The constant term of tempered functions on a real spherical space

TL;DR

This paper develops a theory of constant terms for tempered functions on unimodular real spherical spaces, paralleling Harish-Chandra's work, with a focus on eigenfunctions and their asymptotic behavior.

Contribution

It introduces a systematic framework for constant terms of tempered functions on real spherical spaces, including transitivity and uniform bounds on eigenfunction differences.

Findings

Constant terms are parametrized by subsets of spherical roots.

Constant terms are transitive: $(f_J)_I=f_I$ for $I eq J$.

Established uniform bounds on the difference between eigenfunctions and their constant terms.

Abstract

Let $Z$ be a unimodular real spherical space. We develop a theory of constant terms for tempered functions on $Z$ which parallels the work of Harish-Chandra. The constant terms $f_I$ of an eigenfunction $f$ are parametrized by subsets $I$ of the set $S$ of spherical roots which determine the fine geometry of $Z$ at infinity. Constant terms are transitive i.e. $(f_J)_I=f_I$ for $I\subset J$, and our main result is a quantitative bound of the difference $f-f_I$, which is uniform in the parameter of the eigenfunction.