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 \(\widetilde {\rm{G}}\) be the preimage of G in the metaplectic group \(\widetilde {{\rm{Sp}}}({\rm{W}})\). Given an irreducible unitary representation Π of \(\widetilde {\rm{G}}\) that occurs in the restriction of the Weil representation to \(\widetilde {\rm{G}}\), let Θ_Π denote its character. We prove that, for a suitable embedding T of \(\widetilde {{\rm{Sp}}}({\rm{W}})\) 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 \(\widetilde {{G^\prime}}\) dual to Π, by elementary means.