A geometric criterion for compactness of invariant subspaces

Let M be a non-compact homogeneous Riemannian manifold, and let Ω be a compact subgroup of isometries of M. We show, under general conditions, that the Ω-invariant subspace AΩ of a normed vector space A↪Lq(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A\hookrightarrow L^q(M)}$$\end{document} is compactly embedded into Lq(M) if and only if the group Ω has no orbits with a uniformly bounded diameter in a neighborhood of infinity.