  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.
