Invariant measures on homogeneous spaces, with applications to function spaces and lattice counting

Let G be a real reductive group and G/H a unimodular homogeneous G space with a closed connected subgroup H. We establish estimates for the invariant measure on G/H. Using these, we prove that all smooth vectors in the Banach representation L^p(G/H) of G are functions that vanish at infinity if and only if G/H is of reductive type. An application to lattice counting on G/H is presented.