Theory of branching laws of unitary representations of reductive Lie groups and geometric realization of representations

The branching law means the irreducible decomposition of an irreducible unitary representation of a group when restricted to a subgroup (e.g. decomposition of tensor products, breaking symmetry in physics,…). It is one of principal subjects in representation theory to find branching laws. Nevertheless, very little has been studied on branching laws of unitary representations, except for some special cases until mid-90s, partly because of analytic difficulties arising from infinite dimensions. 1. Our main results during this period are to establish a basic theory of "discrete branching laws of infinite dimensional representations of semisimple Lie groups. Namely, based on new examples that we had found some years ago, we proposed a formulation of discrete branching laws, and proved a criterion for branching laws to be discretely decomposable by using both micro-local analysis and algebraic representation theory. Furthermore, we found new applications of these representation theoretic results to the following problems : i) Non-commutative harmonic analysis. To construct new discrete series representations for homogeneous spaces. ii) Automorphic forms. To prove a vanishing theorem of modular varieties for locally Riemannian symmetric spaces. Moreover, we found explicitly branching laws in certain settings in connection with conformal geometry. On these topics, I gave one-hour lectures in various international conferences, and a plenary lecture at MSJ for the Spring Prize (1999). Also, I gave series of lectures at European School (2000), at Harvard University (2001), and the Winter School at Czech Republic (2002) 2. Since the late 1980s, I have initiated the study of the existence problem of compact CliffordKlein forms of pseudo-Riemannian homogeneous manifolds. Recently, this problem has been studied by different methods such as discrete groups, ergodic theory, symplectic geometry and unitary representation theory, revealing the interactions with other branches of mathematics. I wrote an expository survey on this area and posed some open problems in "Mathematics Unlimited, 2001 and beyond" as a project of the World Mathematical Year 2000.