Lifting of characters on p-adic orthogonal and metaplectic groups

Let F be a p-adic field. Consider a dual pair $({\rm SO}(2n+1)_+, \widetilde {{\rm Sp}}(2n)),$ where SO(2n+1)+ is the split orthogonal group and $\widetilde {{\rm Sp}}(2n)$ is the metaplectic cover of the symplectic group Sp(2n) over F. We study lifting of characters between orthogonal and metaplectic groups. We say that a representation of SO(2n+1)+ lifts to a representation of $\widetilde {{\rm Sp}}(2n)$ if their characters on corresponding conjugacy classes are equal up to a transfer factor. We study properties of this transfer factor, which is essentially the character of the difference of the two halves of the oscillator representation. We show that the lifting commutes with parabolic induction. These results were motivated by the paper ‘Lifting of characters on orthogonal and metaplectic groups’ by Adams who considered the case F=ℝ.