Theta liftings of non-generic representations on double covers of orthogonal groups

TL;DR

This paper investigates theta liftings of non-generic representations on double covers of orthogonal groups, proposing a conjecture on the first non-zero lift occurrence and providing supporting global and local proofs.

Contribution

It introduces a conjecture relating the first non-zero theta lift to unipotent orbits and proves supporting global and local results.

Findings

Conjecture links lift occurrence to unipotent orbit.

Global and local results support the conjecture.

Advances understanding of theta liftings for non-generic reps.

Abstract

We study the generalized theta lifting between the double covers of split special orthogonal groups, which uses the non-minimal theta representations constructed by Bump, Friedberg and Ginzburg. We focus on the theta liftings of non-generic representations and make a conjecture that gives an upper bound of the first non-zero occurrence of the liftings, depending only on the unipotent orbit. We prove both global and local results that support the conjecture.