Kostant Convexity for split real and complex Kac-Moody Groups

The goal of this thesis is a generalization of Kostant’s Convexity Theorem to the class of (split, minimal) real or complex Kac-Moody groups. Originally, this notion refers to an important result in Lie theory that is related to the problem of describing the group multiplication in a semisimple Lie group in terms of Iwasawa coordinates.

In order to prove an analogue of Kostant’s theorem in the Kac-Moody setting, the thesis elaborates on the basic structure and representation theory of Kac-Moody groups & algebras and performs a detailed analysis of the geometry of convex hulls of Weyl group orbits in the Cartan subalgebra of a real or complex Kac-Moody algebra. In the “generic” case of an infinite Weyl group, these convex hulls cannot be investigated using the classical theory of convex polytopes. Instead, the theory of Coxeter groups and the notion of “coloured Coxeter diagrams” are applied in order to describe the face posets of convex hulls of Weyl group orbits and characterize these sets by linear inequalities.

Using these ideas, two similar but unrelated results of Kostant, typically known as the “linear” and the “non-linear” convexity theorem, generalize in a natural way to real and complex Kac-Moody groups. In contrast to the usual analytic techniques of Lie theory, the proof favours combinatorial and representation-theoretic methods that extend more easily to this setting and provide an alternative approach in the classical case of Lie groups as well. As a complement to the main results, the thesis presents various applications of the convexity theorems to algebraic properties of real and complex Kac-Moody groups.

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