The constant term of tempered functions on a real spherical space

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.

Final version. 72 pages. Accepted to IMRN

References in corpus (3)

• Simple compactifications and polar decomposition of homogeneous real spherical spaces
• Reductive group actions
• Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms

Cited by in corpus (1)

• On the regularity of D-modules generated by relative characters