Classical Theta Lifts for Higher Metaplectic Covering Group

TL;DR

This paper extends the classical theta correspondence to higher degree metaplectic covers of symplectic and orthogonal groups, addressing the lack of Weil representation analogues in these cases.

Contribution

It introduces a new framework for theta lifts in higher degree metaplectic covers, overcoming the absence of Weil representations.

Findings

Extended theta correspondence to higher degree covers

Developed new methods without Weil representation

Broadened automorphic form constructions to higher covers

Abstract

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.