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

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 这允许将某些真实的一阶紧致对称空间上的各向同性表示和调和分析联系起来。（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。

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