Wave Front Sets of Reductive Lie Group Representations

TL;DR

This paper characterizes the wave front sets of unitary representations of reductive Lie groups, providing new geometric insights and conditions that advance understanding in harmonic analysis and representation theory.

Contribution

It offers a sufficient condition for wave front sets of induced representations and a geometric description of wave front sets for certain unitary representations of reductive groups.

Findings

Provided a sufficient condition for wave front sets of induced representations.

Gave a geometric description of wave front sets for unitary representations weakly contained in the regular representation.

Extended previous results by Kashiwara-Vergne, Howe, and Rossmann.

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.