The constant term of tempered functions on a real spherical space

Raphaël Beuzart-Plessis (1) , Patrick Delorme (2) , Bernhard Krötz (3) , Sofiane Souaifi (4)

1 I2M - Institut de Mathématiques de Marseille
2 IML - Institut de mathématiques de Luminy
3 Mathematisches Institut der Universität Paderborn
4 IRMA - Institut de Recherche Mathématique Avancée

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.

Domains: General Mathematics [math.GM]; Representation Theory [math.RT]

Raphaël Beuzart-Plessis, Patrick Delorme, Bernhard Krötz, Sofiane Souaifi. The constant term of tempered functions on a real spherical space. international mathematical research notices, In press, ⟨10.1093/imrn/rnaa395⟩. ⟨hal-02150715⟩