Lifting of characters on p-adic orthogonal and metaplectic groups

Abstract

Let F be p-adic field of characteristic zero. Consider a dual pair (Mp(2n), SO(2n+1)+), where Mp(2n) is the metaplectic cover of the symplectic group Sp(2n) and SO(2n+1)+ is the split orthogonal group over F. We show that there is a matching of Cartan subgroups between SO(2n+1)+ and Mp(2n) via stabilized orbit correspondence. We say two representations of SO(2n+1)+ and Mp(2n) correspond, if their characters on matching Cartan subgroups differ by a transfer factor, which is essentially character of the difference of the two halves of the oscillator representation. We show that this correspondence is compatible with parabolic induction: if two representations of Levi factors correspond, then after parabolic induction the two resulting representations also correspond. These results were motivated by the paper "Lifting of characters on orthogonal and metaplectic groups" by J. Adams who considered the case F is the field of real numbers.