Isotropy representations associated with Harish-Chandra modules and nilpotent orbit theory

与 Harish-Chandra 模和幂零轨道理论相关的各向同性表示

• 批准号: 14340001
• 负责人: YAMASHITA Hiroshi
• 金额: $ 5.95万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340001/
• 关键词: semisimple Lie group; nilpotent orbit; unitary representation; Harish-Chandra module; isotropy representation; discrete series representation; moment map; Howe duality correspondence; 既約ユニタリ表現; 単単純リー群; 無限次元表現; 不変微分作用素; 随伴多様体; エルミート対象空間; ユニポテ

The principal purpose of this project is to develop nilpotent orbit theory for Harish-Chandra modules corresponding to irreducible admissible representations of real semisimple Lie groups, in order to get deep understanding on generalized Whittaker models in relation to various nilpotent invariants of representations. We focused our attention to the isotropy representations which give the multiplicities in the associated cycles of Harish-Chandra modules.(1)The isotropy representations have been described explicitly for all singular unitary highest weight modules of simple Hermitian Lie algebras by using the projection to the PRV-components. As a result we have proved the irreducibility of such isotropy representations. A new proof of the Howe duality correspondence for reductive dual pairs is obtained via isotropy representations. For EVII, we have shown that the isotropy representations for certain unitary highest weight modules of non-scalar type give the Dvorski-Sahi correspondence. … More This allows relating the isotropy representations and harmonic analysis on certain compact symmetric spaces of real rank one.(2)General machinery has been established to describe the isotropy representations of Harish-Chandra modules with irreducible associated varieties, by using the principal symbols of differential operators of gradient-type. Applying this machinery, we have revealed an explicit relationship between the isotropy representations for discrete series and the generic fiber of the moment map for the conormal bundle of closed orbits on the flag variety.(3)To identify the generic fiber of the moment map, we have studied the Richardson orbits associated with symmetric pairs. It has been an open problem whether the parabolic subgroups for the Richardson orbits act transitively on the set of Richardson elements in the symmetric part of its nilradical. In this project, we have got a progress to this problem, by giving nice sufficient conditions for the transitivity, and also a counterexample for the Lie groups of type A Less

本项目的主要目的是发展对应于真实的半单李群的不可约容许表示的Harish-Chandra模的幂零轨道理论，以便深入理解广义Whittaker模型与各种幂零不变量的关系。我们把我们的注意力集中到各向同性表示，给出了在相关的循环的Harish-Chandra模块的多重性。（1）利用单Hermitian李代数的PRV-分支的投影，给出了单Hermitian李代数的奇异酉最高权模的各向同性表示。结果证明了这种各向同性表示的不可约性。利用各向同性表示给出了约化对偶对的Howe对偶对应的一个新证明。对于EVII，我们证明了某些非标型酉最高权模的各向同性表示给出了Dvorski-Sahi对应。 ...更多信息 这使得有关的各向同性表示和调和分析某些紧凑的对称空间的真实的秩一。（2）利用梯度型微分算子的主符号，建立了描述具有不可约伴随簇的Harish-Chandra模的各向同性表示的一般机制。应用这一机制，我们已经揭示了一个明确的关系的各向同性表示离散系列和通用纤维的时刻地图的余法线束的封闭轨道的旗品种。(3)To为了确定矩映射的一般纤维，我们研究了与对称对相关的Richardson轨道。Richardson轨道的抛物子群是否传递地作用在其幂零根的对称部分的Richardson元的集合上，一直是一个公开的问题。在这个项目中，我们对这个问题有了一个进展，给出了传递性的充分条件，并给出了A型李群的一个反例。

项目成果

Theta lifting of unitary lowest weight modules and their associated cycles

单一最低重量模块的 Theta 提升及其相关循环

• 发表时间: 2004
• 期刊: Duke Math.J. 125
• 作者: K.Nishiyama;Zhu;Chen-bo
• 通讯作者: Chen-bo

Isotropy representation associated with the discrete series (in Japanese)

与离散序列相关的各向同性表示（日语）

• 发表时间: 2005
• 期刊: Su^rikaisekikenkyusho Ko^kyu^roku 1421
• 作者: Hiroshi Yamashita;山下 博;Hiroshi Yamashita;Hiroshi Yamashita
• 通讯作者: Hiroshi Yamashita

Satoshi Naito: "Three kinds of extremal weight vectors fixed by a diagram automorphism"Jornal of Algebra. 268・1. 343-365 (2003)

Satoshi Naito：“由图自同构固定的三种极值权向量”268・1（2003）。

A note on affine quotients and equivariant double fibrations

关于仿射商和等变双纤维的注释

• 发表时间: 2005
• 期刊: Infinite Dimensional Harmonic Analysis III(World Scientific)
• 作者: Junya Satoh;K.Nishiyama
• 通讯作者: K.Nishiyama

Crossed Burnside rings.II. The Dress construction of a Green functor.

交叉的伯恩赛德环。II。

• 发表时间: 2004
• 期刊: Journal of Algebra 282
• 作者: YOSHIDA;Tomoyuki
• 通讯作者: Tomoyuki