Duke Mathematical Journal

Wave front sets of reductive Lie group representations

Benjamin Harris, Hongyu He, and Gestur Ólafsson

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If G is a Lie group, HG is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξig to be in the wave front set of IndHGτ. 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.

Article information

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

Received: 9 November 2013
Revised: 24 March 2015
First available in Project Euclid: 14 January 2016

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Zentralblatt MATH identifier

Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 43A85: Analysis on homogeneous spaces

wave front set singular spectrum analytic wave front set reductive Lie group induced representation tempered representation branching problem discrete series reductive homogeneous space


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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