
Abstract
We prove an analogue of Kostant's convexity theorem for split real and complex Kac-Moody groups associated to free and cofree 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 Weyl group orbits in the Cartan subalgebra of a real Kac-Moody algebra. It provides an alternative proof of Kostant convexity for semisimple Lie groups and also generalizes a linear analogue of Kostant's theorem for Kac-Moody algebras that has been established by Kac and Peterson in 1984.
Keywords
Cite
@article{arxiv.2312.02584,
title = {On convexity and the Iwasawa decomposition of split real and complex Kac-Moody groups},
author = {Paul Zellhofer and Ralf Köhl},
journal= {arXiv preprint arXiv:2312.02584},
year = {2024}
}
Comments
34 pages, 1 figure, added: classification of GCM's in Section 1; minor change in the proof of Proposition 2.6; discussion including non-split Lie groups in Section 4.3; Corollary 4.4 in Section 4.3