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 G be
the preimage of G in the metaplectic group
Sp​(W). Given an irreducible unitary
representation Î of G that occurs in the restriction
of the Weil representation to G, let ΘΠ​ denote
its character. We prove that, for the embedding T of
Sp​(W) in the space of tempered distributions on
W given by the Weil representation, the distribution T(ΘˇΠ​) has
an asymptotic limit. This limit is an orbital integral over a nilpotent orbit
Om​⊆W. The closure of the image of Om​
in g′ under the moment map is the wave front set of Π′, the
representation of G′ dual to Π.Comment: arXiv admin note: substantial text overlap with arXiv:1405.243