The Wave Front Set Correspondence for Dual Pairs with One Member Compact

Let W be a real symplectic space and (G, G′) an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let \(\widetilde {\rm{G}}\) $$\tilde{\text{G}}$$ be the preimage of G in the metaplectic group \(\widetilde {{\rm{Sp}}}({\rm{W}})\) $$\widetilde{\text{Sp}}(\text{W})$$ . Given an irreducible unitary representation Π of \(\widetilde {\rm{G}}\) $$\tilde{\text{G}}$$ that occurs in the restriction of the Weil representation to \(\widetilde {\rm{G}}\) $$\tilde{\text{G}}$$ , let Θ_Π denote its character. We prove that, for a suitable embedding T of \(\widetilde {{\rm{Sp}}}({\rm{W}})\) $$\widetilde{\text{Sp}}(\text{W})$$ in the space of tempered distributions on W, the distribution T(Θ̌_Π) admits an asymptotic limit, and the limit is a nilpotent orbital integral. As an application, we compute the wave front set of Π′, the representation of \(\widetilde {{G^\prime}}\) $$\widetilde{G^{\prime }}$$ dual to Π, by elementary means.