Realizing the Local Weil Representation over a Number Field

Our main result is that the Weil representation of the symplectic group Sp(2n, F ), where F is a non-archimedian local field of residue characteristic 6= 2, can be realized over a number field K. We take an infinite-dimensional complex vector space V such that the Weil representation is given by ρ : Sp(2n, F ) → PGL(V) and we find a K-subspace V0 of V such that ρ(g)(V0) = V0 for all g ∈ Sp(2n, F ). This answers a question raised by D. Prasad [P]. Indeed, we show that we can take K = Q( √ p, √−p) where p is the residue characteristic of F . We assume that p is odd. A consequence of this, also pointed out by Prasad, is that the local theta correspondence can be defined for representations which are realized over K. Let W be the Weil representation of Sp(2n, F ). The Weil representation can be defined using the Schrödinger representation of the Heisenberg group H . Let λ be a fixed complex character on the additive group of the field F . Suppose that F 2n is the direct sum X ⊕ Y of totally isotropic F -subspaces. The Schrödinger model is realized in the Bruhat-Schwartz space S(X) of locally constant functions f : X → C of compact support. For h ∈ H , there are operators Sλ(h) on S(X) such that Sλ : H → GL(S(X)) is the unique smooth irreducible representation of H with central character λ. The natural action of the symplectic groups extends to an action on H , and the Weil representation is given by operatorsWλ(g) on S(X), g ∈ Sp(2n, F ), such that Wλ(g) Sλ(h)Wλ(g) = Sλ(hg) h ∈ H, g ∈ Sp(2n, F ).