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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 ε ig* to be in the wave front set of IndGH τ. In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a real, reductive algebraic group 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.

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Duke Mathematical Journal

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