Construction of a singular theta-lifting for unitary groups U(p,q)

本项目的目的是构造酉群U（p，q）的具有Millson型核函数的奇异θ-提升，这类提升在正交群中得到了很好的研究，并在算术和几何中得到了广泛的应用.它起源于Kudla和Millson的工作，并由Bruinier和Funke进一步研究。 它采取弱的马斯形式，并提升他们的微分形式上的对称空间的正交群。此外，它是毗邻另一个起重，库德拉米尔森电梯。在埃尔米特的情况下，O（p，2）这个属性也被几何Borcherds提升共享，但在一般情况下不是O（p，q）。对于酉群，这种提升的存在性也是从库德拉和米尔森的工作中得出的。然而，这还没有得到进一步的研究。相比之下，Borcherds提升分别在U（p，1）或U（q，1）的情况下得到了相当好的研究。由霍夫曼建造，其几何性质更密切地审查了布鲁尼耶，霍华德和杨。Hufler也给出了U（p，q）的一个构造，但在这种一般情况下，具有Millson型核的提升将是相当有趣的 用于几何应用。这是该项目的起点：由微分方程定义的核函数将被明确确定。构造奇异θ-升力并计算附绿色流。此外，它预计将研究对称区域的边界组件上的升力的行为。预计这将在使用Kudla程序的应用程序生成系列之间产生相同性。另一个计划的目标是明确地计算升力的傅里叶-雅可比展开。

The aim of the present project is the construction of a singular theta-lifting for unitary groups U(p,q) with a Millson-type kernel function.A lifting of this type is well-studied for orthogonal groups and has found a number of arithmetic and geometric applications. It originates in work of Kudla and Millson and was examined further by Bruinier and Funke. It takes weak Maass forms and lifts them to differential forms on the symmetric space of the orthogonal group. Also, it is adjoint to another lifting, the Kudla Millson lift. In the hermitian case, O(p,2) this property is also shared by the geometric Borcherds lift, not however in the general case O(p,q). For unitary groups, the existence of a lifting of this type also follows from the work of Kudla and Millson. However this has not been studied further as yet. Contrastingly, the Borcherds lift is fairly well-studied for the case U(p,1) or U(q,1), respectively. Constructed by Hofmann, its geometric properties were more closely examined by Bruinier, Howard and Yang. Also, a construction for U(p,q) has been given by Hufler.In this general case, however, a lifting with a Millson-type kernel would be of considerable interest for geometric applications. This is the starting point for the project: The kernel function, which is defined by a differential equation, is to be determined explicitly. The singular theta-lifting is to be constructed and attached Green's currents to be calculated. Further, it is projected to study the behavior of the lift on boundary components of the symmetric domain. This is expected to yield identities between generating series with applications to the Kudla program. A further planned objective is to calculate the Fourier-Jacobi expansion of the lift explicitly.