Normalized local theta correspondence and the duality of inner product formulas

Abstract

We conjecture that local theta correspondence can be normalized by the leading coefficient of a weighted local period integral, and that there exists a duality of local and global inner product formulas. The conjecture is verified for the pair (\(\widetilde{SL}_2 \), PGL 2) and (SL 2, SO(2, 2)). As an application, global inner product formulas are obtained for liftings in the directions PGL 2 → \(\widetilde{SL}_2 \), GSO(2, 2) → GL 2.