The Constant Term of Tempered Functions on a Real Spherical Space

Abstract

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 that 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.