  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.
