Vanishing at infinity on homogeneous spaces of reductive type

Abstract

Abstract. By the collective name of lattice counting we refer to a setup introduced in [10] that aims to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in [10].

Harmonic Analysis for Real Spherical Spaces

“…Theorem 4.3. [18]. Let G be a real reductive group and H < G a connected subgroup of G with real algebraic Lie algebra.…”

Decay of matrix coefficients on reductive homogeneous spaces of spherical type

“…2.1. Further let H < G be a subgroup which is reductive in G (as in [12]). With these data we form the homogeneous space of reductive type Z := G/H .…”

“…Finally, we mention that in [12] we have studied a more qualitative property of decay of smooth L p -functions which are not necessarily matrix coefficients of a Harish-Chandra module. More precisely, we showed that on a reductive homogeneous space the smooth vectors in the Banach representation L p (Z ) all belong to the space of continuous functions vanishing at infinity.…”