Weil Representation, Howe Duality, and the Theta correspondence

We define the γ-factor γ(Q, ψ) more generally for any quadratic form Q on k nto be γ(Q, ψ) = g(Q, ψ)/|g(G, ψ)| where g(Q, ψ) is defined as follows. (Our γ willbe γ(Q, ψ) for Q = ∑ ni=1 x 2 i .)∫∫g(Q, ψ) = P.V. ψ(Q(x))dx def = lim ψ(Q(x))dx,k n m→∞ π −m Oknwhere dx is now chosen so that Fourier inversion holds for B(x, y) = Q(x+y)−Q(x)−Q(y) .2Remark 2.2 : One might wonder why this care about the 8-th root of unityfactor γ when the intertwining operator ω ψ(0 1−1 0)is defined only up to scalaranyway. This has to do with the fact that the metaplectic group ˜ Sp ψ (W ) containsˆ Sp(W ) which is a two sheeted covering of Sp(W ), and therefore the scaling factorcan be normalised up to an ambiguity of sign.Remark 2.3 : It can be proved that for the function ψ Q (x) = ψ(Q(x)) on k n ,ψˆQ = γψQ−1 in the sense of distributions.Remark 2.4 : The construction of the Heisenberg group and the uniquenesstheorem of Stone and von Neumann is also true in the case of finite fields. Thereforeas above, one can construct a projective representation of dimension q n ofSp(n, F q ).This projective representation infact lifts to an ordinary representationof Sp(n, F q ) (in a unique way except if n = 1, q = 3). The representation ofSp(n, F q ) so obtained depends on the choice of the additive chatacter ψ of F q , andthe representation associated to ψ and ψ a (x) = ψ(ax) are isomorphic if and onlyif a is a square in F q .Remark 2.5 : Representation of the Heisenberg group and of the metaplecticgroup can be combined to give a representation of the semi-direct product H(W )ט Sp ψ (W ) where the semi-direct product is via the natural action of Sp(W ) onH(W ).2.6 Lattice Model of the Weil Representation: Let A be a finitelygenerated O k -module of maximal rank in W such that A ⊥ = A. Then in thelattice model of the representation of the Heisenberg group on functions on W ,(g, M[g]) ∈ Sp ˜ ψ (W ) for the operator M[g]:(M[g]f)(w) =∑a∈AgA∩Aψ( < a, w > )f(g −1 (a + w)).26