Vanishing at infinity on homogeneous spaces of reductive type

TL;DR

This paper characterizes when smooth vectors in L^p spaces on certain homogeneous spaces vanish at infinity, showing that this occurs precisely when the space is of reductive type.

Contribution

It establishes a necessary and sufficient condition for the vanishing at infinity property on unimodular homogeneous spaces of reductive type.

Findings

Z satisfies VAI if and only if it is of reductive type

Provides a characterization of vanishing at infinity for smooth vectors in L^p spaces

Focuses on unimodular homogeneous G-spaces with connected H.

Abstract

Let G be a real reductive group and Z=G/H a unimodular homogeneous G-space. The space Z is said to satisfy VAI if all smooth vectors in the Banach representations L^{p}(Z) vanish at infinity, 1 <=p. For H connected we show that Z satisfies VAI if and only if it is of reductive type.