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Howe duality and the trace formula

In [H], Howe conjectured a correspondence between the automorphic representations of SO2n+1 and these of S̃pn. The local and global theory of the Howe correspondence has been established by the works of Furusawa [F], Howe [H2], Piatetski-Shapiro and Soudry [PS-S], Rallis [R], Waldspurger [W], etc. In this paper, we study the correspondence from a new angle, using the method of trace formula. We state two families of trace formulas that yield the Howe correspondence. The local Howe duality play an essential role in our trace formula. Let Fv be a non-Archimedean field with odd residue characteristics. We consider the Weil representation ωψ of the group S̃pn(2n+1)(Fv) (the twofold cover of Spn(2n+1)(Fv)), which is determined by an additive character ψ of Fv. We restrict ωψ to the dual pair SO2n+1 × S̃pn in S̃pn(2n+1), (where the symplectic form is given by the tensor product Q ⊗ A, Q being a symmetric nondegenerate bilinear form in 2n + 1 variables, A being an alternate nondegenerate bilinear form in 2n variables). One version of the (local) Howe duality conjecture is: there exists a surjective homomorphism