If is a Lie group, is a closed subgroup, and is a unitary representation of , then the authors give a sufficient condition on to be in the wave front set of . In the special case where is the trivial representation, this result was conjectured by Howe. If is a real, reductive algebraic group and is a unitary representation of that is weakly contained in the regular representation, then the authors give a geometric description of in terms of the direct integral decomposition of 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.
Benjamin Harris. Hongyu He. Gestur Ólafsson. "Wave front sets of reductive Lie group representations." Duke Math. J. 165 (5) 793 - 846, 1 April 2016. https://doi.org/10.1215/00127094-3167168