Wave Front Sets of Reductive Lie Group Representations

If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $\tau$ is a unitary representation of $H$, then the authors give a sufficient condition on $\xi \in i{\mathfrak{g}}^{\ast }$ to be in the wave front set of ${\mathrm{Ind}}_H^G\tau$. In the special case where $\tau$ is the trivial representation, this result was conjectured by Howe. If $G$ is a real, reductive algebraic group and $\pi$ is a unitary representation of $G$ that is weakly contained in the regular representation, then the authors give a geometric description of $\mathrm{WF}(\pi )$ in terms of the direct integral decomposition of $\pi$ into irreducibles. Special cases of this result were previously obtained by Kashiwara-Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.