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

Abstract

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 $\tilde{\mathrm{G}}$ be the preimage of G in the metaplectic group $\widetilde{\mathrm{Sp}}(\mathrm{W})$. Given an irreducible unitary representation $\Pi$ of $\tilde{\mathrm{G}}$ that occurs in the restriction of the Weil representation to $\tilde{\mathrm{G}}$, let ${\Theta }_{\Pi }$ denote its character. We prove that, for the embedding $T$ of $\widetilde{\mathrm{Sp}}(\mathrm{W})$ in the space of tempered distributions on W given by the Weil representation, the distribution $T({\check{\Theta }}_{\Pi })$ has an 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 ${\mathfrak{g}}^{\prime }$ under the moment map is the wave front set of ${\Pi }^{\prime }$, the representation of $\widetilde{{\mathrm{G}}^{\prime }}$ dual to $\Pi$.

The present paper subsumes and extends sections 5 and 8 of the (unpublished) preprint arXiv:1405.2431