Speaker: Ralf Köhl (University of Kiel)

Title: Topological Kac-Moody groups -- Discussing the topology proposed by Kac and Peterson

Abstract: Given a (minimal) Kac-Moody group over a local field k (say of characteristic 0), for instance the subgroup of $Aut(g)$, g a Kac-Moody algebra over k, generated by the groups $(P)SL_2(k)$ belonging to the simple roots, one can endow it with the finest group topology such that the embeddings of the Lie groups $(P)SL_2(k)$ become continuous.

This turns out to be a Hausdorff group topology, and actually equal to the group topology that Kac and Peterson suggested for such Kac-Moody groups in the 1980s.

This topology is always $k_omega$, and it is locally compact if and only if the Kac-Moody group actually is a finite-dimensional Lie group.

This topology induces the Lie topology on the Levi factors of parabolics of spherical type, in the indefinite cases it provides new examples for Kramer's theory of topological twin buildings (which he developed in 2002 for loop groups), and in the Archimedian case it is possible to determine their fundamental groups, actually providing a structural explanation for the (known by classification) fundamental groups of semisimple split real Lie groups.

Moreover, in the 2-spherical situation these topological groups turn out to satisfy Kazhdan's Property (T), and allow Mostow-Margulis-type rigidity results for (S-)arithmetic subgroups.

Kac-Moody groups also admit symmetric spaces. In the non-spherical situation, these symmetric spaces have a causal structure with the two halves of the twin building visible at infinity in the future, resp. past directions. One can prove that either time-travel is impossible on Kac-Moody symmetric spaces (i.e., there are no non-trivial closed causal piecewise geodesic curves) or all points of the Kac-Moody symmetric space are causally equivalent. It turns out that the question which of the two cases occurs is equivalent to the question whether (global) Kostant convexity holds for Kac-Moody groups; it also seems, by an observation that Hartnick and Damour pointed out to me, that this question is closely related to what physicists call "cosmological billards", so from a physical point of view one should expect Kostant convexity to hold and, thus, time travel to be impossible on Kac-Moody symmetric spaces.

(Global) Kostant convexity is the question how the A-part in the Iwasawa decomposition KAN changes if one multiplies with an element in K from the wrong side. There is a local version concerning the adjoint action on the Kac-Moody algebra, and this holds by a result by Kac and Peterson from 1984.

I started thinking about Kac-Moody groups as a postdoc in 2005 (when together with three peers at TU Darmstadt we founded what we then called the "anonymous Kac-Moody theorists"), and this is a report on various things we encountered along the way; two of the three other anonymous Kac-Moody theorists became key collaborators of mine over the years. My collaborators and students during various stages of my work will be mentioned explicitly as we make our way through the various observations.