Geometric invariants of representations of real reductive groups and integral transformations

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The principal investigator Ochiai discusses the following problems and obtains the following results in the four-year research period. The associated variety and the associated cycles of unitary representations of real reductive groups are one of the important geometric invariants of representations. In the joint work with Kyo Nishiyama (Kyoto University) and C.B. Zhu (National University of Singapore), we deal with the unitary representations in the context of the reductive dual pairs. For the representations obtained by theta lifting from the holomorphic discrete series representations, we describe the associated varieties, which turn out to be the closure of a nilpotent orbit, and their multiplicities (Bernstein degrees) in terms of the definite integrals.The non-commutative harmonic oscillator is the Weyl quantization of the multi-component harmonic oscillators. The model has two non-commutativities: differential operators and matrices. For the rank two cases, by a representation theoretical approach, the spectral problem for the non-commutative harmonic oscillator is proved to be equivalent to the existence of the global solutions of the Heun differential equation on the complex domain.We also find the hidden symmetry in the space of invariant eigen distributions on the global affine symmetric spaces of a specific type as well as on their tangent spaces. In the joint work with Michihiko Fujii (Kyoto University), we give a decomposition of the D-modules defining the spaces of the vector-valued harmonic forms on the hyperbolic cone manifold into the scalar-valued systems.Related with the above activities, the investigator Kobayashi made an extensive research on the realization of the minimal representations and operators, on the proper actions of the discontinuous groups and on the visible actions on the homogeneous spaces.