At Infinity of Symmetric Spaces

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The purpose of the proposed research project is, as a first step, to revise the theory of Kac-Moody symmetric spaces in order to also cover the affine case and the field of complex numbers, in particular to describe the causal structure in this more general setting, to identify the twin building at infinity with respect to this causal structure, and to establish boundary rigidity.Boundary rigidity, as a second step, will then allow us to extend Galois descent of complex Kac-Moody groups, resp. complex Kac-Moody twin buildings to complex Kac-Moody symmetric spaces by extending the Galois automorphism at infinity to a Galois automorphism of the symmetric space, thus enabling us to initiate the study of almost split real Kac-Moody symmetric spaces. Along the way we will establish some (partial) classification results for connected topological Moufang twin buildings and extend the theory of topological Kac-Moody twin buildings also to the symmetrizable non-two-spherical situation.As a third step, we will investigate Kostant convexity properties related to the question whether the causal structure on the symmetric space actually defines a partial order.The proposed research is set within the rigidity theme of the priority programme and will benefit from the interaction with other projects concerned with buildings, symmetric spaces, Lorentzian geometry, and spin structures.