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Let W be a real symplectic space and (G,G') an irreducible dual pair inSp(W), in the sense of Howe, with G compact. Let $\widetilde{\mathrm{G}}$ bethe preimage of G in the metaplectic group$\widetilde{\mathrm{Sp}}(\mathrm{W})$. Given an irreducible unitaryrepresentation $\Pi$ of $\widetilde{\mathrm{G}}$ that occurs in the restrictionof the Weil representation to $\widetilde{\mathrm{G}}$, let $\Theta_\Pi$ denoteits character. We prove that, for the embedding $T$ of$\widetilde{\mathrm{Sp}}(\mathrm{W})$ in the space of tempered distributions onW given by the Weil representation, the distribution $T(\check\Theta_\Pi)$ hasan asymptotic limit. This limit is an orbital integral over a nilpotent orbit$\mathcal O_m\subseteq \mathrm{W}$. The closure of the image of $\mathcal O_m$in $\mathfrak{g}'$ under the moment map is the wave front set of $\Pi'$, therepresentation of $\widetilde{\mathrm{G}'}$ dual to $\Pi$.

The wave front set correspondence for dual pairs with one member compact