Infinite dimensional quaternionic representations and nilpotent orbits

不可约Harish-Chandra模的相关变量给出了实约群的不可约容许表示的基本幂零不变量。此外，相关变化的不可约分量的Harish-Chandra模中的多重度可以看作是某个有限维表示的维数，称为各向同性表示。首席研究员Yamashita已经证明，在许多情况下，各向同性表示原则上可以用梯度型微分算子的主符号来描述，其核实现对偶Harish-Chandra模。在本研究项目中，我们开始系统地研究了具有不可约相关变量的Harish-Chandra模的各向同性表示，包括四元数表示、离散级数和幺正最高权模。结果总结如下：从Vogan相关旋回理论出发，对各向同性表示提出了一般性的理论。特别地，给出了各向同性表示不可约性的一个判据。此外，我们还研究了何时各向同性表示可以用梯度型微分算子来描述。对于离散级数，统一构造了各向同性表示的非零商。看来这个商表示在整个各向同性模块中是足够大的。我们已经证明，如果从所讨论的离散级数正规确定的稳定抛物子群相对于复化对称对允许Richardson幂零轨道，则会出现这种情况。对厄密李代数BI、DI和evi的每一个奇异酉最高权模的各向同性表示进行了显式描述。这使得我们可以推导出各向同性模对于任意简单厄米李代数的所有奇异酉最高权模都是不可约的。研究人员的主要贡献：Saito发展了他的a -超几何系统的研究，这与实现幺正最高权模密切相关。他建立了齐次a -超几何系统的秩公式。Wachi构造了标量型广义Verma模的Capelli恒等式的类比。Nishiyama和Ohta通过约化对偶对的矩映射给出了与对称对相关的幂零轨道的对应关系。少

The associated variety of an irreducible Harish-Chandra module gives a fundamental nilpotent invariant for the corresponding irreducible admissible representation of a real reductive group. Moreover, the multiplicity in the Harish-Chandra module of an irreducible component of the associated variety can be regarded as the dimension of a certain finite-dimensional representation, called the isotropy representation.The head investigator, Yamashita, has already shown that, in many cases, the isotropy representation can be described, in principle, by means of the principal symbol of a differential operator of gradient-type whose kernel realizes the dual Harish-Chandra module. In this research project, we have begun a systematic study of the isotropy representations attached to Harish-Chandra modules with irreducible associated varieties, including quaternionic representations, discrete series and unitary highest weight modules.The results are summarized as follows:We developed a general the … More ory for the isotropy representations, starting from the Vogan theory on associated cycles. In particular, a criterion for the irreducibility of an isotropy representation is presented. Also, we looked at when the isotropy representation can be described in terms of a differential operator of gradient-type.As for the discrete series, a nonzero quotient of the isotropy representation has been constructed in a unified manner. It seems that this quotient representation is large enough in the whole isotropy module. We have shown that this is the case if the theta-stable parabolic subgroup canonically determined from the discrete series in question admits a Richardson nilpotent orbit with respect to the complexified symmetric pair.The isotropy representation is explicitly described for every singular unitary highest weight module of Hermitian Lie algebras BI, DI and EVII. This allows us to deduce that the isotropy modules are irreducible for all singular unitary highest weight modules of arbitrary simple Hermitian Lie algebra.Principal contribution by the investigators : Saito developed his research on A-hypergeometric system, which is closely related to a realization of unitary highest weight modules. He has established a formula for the rank of a homogeneous A-hypergeometric system. Wachi constructed an analogue of the Capelli identity for generalized Verma modules of scalar type. Nishiyama and Ohta gave a correspondence of nilpotent orbits associated to a symmetric pair, by menas of the moment map with respect to a reductive dual pair. Less