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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theta对应建立了研究半单李群的酉表示的伴随圈的机器，更明确地说，设(G_1，G_2)是一个不可约的、可约的第I型对，它是以G_2为较小成员的稳定范围内的。取G_2的一个不可约表示π_2(或它的亚可解双覆盖)，并将其提升到G_1(或它的亚可解覆盖)的表示π_1，称为π_2的theta提升。我们研究了π_2是有限维么正表示或全纯离散级数表示的情况。即使π_2是平凡表示，π_1也是无限维么正表示，它被认为是一个幂等表示，这是表示论中的一个重要对象。在下面，我们将列出我们的结果，其中π_2是有限维么正表示，或者是全纯离散级数的表示，我们得到了π_1的K型公式，并明确地描述了(π_1，π_2)的相关簇之间的对应关系。此外，我们成功地得到了它们的关联圈中的重数，并得到了它们之间的一个简单的对应关系(一般仍是猜想的)。一个关联簇往往是不可约的，并且是单个幂零轨道的闭包。利用表示的theta对应的结果，我们研究了幂零轨道的对应。主要结果是，刻画了函数环的结构，给出了幂零轨道次数的积分公式。