Let W be a real symplectic space and (G, G0) an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let G be the preimage of G in the emetaplectic group Sp(W). Given an irreducible unitary representation Π of f G that occurs e in the restriction of the Weil representation to G, let Θ e Π denote its character. We prove that, for a suitable embedding T of Sp(W) in the space of tempered distributions on W, f the distribution T(Θˇ Π) admits an asymptotic limit, and the limit is a nilpotent orbital integral. As an application, we compute the wave front set of Π0, the representation of Gf0 dual to Π, by elementary means