Wave Front Sets of Reductive Lie Group Representations III

TL;DR

This paper provides an explicit description of the asymptotics of the tempered support of $L^2$ spaces on homogeneous spaces for real reductive groups, advancing understanding of wave front sets in harmonic analysis.

Contribution

It offers a detailed characterization of wave front sets for representations of reductive Lie groups acting on homogeneous spaces, extending previous results to vector bundle contexts under certain conditions.

Findings

Explicit description of a Zariski open subset of asymptotics of tempered support.

Upper bound on wave front set of induced Lie group representations.

Extension of results to vector bundle valued harmonic analysis.

Abstract

Let $G$ be a real, reductive algebraic group, and let $X$ be a homogeneous space for $G$ with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of $L^2(X)$. Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on $X$. These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition.