We consider a dual pair (G,G′), in the sense of Howe, with G compact
acting on L2(Rn) for an appropriate n via the Weil
Representation. Let G be the preimage of G in the metaplectic
group. Given a genuine irreducible unitary representation Î of
G we compute the Weyl symbol of orthogonal projection onto
L2(Rn)Π​, the Π-isotypic component. We apply the result to
obtain an explicit formula for the character of the corresponding irreducible
unitary representation Π′ of G′ and to compute of the wave
front set of Π′ by elementary means