相似类约化群上的Theta对应

Roughly speaking, Weil representation is closely related to Fourier transformation. This kind of representation plays an important role in Representation theory, Number theory and Algebraic Geometry. For instance, its restriction to a reductive dual pair ( in the sense of Howe) yields a correspondence, called Theta correspondence, between certain representations of the pair. Theta correspondences provide a unified way to explain classical theta series, theta functions, etc. One way to approach Langlands Functoriality is using theta correspondences, although the exact relation between this two parts are very apparent. In this project, we will study theta correspondences for the similitude reductive groups and focus our research on local field cases, i.e., p-adic field, real number field, or complex number field. It is also beneficial to compare our results with other research fields, such as Local Langland correspondences, L-functions, Shimura variety and so on.

Weil 表示是隐藏在Fourier 变换背后的一类表示。这类表示在诸多领域中具有广泛应用。例如在著名的Langlands纲领中，可以通过研究其限制到对偶约化群来实现部分函子性。这类研究一般都可统一在被称为Howe 对偶或Theta 对应框架下进行。 . 在此项目中我们将考虑推广经典的Howe 对应到相似类约化群上。 该研究更具有几何意义和前瞻性， 因为Langlands纲领 中核心的几何对象之一---Shimura 簇， 就是定义在约化群的相似类群上的。. 鉴于这类问题的研究历史，现状及相关数学发展的成熟度，我们将集中研究局部域上的Theta 对应， 同时也会涉及些相关领域的应用。 例如， 局部Langlands对应， 具体的L-函数构造等。