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Theta correspondence and associated cycles

Theta 对应和相关周期

基本信息

批准号：
11640025
负责人：
NISHIYAMA Kyo
金额：
$ 2.3万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640025/
关键词：
semisimple Lie group unitary representation associated variety associated cycle nilpotent orbit theta correspondence geometric invariant theory 半単純群の表現論 reductive dual pair 概均質ベクトル空間球等質空間

项目摘要

We developed certain machinery for studying associated cycles of unitary representations of semisimple Lie groups by means of theta correspondence.To be more explicit, let (G_1 , G_2) be an irreducible, reductive dual pair of type I, which is in the stable range with G_2 as the smaller member. Take an irreducible representation π_2 of G_2 (or of its metaplectic double cover), and lift it to the representation π_1 of G_1 (or its metaplectic cover), called the theta lift ofπ_2. We study the cases where π_2 is a finite-dimensional unitary representation, or a representation in the holomorphic discrete series. Even if π_2 is the trivial representation, π_1 is an infinite dimensional unitary representation, which is supposed to be a unipotent representation, one of important objects in representation theory. In the following, we shall list up our results, where π_2 is a finite-dimensional unitary representation, or a representation in the holomorphic discrete series.We get the K-type formula for π_1.We describe the correspondence of associated varieties of (π_1, π_2) explicitly. Moreover, we succeed to get the multiplicities in their associated cycles and get a simple correspondence (still conjectural in general) between them.An associated variety can often be irreducible, and is the closure of a single nilpotent orbit. Using the results on the theta correspondence of representations, we investigate the corre spondence of nilpotent orbits. The main results are, a description of the structure of the function ring, an integral formula of degree of nilpotent orbits.

我们利用θ对应研究了半单李群酉表示的伴随圈，更明确地说，设（G_1，G_2）是I型不可约的可约对偶对，它在稳定范围内，G_2是较小的成员。取G_2（或其亚复盖）的一个不可约表示π_2，将其提升到G_1（或其亚复盖）的表示π_1，称为π_2的θ提升。研究了π_2是有限维酉表示或全纯离散级数表示的情形。即使π_2是平凡表示，π_1也是无穷维酉表示，它被认为是幂幺表示，是表示论的重要研究对象之一。下面，我们将列出我们的结果，其中π_2是有限维酉表示，或全纯离散级数的表示，我们得到了π_1的K型公式，明确地描述了（π_1，π_2）的相关簇的对应。此外，我们还成功地得到了它们相关联的圈中的重数，并得到了它们之间的简单对应（一般来说仍然是拓扑的），相关联的簇通常是不可约的，并且是单个幂零轨道的闭包。利用有关表示的θ对应的结果，研究了幂零轨道的对应性。主要结果有：函数环的结构描述，幂零轨道次数的积分公式。