Symmetric spaces of Kac-Moody type

• Applicant: Professor Dr. Ernst Heintze
• Subject Area: Mathematics
• Term: from 2003 to 2009
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5406928

The aim of this project is to study special submanifolds and foliations which resemble orbits of isometric group actions. A major problem is to decide whether they are homogeneous. Typical examples of such submanifolds are the isoparametric hypersurfaces and their various generalizations. These are known to be homogeneous under certain restrictions. Our search focusses on submanifolds of Euclidean spaces, symmetric spaces and Hilbert spaces. In the last case the submanifolds are strongly related to loop groups and affine Kac-Moody groups. The problem whether they arise as orbits of the isotropy representation of certain infinite dimensional symmetric spaces of Kac-Moody type will be attacked. In addition, these symmetric spaces will be investigated themselves as they promise to have an interestering geometry. To understand the homogeneous examples, special classes of group actions which occur in this context shall be classified. These are the polar actions on symmetric spaces, as well as affine polar representation on Hilbert spaces. Moreover linear representations whose orbit spaces have a boundary will be investigated.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1154: Global Differential Geometry