在本项目中，我们计划研究theta对应与Galois周期的关系，这是相对朗兰兹纲领范畴内的一个问题。具体而言，我们将研究局部theta对应下的如下问题：对于正则supercuspidal表示的具体刻画，discrete series表示之间spherical特征标的关系，以及tempered表示的局部周期的构造。我们也将考虑整体theta提升和它们的Galois自守周期。

In this project, we plan to study the relation between the theta correspondence and Galois periods, which lands in the scope of the relative Langlands program. Conretely, we will study the following problems under the local theta correspondence: an explicit description for regular supercuspidal representations, the spherical character relations between discrete series representations, and the construction of local periods for tempered representations. We will also consider the global theta lifts and their Galois automorphic periods.