Theta correspondence and associated cycles

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。

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.