Vanishing at infinity on homogeneous spaces of reductive type

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.

An assumption that G/H is algebraic is added to deduce the volume estimate needed for the if direction of the main result. A shorter proof and improved exposition of the "only if direction" of the main result