Kostant convexity for Kac-Moody groups

The notion of Kostant Convexity refers to a wide range of similar and related theorems in different areas of mathematics. In its original form, it is a classical result in Lie theory that was established by B. Kostant in 1973. It states that if G=KAU is an Iwasawa decomposition of a semisimple Lie group, then for any fixed a in A, the set of A-components of all elements in the left coset aK coincides with the convex hull of the Weyl group orbit of that chosen element. While this statement is completely algebraic, the theorem can be reformulated in many ways and has been transferred to various settings, e.g. symplectic geometry or the theory of buildings. In this talk, I will present a natural generalization of Kostant’s Convexity Theorem to the class of split real and complex Kac-Moody groups and some of its applications to their algebraic structure. Most notably, the result can be used to show that in the non-spherical case, the causal pre-order on Kac-Moody symmetric spaces introduced by Freyn, Hartnick, Horn and Köhl is a partial order.