The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issue here is that for higher degree covers there is no analogue of the Weil representation, and additional ingredients are needed. Our work reflects a broader paradigm: constructions in automorphic forms that work for algebraic groups or their double covers should often extend to higher degree metaplectic covers.

经典的theta对应关系建立了特殊正交群上的自同构表示和辛群或其双封面上的同构表示之间的关系。通过使用由Weil表示构造的辛群的辛双覆盖上的theta系列作为积分内核，可以实现这种对应关系。还有一个类似的本地通信。在这项工作中，我们提出了将经典theta对应扩展到辛和特殊正交群的高次辛覆盖。这里的关键问题是，对于更高学历的覆盖，没有任何类似于Weil的表示形式，并且需要其他成分。我们的工作反映了更广泛的范例：