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

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