Theory of branching laws of unitary representations and non-commutative harmonic analysis by transformation groups of geometric structures

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The branching law is the irreducible decomposition of a group representation when restricted to a subgroup (e.g. decomposition of tensor products, breaking symmetry in physics,...). Analysis of branching laws is one of principal subjects in representation theory. Nevertheless, very little has been studied on branching laws of unitary representations of reductive groups, except for some special cases until mid 1990s, partly because of analytic difficulties arising from infinite dimensions.Our main results during this period are :1.To publish the basic theory of branching laws of unitary representations with emphasis on discrete decomposable cases and its applications to representation theory itself, non-commutative harmonic analysis, and topology of locally symmetric spaces.2.To construct the canonical representations of the conformal groups of pseudo-Riemannian manifolds by means of the Yamabe operator. In particular, we use this machinery for the study of minimal representations of in … More definite orthogonal groups. Besides, we find (conformal) conservation laws of solutions to certain ultra-hyperbolic equations. Branching laws when restricted to isometry groups play an important role in our analysis (joint with B.Orsted).3.To make a unified approach to multiplicity-free theorems of the biholomorphic transformation groups of complex manifolds by means of orbit-preserving anthi-holomorphic maps. In particular, we use this machinery for the study of multiplicity-free tensor product representations of general linear groups.4.In the late 1980, I initiated the study of discontinuous groups for pseudo-Riemannian homogeneous spaces. In the memorial volume of A.Borel, I published a survey article on this area, and also gave new criteria for the existence of co-compact discontinuous groups for semisimple symmetric spaces and their tangential symmetric spaces (joint with T.Yoshino). Besides, the deformation spaces of discontinuous groups are investigated.5.On these topics, I gave one-hour invited lectures in various international conferences, including ICM 2002, Pan-African Congress 2004 (plenary address), and Asian Congress 2005. Less