Duke Mathematical Journal

Wave front sets of reductive Lie group representations

Abstract

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 $\xi\in i\mathfrak{g}^{*}$ to be in the wave front set of $\operatorname{Ind}_{H}^{G}\tau$. 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 $\operatorname{WF}(\pi)$ 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.

Article information

Source
Duke Math. J., Volume 165, Number 5 (2016), 793-846.

Dates
Revised: 24 March 2015
First available in Project Euclid: 14 January 2016

https://projecteuclid.org/euclid.dmj/1452780579

Digital Object Identifier
doi:10.1215/00127094-3167168

Mathematical Reviews number (MathSciNet)
MR3482333

Zentralblatt MATH identifier
1341.22008

Citation

Harris, Benjamin; He, Hongyu; Ólafsson, Gestur. Wave front sets of reductive Lie group representations. Duke Math. J. 165 (2016), no. 5, 793--846. doi:10.1215/00127094-3167168. https://projecteuclid.org/euclid.dmj/1452780579

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