Geometric invariants of representations of real reductive groups and integral transformations

首席研究员落合在四年的研究期间讨论了以下问题，并获得了以下结果。真实的约化群的酉表示的相伴簇和相伴圈是表示的重要几何不变量之一。在与Kyo Nishiyama（京都大学）和C.B.朱（新加坡国立大学），我们在约化对偶对的背景下处理酉表示。对于从全纯离散级数表示中提升θ得到的表示，我们用定积分描述了相应的簇，这些簇是幂零轨道的闭包及其重数（伯恩斯坦度），非对易谐振子是多分量谐振子的Weyl量子化.该模型具有两个不可交换性：微分算子和矩阵。对于秩为2的情形，利用表示论方法，证明了非对易谐振子的谱问题等价于复域上Heun微分方程整体解的存在性，并在特定类型的整体仿射对称空间及其切空间上，发现了不变本征分布空间的隐对称性.在与藤井裕彦的合作中，（京都大学），我们给出了双曲锥流形上定义向量值调和形式空间的D-模到标值系统的分解.与上述活动相关，研究者小林对极小表示和算子的实现进行了广泛的研究，不连续群的真作用和齐性空间上的可见作用。

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.