Finite multiplicities beyond spherical pairs

Let $G$ be a real reductive algebraic group, and let $Hsubset G$ be an algebraic subgroup. It is known that the action of $G$ on the space of functions on $G/H$ is tame if this space is spherical. In particular, the multiplicities of the space $mathcal{S}(G/H)$ of Schwartz functions on $G/H$ are finite in this case. In this paper we formulate and analyze a generalization of sphericity that implies finite multiplicities in $mathcal{S}(G/H)$ for small enough irreducible representations of $G$.