Wave Front Sets of Reductive Lie Group Representations

If G is a Lie group, H ⊂ G is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξ ∈ ig∗ to be in the wave front set of IndH τ . In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a reductive Lie group of Harish-Chandra class and π is a unitary representation of G that is weakly contained in the regular representation, then the authors give a geometric description of WF(π) in terms of the direct integral decomposition of π 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.