Mini-Workshop: Generalizations of Symmetric Spaces

Abstract

This workshop brought together experts from the areas of algebraic Lie theory, invariant theory, Kac–Moody theory and the theories of Tits buildings and of symmetric spaces. The main focus was on topics related to symmetric spaces in order to stimulate progress in current research projects or trigger new collaboration via comparison, analogy, transfer, generalization, and unification of methods. Specific topics that were covered include Kac– Moody symmetric spaces, double coset decompositions of (groups of rational points of) algebraic groups and Kac–Moody groups, and symmetric/Gelfand pairs in Lie algebras. Mathematics Subject Classification (2000): 17Bxx, 20Gxx, 51E24, 53C35. Introduction by the Organisers The topics of this workshop all in some way evolved from the classical theory of real and complex Lie groups. One of the important mathematical goals during the 1950’s was to find analogs of the semisimple Lie groups of exceptional type over arbitrary fields. Chevalley completed the first crucial step by producing his famous basis theorem for simple complex Lie algebras, and later Steinberg succeeded in describing these analogs group-theoretically. An important theory developed by Tits was the theory of groups with a BN -pair and the invention of buildings; these buildings belong to arbitrary Chevalley groups as naturally as the projective spaces belong to the special linear groups. Certain S-arithmetic groups in positive characteristic and the Kac–Moody groups also belong to the class of groups admitting BN -pairs. Since then the various disciplines developed into different directions. However, due to their common origin the different theories often lead naturally to similar 1108 Oberwolfach Report 18/2012 questions. The discussions during the workshop concerning Kac–Moody symmetric spaces may serve as a suitable example how different approaches and backgrounds can interact: the absence of a KAK decomposition for real Kac–Moody groups implies that real Kac–Moody symmetric spaces suffer from the shortcoming that there exist pairs of points that do not lie in a common flat; an identification of the set of points of a real Kac–Moody symmetric space with the set of anisotropic involutions of the corresponding real Kac–Moody group immediately shows that nevertheless each point of the Kac–Moody symmetric space can be joined with each point at infinity by a geodesic ray; investigations as to whether KNK (Kostant) decompositions hold in real Kac–Moody groups will provide insight on whether a reasonable concept of horospheres exists in Kac–Moody symmetric spaces; connections on real Kac–Moody symmetric spaces arise by abstract means from the underlying Kac–Moody Lie triple systems. In total there have been 16 talks by 12 of the participants of the workshop that gave insight into different aspects of the theory of symmetric spaces, its generalizations, and neighbouring fields. These 16 talks are represented by the 15 attached reports, the two talks on hyperbolic Kac–Moody geometry having been subsumed into one report. We are particularly pleased by the lively interaction between the participants during the long afternoon breaks (each morning’s lectures finished at 11.30 a.m. while the afternoon sessions only started at 4 p.m.) and during the evening. Mini-Workshop: Generalizations of Symmetric Spaces 1109 Mini-Workshop: Generalizations of Symmetric Spaces