Geometry and Harmonic Analysis on Nilpotent Orbits

幂零轨道的几何与调和分析

基本信息

批准号：
13440046
负责人：
NISHIYAMA Kyo
金额：
$ 5.12万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

(1)We describe the theta lifting of nilpotent orbits in the terms of invariant theory, and prove that (the closures of) some good liftings are normal variety and the action of the algebraic groups on them are multiplicity free. We also obtain a degree formula of the nilpotent orbits expressed by integrals.A general theory of the lifting of coherent sheaves are constructed, and in particular we prove the preservation of the multiplicities of the support of coherent sheaves.(2)For the indefinite unitary group U(p,p), we completely classify the spherical nilpotent orbits and at the same time we describe the structure of the function rings on them.(3) We generalize the notion of theta lifting to general orbits other than nilpotent ones, and shed a light on the research of the complete understanding of the orbit correspondence. This includes an example of the unimodular congruence classes of the bilinear forms studied by Sekiguchi, Djokovic, Zhao.At last, we summarize the research of each investigator.Sekiguchi has studied the unimodular congruence classes of the bilinear forms from the view point of the invariant theory.Ohta has clarified the correspondence between the orbits of complex reductive algebraic groups and its real forms.More generally, he extends his research to the orbits of symmetric pairs.Yamashita has investigated the relations between associated cycles of the unitary representations of semisimple Lie groups, their isotropy representations and generalized Whittaker vectors.

（1）我们用不变理论的术语描述了nilpotent轨道的theta提升，并证明（封闭）某些良好的升降机是正常的变化，而代数群对它们的作用是无数次的。 We also obtain a degree formula of the nilpotent orbits expressed by integrals.A general theory of the lifting of coherent sheaves are constructed, and in particular we prove the preservation of the multiplicities of the support of coherent sheaves.(2)For the indefinite unitary group U(p,p), we completely classify the spherical nilpotent orbits and at the same time we describe the structure of the function rings on （3）我们将theta提升的概念推广到除nilpotent的轨道以外的其他轨道，并阐明了对轨道对应关系的完全理解的研究。这包括一个由Sekiguchi，Djokovic，Djokovic，Zhao.at研究的双线性形式的单线性形式的示例，我们总结了每个研究者的研究。Sekiguchi研究了单型的一致性类别的单型一致性班级，从无效的理论的角度来看。通常，他将研究扩展到对称对的轨道。Yamashita调查了半岛谎言群体的单一表示的相关周期之间的关系，他们的各向同性表示和广义的Whittaker载体。