The local to global study of geometries was a major trend of 20th century, with remarkable developments achieved particularly in Riemannian geometry. In contrast, surprising little was known 30 years ago about global properties of locally homogeneous spaces with indefinite-metric, which led us to the theory of discontinuous groups beyond the Riemannian setting.
Concerning linear actions (representation theory), one of fundamental problems is to understand how things are built from smallest objects (irreducible decomposition). Branching problems are typical case, but were supposed to be out of control for reductive Lie groups. Breakthrough ideas for branching problems in representation theory emerged partially from the study of discontinuous groups beyond Riemannian setting, and conversely, they have opened new research such as global analysis of indefinite-Riemannian locally symmetric spaces (e. g. anti-de Sitter manifolds).
Based on the developments over the last two decades, we present a program on branching problems, from the general theory to concrete construction of symmetry breaking operators.
© Toshiyuki Kobayashi