Infinite dimensional quaternionic representations and nilpotent orbits

无限维四元数表示和幂零轨道

• 批准号: 12640001
• 负责人: YAMASHITA Hiroshi
• 金额: $ 2.3万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640001/
• 关键词: semisimple Lie group; Harish-Chandra module; nilpotent orbit; quaternionic symmetric space; unitary highest weight module; discrete series; isotropy representation; invariant differential operator; 無限次元表現; 随伴サイクル

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

不可约Harish-Chandra模的伴随簇给出了真实的约化群的不可约容许表示的基本幂零不变量。此外，相关簇的不可约分量的Harish-Chandra模中的重数可以被视为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被描述为：借助于核实现对偶Harish-Chandra模的梯度型微分算子的主符号.在本研究计划中，我们系统地研究了带有不可约伴随簇的Harish-Chandra模的各向同性表示，包括四元数表示、离散级数和酉最高权模，得到了如下结果：我们发展了一个一般的 ...更多信息 各向同性表示的理论，从相关循环的沃根理论开始。特别地，给出了各向同性表示不可约的一个判据。此外，我们还研究了各向同性表示何时可以用梯度型微分算子来描述。对于离散级数，我们以统一的方式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中是足够大的。我们证明了如果由离散级数正则确定的θ-稳定抛物子群关于复化对称对有一个Richardson幂零轨道，则是这种情况.对于Hermitian李代数BI，DI和EVII的每一个奇异酉最高权模，都明确地描述了各向同性表示.这使得我们可以推出，对于任意单Hermitian李代数的所有奇异酉最高权模，迷向模都是不可约的。主要贡献：Saito发展了他对A-超几何系统的研究，这与酉最高权模的实现密切相关。他建立了一个公式的秩齐次A-超几何系统。Wachi构造了一个类似的Capelli身份的广义Verma模块的标量类型。Nishiyama和Ohta利用关于约化对偶对的矩映射给出了与对称对相关联的幂零轨道的对应。少

项目成果

Hiroshi Yamashita: "Cayley transform and generalized Whittaker models for irreducible highest weight modules"Asterisque. 273. 81-137 (2001)

Hiroshi Yamashita：“不可约最高重量模块的凯莱变换和广义 Whittaker 模型”Asterisque。

Kyo Nishiyama: "Multiplicity-free actions and the geometry of nilpotent orbits"Mathematische Annalen. 318・4. 777-793 (2000)

西山京：“多重自由作用和幂零轨道的几何”《数学年鉴》318・4（2000）。

Hiroshi Yamashita: "Associated cycles of Harish-Chandra modules and differential operators of gradient type"RIMS Kokyuroku. Vol. 1183. 157-167 (2001)

Hiroshi Yamashita：“Harish-Chandra 模和梯度型微分算子的关联循环”RIMS Kokyuroku。

Kyo Nishiyama: "Bernstein degree and associated cycles of Harish-Chandra modules-Hermitian symmetric case"Asterisque. 273. 13-80 (2001)

Kyo Nishiyama：“伯恩斯坦度和 Harish-Chandra 模的相关循环 - 埃尔米特对称情况”星号。

Kyo Nishiyama: "Theta lifting of holomorphic discrete series. The case of U(p, q)xU(n, n)"Transactions of AMS. 353・8. 3327-3345 (2001)

Kyo Nishiyama：“全纯离散级数的 Theta 提升。U(p, q)xU(n, n) 的情况”AMS 353・8 (2001)。