My research interests focus on the analysis of“symmetries”in mathematics. Currently I am working on representation theory and a theory of discontinuous groups including the following topics: 1 Analysis of minimal representations: Minimal representations are special irreducible representations, which are a building block of linear symmetries. My guiding hypothesis minimal representations (algebra) = maximal symmetries (function spaces) is a driving force for a new theory of global analysis based on non-commutative symmetries of minimal representations.

2 Spectral analysis on locally symmetric spaces: For spaces of indefinite metric, intrinsic differential operators (e.g., Laplacian) are not necessarily elliptic. As a first step of spectral theory in this new general setting, I am working on the construction of discrete spectrum of such operators, and studying its stability under the deformation of geometric structure.

My research achievements include

3 pioneering works on the theory of discontinuous groups for homogeneous spaces beyond the classical Riemannian setting,

4 pioneering works on the theory of discretely decomposable restrictions of representations (discrete symmetry breaking), and

5 an original theory of visible actions on complex manifolds, and its systematic and synthetic application to multiplicity-free theorems on both finite and infinite dimensional representations.