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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)We用不变量理论描述了幂零轨道的θ提升，证明了一些好的提升的闭包是正规簇，代数群在它们上的作用是多重自由的.得到了用积分表示的幂零轨道的度公式，构造了相干层提升的一般理论，特别证明了相干层支撑重数的保持性.（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，并给出了球面幂零轨道上函数环的结构。(3)我们将θ提升的概念推广到一般轨道而不是幂零轨道，从而为轨道对应的完整理解研究提供了一个思路。最后，我们总结了各位学者的研究成果：关口从不变论的角度研究了双线性型的幺模同余类，太田阐明了复约化代数群的轨道与其真实的形式之间的对应关系，Yamashita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向量的相关圈之间的关系。