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

TL;DR

This paper establishes a connection between wave front sets of dual pairs with one compact member in the symplectic group, using asymptotic limits of distributions derived from the Weil representation and orbital integrals.

Contribution

It proves that the asymptotic limit of certain distributions associated with dual pairs corresponds to the wave front set of the dual representation, linking representation theory and geometric orbital data.

Findings

Asymptotic limits of distributions relate to nilpotent orbits.

Wave front set of dual representation is described via orbital integrals.

Results apply to dual pairs with one compact member in symplectic groups.

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 $\widetilde{\mathrm{G}}$ be the preimage of G in the metaplectic group $\widetilde{\mathrm{Sp}}(\mathrm{W})$. Given an irreducible unitary representation $\Pi$ of $\widetilde{\mathrm{G}}$ that occurs in the restriction of the Weil representation to $\widetilde{\mathrm{G}}$, 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 $\mathfrak{g}'$ under the moment map is the wave front set of $\Pi'$, the representation of $\widetilde{\mathrm{G}'}$ dual to $\Pi$.