At Infinity of Symmetric Spaces

作为第一步，修正Kac-Moody对称空间的理论，以涵盖仿射情形和复数领域，特别是描述在这种更一般的背景下的因果结构，识别关于这种因果结构的无穷远处的孪生建筑，并建立边界刚性。作为第二步，边界刚性将允许我们扩展复Kac-Moody群的Galois下降。通过将无穷远处的Galois自同构推广到对称空间的Galois自同构，从而将复Kac-Moody孪生建筑推广到复Kac-Moody对称空间，从而使我们能够开始研究几乎分裂的实Kac-Moody对称空间。在此过程中，我们将建立连通拓扑Moufang孪生建筑的一些(部分)分类结果，并将拓扑Kac-Moody孪生建筑的理论推广到可对称化的非两球情形。作为第三步，我们将研究与对称空间上的因果结构是否实际上定义了偏序这一问题有关的Kostant凸性性质。所提出的研究设定在优先方案的刚性主题内，并将受益于与建筑、对称空间、洛伦兹几何和自旋结构相关的其他项目。

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.