Vanishing at infinity on homogeneous spaces of reductive type (Q2816793)

Let \(G\) be a real Lie group and \(H\subset G\) a closed subgroup. Consider the homogeneous space \(Z=G/H\) and assume that it is unimodular, that is, it carries a \(G\)-invariant measure \(\mu_Z\). Such a measure is unique up to a scalar multiple. In the present paper the decay is studied of smooth \(L^p\)-functions on \(G/H\). The homogeneous space \(Z\) has the property VAI (vanishing at infinity) if for all \(1\leq p<\infty\) one has \(L^p(Z)^\infty \subset C^\infty_0(Z)\). The VAI property is valid for \(G\) unimodular and \(H=\{1\}\). All reductive symmetric spaces admit VAI. A non-compact homogeneous space with finite volume cannot have VAI. The authors obtain the following main result. Theorem 1.2. Let \(G\) be a connected real reductive group and \(H\subset G\) a closed connected subgroup such that \(Z=G/H\) is unimodular and of algebraic type. Then VAI holds for \(Z\) if and only if it is of reductive type. After introducing some notations, the authors study VAI versus volume growth and obtain a certain algebraic lower bound of the volume function. Next it is shown that reductive spaces are VAI and non-reductive spaces are not VAI.