  Let $G$ be a real reductive algebraic group, and let $H\subset G$ 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 $G$.
