At Infinity of Symmetric Spaces

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.

作为第一步，提议的研究项目的目的是修改kac-moody对称空间的理论，以涵盖仿射案例和复数的领域，特别是在这种更一般的环境中描述因果结构，以确定相对于这种因果结构的无限属性的双胞胎建筑，并允许建立边界刚性。解答复杂的Kac-Moody双胞胎建筑物与复杂的Kac-Moody对称空间通过将无穷大的Galois自动形态扩展到对称空间的Galois自动形态，从而使我们能够启动几乎分裂的真实Kac-Moody对称空间的研究。一路走来，我们将为连接的拓扑穆法双胞胎建筑物建立一些（部分）分类结果，并将拓扑kac-moody双胞胎建筑的理论扩展到同样可用的非两次态势的情况下，作为第三步，我们将研究与kostant converities的启动性的启动性相关的启动性的启动阶层是否符合对称性的启动，是否符合对称性的依据，是否符合对称性的空间。并将受益于与建筑物，对称空间，洛伦兹几何形状和旋转结构有关的其他项目的互动。