Decay on homogeneous spaces of reductive type

TL;DR

This paper studies the decay properties and L^p-integrability of functions on homogeneous spaces of reductive Lie groups, with applications to lattice point density and new examples provided.

Contribution

It introduces new decay and integrability results for functions on reductive homogeneous spaces, expanding understanding of their asymptotic properties.

Findings

Decay rates at infinity for smooth functions on Z

L^p-integrability conditions for matrix coefficients

New examples of spaces with these properties

Abstract

In this paper we explore homogeneous spaces Z=G/H of a a real reductive Lie group G with a closed connected subgroup H. The investigation concerns the decay at infinity of smooth functions on Z, and L^p-integrability of matrix coefficients. These results are used in a study of the asymptotic density of lattice points on Z. Explicit examples are given of spaces for which results are new.