Wave front sets of reductive Lie group representations

Abstract

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 Ind G H 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.

Wave front sets of reductive Lie group representations III

“…In this article, we are interested in relating WF(τ ) and WF(Ind In Theorem 1.1 of [HHO16], it is shown that WF(Ind This result is proven in Section 3. Observe…”

“…For instance, they play a fundamental role in the seminal series of papers [Kob94], [Kob98a], [Kob98b], which give necessary and sufficient conditions for discrete decomposability of Lie group representations. This is the third in a series of papers on wave front sets of Lie group representations [HHO16], [Har]. The ideas in this series are heavily influenced by the earlier work of Kobayashi. Acknowledgements: B. Harris is indebted to Hongyu He and GesturÓlafsson, his former postdoctoral advisers at Louisiana State University, for their support, encouragement, knowledge, and wisdom.…”

“…Let η 0 / ∈ Ad * (G) · i(g/h) * . We have to show (see for instance Definition 2.3 of [HHO16]) that for fixed…”

“…As in the introduction, we form the unitary representation Ind G H τ of G. In Theorem 1.1 of [HHO16], it is shown that…”

A smooth introduction to the wavefront set

“…Many variations have been devised: analytic wavefront set (see [69] and [70] for a recent comparison of various definitions), homogeneous wavefront set [71], Gabor wavefront set [72], global wavefront set [73,74], etc.…”