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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)我们用不变量理论描述了幂零轨道的theta提升，并证明了一些好的提升（的闭包）是正规簇，并且代数群对它们的作用是无重数的。我们还得到了用积分表示的幂零轨道的度数公式。构建了相干滑轮提升的一般理论，特别证明了相干滑轮支撑多重性的保存性。(2)对于不定酉群U(p,p)，我们对球面幂零轨道进行了完全分类，同时描述了函数环的结构 (3)我们将θ提升的概念推广到幂零轨道以外的一般轨道，并为轨道对应关系的完整理解的研究提供了线索。其中包括关口、德约科维奇、赵研究的双线性形式的单模同余类的例子。最后，我们总结了每位研究者的研究。关口从不变理论的角度研究了双线性形式的单模同余类。太田阐明了复还原代数群的轨道与其实数群的对应关系。更一般地说，他将他的研究扩展到对称对的轨道。Yamashita 研究了半单李群的酉表示、它们的各向同性表示和广义 Whittaker 向量的关联循环之间的关系。